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(p. 147) Intervention: Modelling, Demographic Impact and the Public Health

Intervention: Modelling, Demographic Impact and the Public Health
Chapter:
Intervention: Modelling, Demographic Impact and the Public Health
DOI:
10.1093/med/9780199596614.003.0006
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date: 29 July 2021

2. 6.2 The Basic Reproduction Number, R0 [link]

• Spanish Influenza: Australia, 1918–19 [link]

• Pandemic Influenza A/Swine/H1N1, 2009 [link]

3. 6.3 The Spatial Basic Reproduction Number, R0A [link]

• Cyclical Re-Emergence: Spotting Influenza Pandemics [link]

• Swash Model and Other Infectious Diseases [link]

4. 6.4 Forecasting for Intervention [link]

5. 6.5 Intervention, Value-for-Money and Demography [link]

• Global Immunization Vision and Strategy: Intervention Costs and Benefits [link]

• Appendix 6.1: Swash–Backwash Model Equations [link]

(p. 148) 6.1 Introduction

…It follows that epidemic theory should certainly continue to search for new insights into the mechanisms of the population dynamics of infectious diseases, especially those of high priority in the world today, but that increased attention should be paid to formulating applied models that are sufficiently realistic to contribute directly to broad programmes of intervention and control.

N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and its Application. London: Griffin, 1975, p. 27

…The history of malaria contains a great lesson for humanity – that we should be more scientific in our habits of thought, and more practical in our habits of government. The neglect of this lesson has already cost many countries an immense loss in life and in prosperity.

Ronald Ross, The Prevention of Malaria. London: John Murray/New York: Dutton, 1910, p. 48

Figure 6.1
The Plumb Pudding in Danger. Pitt and Napoleon exercise their own version of intervention at the global scale in James Gillray’s famous political caricature of 1805. In the image, Pitt slices a chunk of ocean, while Napoleon helps himself to Europe – allegories of the respective areas of influence of the two countries.

Source: Reproduced with permission from Wellcome Library, London.

In this chapter, we discuss and illustrate modelling for communicable diseases which may provide some answers to these questions. We begin in Sections 6.2 and 6.3 with the most basic question of all: will this infectious disease outbreak burgeon over time and space, or will it simply fade away? The models are illustrated by application to twentieth-century influenza and poliomyelitis data from a variety of different geographical locations and spatial scales (the United States, France, Iceland, the United Kingdom and Australia). We then move on in Section 6.4 to look at spatial forecasting models which may be deployed to answer the question of “where from and where to?” for a growing epidemic, drawing upon examples from the epidemiological laboratory of Iceland.

The book is concluded in Sections 6.5 and 6.6. Here we review global aspirations through to 2015 for controlling the spread of communicable diseases, including a cost–benefit analysis of the programme and the likely demographic impacts of control.

6.2 The Basic Reproduction Number, R0

The basic reproduction number, R0, is one of the key concepts in epidemiology. Critical reviews of the history, development and use of the concept are provided by (Dietz 1993), (Heesterbeek 2002) and (Heffernan, et al. 2005). (Heesterbeek and Dietz 1996) regard the basic reproduction number as “one of the foremost and most valuable ideas that mathematical thinking has brought to epidemic theory”. As described in Heffernan, et al., R0 was originally developed for the study of demographics (Böckh, 1886; Sharp and Lotka, 1911; Dublin and Lotka, 1925; Kuczynski, 1928). It was independently studied for vector-borne diseases such as malaria (Ross, 1911; MacDonald, 1952) and directly transmitted human infections (Kermack and McKendrick, 1927; Dietz, 1975; Hethcote, 1975). It is now widely used in the study of infectious disease.

We use the concepts and notation of the SIR models described in Section 1.4. In a population which is entirely susceptible, R0 is the number of other individuals infected by a single infected individual during his or her entire infectious period. However, only very rarely will a population be totally susceptible to an infection in the real world. Many human communicable diseases (e.g. measles) confer lifelong immunity upon those attacked. In these circumstances, the effective reproduction number estimates the average number of secondary cases per infectious case in a population made up of both susceptible and non-susceptible hosts. It can be thought of as the number of secondary infections produced by a typical infective, and it is the basic reproduction number discounted by the fraction of the host population that is susceptible.

From these definitions, it is clear that when R0 < 1, each infected individual produces, on average, less than one new infected individual, and we therefore predict that the infection will be cleared from the population, or the microparasite will be cleared from the individual. If R0 > 1, the pathogen is able to invade the susceptible population. This threshold behaviour is the most important and useful aspect of the R0 concept (see Appendix 6.1). In an epidemic, we can determine which control measures, and at what magnitude, would be most effective in reducing R0 below one, providing important guidance for public health initiatives. The magnitude of R0 is also used to gauge the risk of an epidemic or pandemic in emerging infectious disease. For example, the estimation of R0 was of critical importance in understanding the outbreak and potential danger from SARS (Choi and Pak, 2003; Lipsitch, et al., 2003; Lloyd-Smith, et al., 2003; Riley, et al., 2003). It has been likewise used to characterise bovine spongiform encephalitis (BSE) (Woolhouse and Anderson, 1997; Ferguson, et al., 1999; de Koeijer, et al., 2004), foot-and-mouth disease (FMD) (Ferguson, et al., 2001; Matthews, et al., 2003), novel strains of influenza (Mills, et al., 2004; Stegeman, et al., 2004) and West Nile virus (Wonham, et al., 2004). The incidence and spread of dengue (Luz, et al., 2003), malaria (Hagmann, et al., 2003), Ebola (Chowell, et al., 2004) and scrapie (Gravenor, et al., 2004) have also been assessed using R0 in recent literature. Topical issues such as the risks of indoor airborne infection (Rudnick and Milton, 2003) and bioterrorism (Kaplan, et al., 2002; Longini, et al., 2004) also draw upon this concept.

Herd Immunity

For a communicable disease to be controlled, by whatever means, R0 in the population must be maintained below unity. Today, this is generally by vaccination. Herd immunity occurs when a significant proportion of the population (or the herd) has been vaccinated, and this provides protection for unprotected individuals. See Fine (1993) for a review of the concept. The larger the number of people in a population who are vaccinated, the lower the likelihood that a susceptible (unvaccinated) person will come into contact with the infection. It is more difficult for diseases to spread between individuals if large numbers are already immune, and the chain of infection is broken (cf. Section 1.4).

The herd immunity threshold, HIT, is the proportion of a population that needs to be immune in order for an infectious disease to become stable in that community. If this is reached, for example as a result of immunisation, then each case leads to a single new case and the infection will become stable within the population (R0 = 1). If the threshold is bettered, then R0 < 1 and the disease will die out. HIT is defined as: (6.1) $H I T = ( R 0 − 1 ) / R 0 = 1 − ( 1 / R 0 )$

This is an important measure used in infectious disease control and in immunisation and eradication programmes.

In the first decade of the new millennium, there have been several global alerts about the health risks posed by newly-emerging and re-emerging diseases (Cliff, Smallman-Raynor, Haggett, et al., 2009, p. 668). Especial concern has been expressed about both SARS and the likelihood of a new global pandemic of influenza with the killing power of the 1918–19 H1N1 pandemic – a concern heightened by the emergence of two novel strains of the influenza virus, highly pathogenic avian influenza (HPAI/A/H5N1) in 2003–5 and swine flu (A/swine/H1N1) in 2009. In the event, all have thankfully had limited impact in terms of human mortality although the World Health Organization raised the alert for H1N1 to the final phase of its six-phase severity scale. As Anderson (Times Higher, 9 January 2004) has drily remarked in the context of SARS, “Next time we may not be so lucky”, and so we look here at the values for R0 for the 1918–19 and 2009 influenza outbreaks, not least because both were caused by varieties of H1N1.

Spanish Influenza: Australia, 1918–19

The great influenza pandemic of 1918–19, caused by a new virus strain H1N1, occurred in three waves: spring and early summer 1918 (Wave I); autumn 1918 (Wave II); and winter 1918–19 (Wave III); see Figure 6.2. Here, we provide a brief overview of the global dispersal of the three waves. Our account draws on the study of (Patterson and Pyle 1991).

Figure 6.2
Spanish influenza, 1918–19. (A) Weekly influenza death rate per 100,000 population, London and the county boroughs of England and Wales, June 1918–April 1919. The three wave form of the pandemic is clear. The global diffusion of the first two waves is mapped in (B) and (C). (B) Wave I, spring and early summer 1918. (C) Wave II, Autumn 1918.

Wave I (Spring and Early Summer 1918)

Wave I of the influenza pandemic was attributed at the time to Spain by France and vice versa, and to Eastern Europe by the Americans. But, whatever the truth, the disease acquired its popular name of Spanish influenza at this stage. Some of the first records of influenza activity can be traced to US Army recruits at Camp Funston, Kansas, where an epidemic of influenza first manifested in early March 1918. By the end of July, all continents had been reached. Wave I was comparatively mild.

Wave II (Autumn 1918)

During the summer of 1918, the virus mutated into a more lethal strain and a second, more severe, form of the disease emerged. Pneumonia often developed quickly, with death usually coming two days after the first indications of influenza. The exact geographical origins of Wave II are, like Wave I, unknown although western France is generally viewed as the source. The first reports can be traced to the French Atlantic port of Brest, a landing point for American troops, in late August. From here, ships appear to have carried the disease to coastal locations of North America, Africa, Latin America, South Asia and the Far East by September. New Zealand was infected in October by ships from the United States while Australia remained largely free of the disease until January 1919.

Wave III (Winter 1918–19)

In the aftermath of Wave II, a third influenza wave – of intermediate severity to the preceding waves – appeared in the winter of 1918–19. While relatively little is known of the origin and spread of this tertiary wave, the pandemic appears to have finally run its course by the spring of 1919.

Australia

Australia’s involvement in the First World War was on a massive scale in relation to its small size. From a population of some five million, over 300,000 troops served in Europe, and the problems of bringing the survivors home in the autumn and winter of 1918 – at the very height of the influenza pandemic – was on a similarly massive scale. In the early part of October 1918, with the health authorities of Australia cognisant that a virulent form of influenza was within striking distance of the country, instructions were issued for the port quarantining of all vessels with cases of influenza aboard (Cumpston, 1919, p. 7). So began Australia’s six-month maritime defence against the great influenza pandemic. The medical officer who successfully led the strategy was J.H.L. Cumpston (Figure 6.3), Director of Quarantine and later Australia’s first Commonwealth Director-General of Health.

Figure 6.3
Quarantine administration, Commonwealth of Australia, at the end of the First World War. Australia’s principal and secondary quarantine stations are marked, along with other ports of entry at which quarantine officers were stationed. The photograph is of J.H.L. Cumpston (1880–1954), Australia’s first Commonwealth Director-General of Health, 1921–45. Cumpston oversaw the health of Australia and the troops she put in arms in two world wars. He produced a number of service handbooks on diseases in Australia from 1788 to 1920. The findings were summarised in a book, Health and Disease in Australia, completed in 1928. This was not published until 1989, when the work was edited and introduced by Milton Lewis as part of Australia’s bicentennial celebrations.

Sources: (Map) drawn from The Quarantine Annual, 1930, Office International d’Hygiène Publique, League of Nations; (photograph) National Library of Australia.

The quarantine record

Records have survived for 228 vessels arriving in Australia between October 1918 and April 1919, of which 79 (35 percent) documented cases of influenza (Smallman-Raynor and Cliff, 2004, Table 11.7, p. 584). Figure 6.4 plots, by week of arrival in Australia, the distribution of the 79 infected vessels according to the inferred place of virus acquisition: Pacific Rim (chart A); Europe (B); and other locations (C). Vessels on which cases of influenza were documented after arrival in Australia and which, in the absence of quarantine, would have served as potential sources of infection for the population of Australia, are indicated by the black shading. Figure 6.4 suggests a temporal shift in the geographical source of maritime influenza, from the Pacific Rim in October–December 1918 (chart A) to Europe in January–April 1919 (chart B). This temporal shift was associated with the arrival of generally larger, and more heavily crowded, troopships from Europe via Suez in the latter period.

Figure 6.4
Maritime quarantine and ‘Spanish influenza’, Australia, 1918–19. Graphs plot, by overseas source of infection, the weekly count of vessels infected with influenza and quarantined on arrival at Australian ports, October 1918–May 1919. (A) Pacific Rim. (B) Europe. (C) Other. Vessels on which cases of influenza were recorded after arrival in Australia, and which may therefore be assumed to have maintained an unbroken chain of infection aboard ship, are indicated on each graph by the black shading.

Source: Reproduced from Smallman-Raynor, M.R. and Cliff, A.D, War Epidemics: An Historical Geography of Infectious Diseases in Military Conflict and Civil Strife, Figure 11.5, p. 585, Oxford University Press, Oxford, UK, Copyright © 2004, by permission of Oxford University Press. Data from Cumpston, J. H. L., Influenza and Maritime Quarantine in Australia, Sydney, Commonwealth of Australia, Quarantine Service Publication, No. 18, Copyright © 1919.

Evidence from infected troopships

The influenza records for four troopships (Devon, Medic, Boonah and Ceramic), arriving at Australian ports between November 1918 and March 1919, are shown in Figure 6.5; the vessels involved are illustrated in Figure 6.6. Some impression of the conditions that prevailed on the ships can be gained from the eyewitness accounts of senior officers. According to one anonymous officer aboard the Medic, the scene

…was remarkable. A score or more of stalwart young men lay helpless about the after well-deck, awaiting transport to the improvised and overflowing hospitals in the troop-decks. That they lay there was not due to any neglect or delay on the part of the busy stretcher-bearers, but to the extraordinarily sudden and disabling onset of the disease. One smart, well set-up young soldier came up a companion-ladder close to where I stood, held on for a few seconds to a rail, and then sagged slowly down till he assumed the characteristic flattened sprawl on the deck. There was no pretence or “old-soldiering” about it. The men were literally being knocked down by a profound systematic intoxication of extraordinarily rapid onset

(Anonymous, cited in Cumpston, 1919, p. 53).

Figure 6.5
The role of ships in dispersing ‘Spanish’ influenza, 1918–19. Number of cases of influenza reported daily on four troopships repatriating troops from Europe to Australia in the aftermath of the First World War. (A) Devon. (B) Medic. (C) Boonah. (D) Ceramic.

Source: Reproduced from Cliff et al, Island Epidemics, Figure 5.12, pp. 194, Oxford University Press, Oxford, UK, Copyright © 2000 with permission from Oxford University Press.

Figure 6.6
Australia and the influenza pandemic of 1918–19. Four troopships which carried ‘Spanish’ influenza from the European Theatre or South Africa to Australia in the aftermath of the First World War. (Upper left) Troopship Devon (foreground) in First World War camouflage with troops aboard, 1915. (Upper right) Troopship Medic. HMAT (His Majesty’s Australian Transports) Medic (A7) departs Melbourne assisted by a tug, and watched by a crowd of well-wishers on the wharf, 16 December 1916. (Lower left) Troopship Boonah (HMAT A36) departing from Brisbane during World War One, 1916. Image shows a large crowd gathered to wave farewell to the troops. The ship is decorated with streamers. (Lower right) Signed photograph of the troopship HMAT A40 Ceramic, the troopship that took the 16th Battalion from Melbourne to Egypt in late December 1914. On this, the Ceramic’s first trip as a troopship, the dangers of epidemics of infectious diseases in confined spaces was illustrated. “The first death of a member of the batallion occurred this day [20 January 1915] when Private C.R…. died of pneumonia following measles” (Longmore, 1929, p. 29), and “On January 20, a sick return showed that there were 10 cases of pneumonia, 13 of measles, 7 of influenza, and 15 with other diseases in hospital, a total of 45 or 1.7 per cent. of the troops on board. During this period of the trip all hands were vaccinated and inoculated, to the obvious disgust and temporary discomfort of the troops” (Longmore, 1929, p. 21).

Sources: (Upper left) National Maritime Museum, Greenwich, UK (Upper right) Reproduced with permission from Australian War Memorial website, photographer Josiah Barnes; (Lower left) Reproduced with permission from John Oxley Library, State Library of Queensland, Australia; and (Lower right) Reproduced from Longmore, C., The Old Sixteenth: Being a Record of the 16th Battalion, AIF [Australian Imperial Force], during the Great War 1914–1918, pp. 18-19, History Committee of the 16th Battalion Association, Perth, Australia, 1929.

As Figure 6.5 shows, the influenza curve for the Medic followed a broadly similar course to other troopships, with two marked features of the frequency distribution of cases:

• a generally log-normal nature. Cases peaked within three to 10 days of start of voyage, with a long positive tail to the distribution;

• recurring secondary cycles of cases within general lognormal shape. Statistical analysis using the autocorrelation function (ACF) suggests that there is a periodicity in the time series of about four days, roughly corresponding with the known serial interval of the disease.

The long journey from Europe gave Cumpston time to organise a quarantine system. Boats reporting influenza were isolated in harbour, and troops were not allowed ashore until they were free of the disease. Once soldiers were landed and returned to their homes, they found that interstate travel was also restricted to inhibit transmission – by armed guards in the case of the land border between Queensland and New South Wales. The effect of these measures was to ensure that peak death rates in Australia were about an eighth of those in the United States, and a quarter of those in South Africa and New Zealand. Such was the success of the influenza control measures, concluded Cumpston, that

during the last three months of 1918, maritime quarantine had the effect of holding at the sea frontiers an intensely virulent and intensely infective form of influenza, which not only caused disastrous epidemics in New Zealand and South Africa, but actually arrived at the maritime frontiers of Australia, and caused alarming epidemics amongst the personnel of vessels detained in quarantine

(Cumpston and Lewis, 1989, pp. 318–19).

Although the effectiveness of the quarantine measures of late 1918 has been questioned (McQueen, 1975), epidemiological evidence does point to a delay of several months in the spread of virulent influenza in Australia. So, on the basis of recorded mortality, Cumpston traced the start of the Australian epidemic to early 1919. The first influenza deaths were reported in the state of Victoria (January), to be followed thereafter by New South Wales (February), South Australia (March), Queensland (April), Western Australia (June) and Tasmania (August). In explaining the timing of the epidemic sequence, Cumpston noted that:

Western Australia and Tasmania are the two States having the least human contact with the two [earliest] infected States, New South Wales and Victoria; they were the two States which most vigorously applied inter-State quarantine restrictions; and they were the States latest infected in the series

(Cumpston and Lewis, 1989, p. 119).

Lessons from Epidemics at Sea

A number of writers have commented on the potential epidemiological value of the Australian experience in sea transport and naval vessels in the First World War. (Cumpston 1919) argued that a troopship is a self-contained highly insulated and concentrated herd. In Australian transports, this herd remained together on the voyage to or from England for a period of eight weeks, far longer than other transports (for example, US and Canadian troops crossing the Atlantic). The herd was reasonably homogeneous in its susceptible history, and used to suffering a wide range of protective and prophylactic expedients and experiments (for example, shore quarantine, prompt diagnosis and isolation, immunisation and treatment). As (Butler 1943, p. 671) has commented:

Every new case was reported on pain of severe disciplinary action, a skilled staff, professional and clerical, was often available; and provision could easily have been made for the collecting, assembling, and manipulation of a large number of reasonably comparable experiences.

Cumpston’s assertion is borne out by (Vynnycky, et al. 2007). Using the morbidity data from the Devon, Boonah and Medic as confined settings, they compared the estimated basic and effective reproduction numbers for the 1918 pandemic with the numbers calculated for open settings in European and American cities. Several different estimation methods based on the growth rate and the final size of the pandemic were used. They found that the effective reproduction number was in the range 2.1–7.5 for confined settings like the Devon and 1.2–3.0 in community-based settings (Figure 6.7). Further, assuming that 30 percent and 50 percent of individuals were immune to Spanish influenza after the first and second waves respectively, Vynnycky, et al. estimated that, in a totally susceptible population, an infectious case could have led to 2.6–10.6 further cases in confined settings and 2.4–4.3 in community-based settings. These findings confirm the greater dangers of transmission in confined environments and the relatively low transmissibility of the 1918 Spanish influenza virus in open settings which has been found by other studies (for example, Mills, et al., 2004; Chowell, et al., 2006).

Figure 6.7
Effective and basic reproduction numbers for Spanish influenza, 1918. Morbidity data from cities in Europe and America for community-based and confined settings. Several different methods based on the growth rate and final size of the epidemic have been used to estimate the effective and basic reproduction numbers for the 1918 (Spanish) influenza virus. The effective reproduction number (the average number of secondary infectious cases produced by a typical infectious case in a given population) for the 1918 influenza virus was in the range 1.2–3.0 and 2.1–7.5 for community-based and confined settings, respectively. The basic reproduction number (the average number of secondary infectious cases resulting from a typical infectious case in a totally susceptible population) was in the range 2.4–4.3 and 2.6–10.6 cases in community-based and confined settings, respectively.

Source: Reproduced from Vynnycky et al, Estimates of the reproduction numbers of Spanish influenza using morbidity data, International Journal of Epidemiology, Volume 36, Issue 4, pp. 881-9, Copyright © 2007 International Epidemiological Association, by permission of Oxford University Press.

Pandemic Influenza A/Swine/H1N1, 2009

The novel strain of influenza, A/swine/H1N1 claimed its first suspected cases in Mexico during March, 2009, and spread to the United States (California) in early April. By 12 May 2009, 5,251 cases of the new influenza had been officially reported to the World Health Organization (WHO) from 30 countries, with most of the identified cases exported from Mexico (Fraser, et al., 2009). (Boëlle, et al. 2009) estimated the basic reproduction number from the epidemic curve in Mexico using several methods and concluded that the number was less than 2.2–3.1 in Mexico. (Fraser, et al. 2009) arrived at a value in the range 1.34–2.04 using data for La Gloria, Mexico. These comparatively low rates are in line with or slightly below the rate estimated by Vynnycky, et al. for the 1918 pandemic in community settings. This seems to imply that it is sustained person-to-person transmission which is essential for pandemic development of influenza rather than catastrophically high basic reproduction numbers. We look more closely at the development of the swine flu pandemic in the US in Section 6.3. See (Anderson 2009) for a discussion of the 2009 pandemic in the United Kingdom.

6.3 The Spatial Basic Reproduction Number, R0A

Swash–Backwash Models

If we are dealing with the spread of a communicable disease through a system of geographical areas like counties or countries, we need to be aware of Tobler’s (1970, p. 236) First Law of Geography – “everything is related to everything else, but near things are more related than distant things”, a point reinforced by (Gould 1970, pp. 443–44):

Why we should expect [spatial] independence of observations that are of the slightest intellectual interest or importance in geographic research I cannot imagine. All our efforts to understand spatial pattern, structure and process have indicated that it is precisely the lack of independence – the interdependence – of spatial phenomena that allows us to substitute pattern, and therefore predictability and order, for chaos and apparent lack of interdependence – of things in time and space.

Unfortunately, this leaves us with a problem as far as R0 is concerned because its formulation is aspatial. The best we can do when dealing with a system of areas is either to evaluate it for the set of areas taken as a whole, or independently for each geographical unit in the set. We thus need a spatial analogue of R0 which takes account of the First Law and which gives a sense of whether an infectious disease will propagate epidemically through a system of areas or die out.

We have demonstrated elsewhere using measles and poliomyelitis data that the spatial extent of infected units within a geographical area, and the time taken from the start of an epidemic for a communicable disease to reach each unit, may be used to measure the spatial velocity of an epidemic wave (Cliff and Haggett, 1981; Trevelyan, et al.,2005; Cliff and Haggett, 2006). In this section, we propose a robust spatial version of R0, which we denote by R0A, for estimating the spatial velocity of a communicable disease as it passes through a system of areas. It enables us to determine whether a disease will grow spatially or fade out. R0A is robust because it uses only the binary presence/absence of disease reports rather than the actual number of reported cases/mortality.

The Model

Our account is taken from (Cliff, Smallman-Raynor, Haggett, et al. 2009, pp. 555–58). A single epidemic wave in any large geographical area is a composite of the waves for each of its constituent sub-areas. That is, the composite wave at the larger geographical scale (say, a country) can in principle be broken down into a series of multiple waves for its constituent sub-areas (say, its regions) at the smaller geographical scale.

Assuming that both the sub-areas and the time periods are discrete, we use the following notation:

A = Area covered by an epidemic wave in terms of the number of sub-areas infected where ai is a sub-area in the sequence 1, 2,…aiA

T = Duration of an epidemic wave, defined in terms of a number of discrete time periods, tj, in the sequence 1, 2,…tjT.

Q = Total cases of a disease recorded in a single epidemic wave measured over all sub-areas and all time periods. Thus qij is the number of cases recorded in the cell formed by the i-th sub-area and the j-th time period of the A × T data matrix.

To illustrate the method, we assume in Figure 6.8 a simple epidemic wave where A = 12, T = 10 and Q = 122. Thus in Figure 6.8A we begin with a hypothetical map which is converted into a 12 × 10 space–time data matrix in which 122 recorded cases of a disease are distributed to simulate an array typical of an epidemic wave. Note that, whereas this overall wave is continuous (no time periods with zero cases), for individual sub-areas the record may be discontinuous with one or more time periods with zero cases.

Figure 6.8
Calculating R0A: hypothetical 12-area example. (A) Base map with areas reference coded as a1a12, and associated space–time data matrix with reported cases for each area given in cells. (B) Matrix rearranged to order leading edges, LE and (C) following or trailing edges, FE. (D) Leading and following edges plotted as a phase-transition diagram with susceptible (S), infected (I) and recovered (R) integrals for areas.

Source: Reproduced from Cliff et al, Infectious Diseases: Emergence and Re-Emergence: A Geographical Analysis, Figure 10.3, p.556, Oxford University Press, Oxford, UK, Copyright © 2009 by permission of Oxford University Press.

For any one of the rows in the data matrix in Figure 6.8A, two cells can be identified which mark the ‘start cell’ and the ‘end cell’ of a recorded outbreak; if the infection only lasts for one time period, the start and end cell are the same. Figures 6.8B and C analyse these start and end cells. In Figure 6.8B the 12 × 10 matrix is rearranged so as to position the start cells in an ascending temporal order. This line of cells (dark shading) defines the position of the leading edge (LE) marking the start of the epidemic wave in the different sub-areas. To the left and above this line lies a zone of cells (light shading) which have yet to be infected and thus may be regarded as areas to which the epidemic has yet to spread.

Equally, the 12 × 10 matrix can be organised as in Figure 6.8C, so as to arrange the end cells in ascending temporal order. This line of cells (dark shading) defines the position of the trailing edge or following edge (FE), marking the completion of the epidemic waves in each of the different sub-areas. To the right and below this line lies a zone of cells which have ceased to be infected and which thus may be regarded as areas which have recovered from infection.

Both the edges, LE and FE, can be combined as in Figure 6.8D to identify cells which are in susceptible, S, infected, I, and recovered, R, states. The resulting graph may be regarded as a phase transition or SIR diagram. It has two roles: first, it defines the boundaries of the two phase shifts from susceptible to infective status (SI) and from infective to recovered status (IR) and second, it integrates the three phases, S, I, and R, as areas within the graph. As discussed in (Cliff and Haggett 2006), the phase diagram assumes characteristic configurations depending upon the velocity, duration and ultimate spatial extent of an epidemic wave as it passes through a region.

Three model parameters relating to the phase diagram are especially useful. We refer to these here as VLE, and VFE, the velocities of the leading and following edges of the wave which are functions of the average temporal position of the edges in the phase diagram; and R0A, here defined as the spatial basic reproduction number. The relevant equations are summarised in Appendix 6.1.

Spatial Basic Reproduction Number

In the notation of Figure 1.17, the aspatial basic reproduction number, R0, is defined as the ratio between the infection rate, β, and the recovery rate, μ. That is, R0 = β/μ and, as we have noted, it is interpreted as the average number of secondary infections produced when one infected individual is introduced into a wholly susceptible population. In the spatial domain, as shown in Figure 6.8D, the S, I and R integrals define the boundaries of the two phase shifts from susceptible to infective status (SI) and from infective to recovered status (IR). The spatial basic reproduction number, R0A, is the average number of secondary infected areas produced from one infected area in a virgin region. The integral S parallels β in that a small value indicates a very rapid spread. The integral R parallels the reciprocal of μ in that a large value indicates very rapid recovery. As detailed in Appendix 6.1, we can compute R0A as the ratio of SA:RA. Values of R0A calibrate the velocity of spread (the larger the value, the greater the rate of spread).

Cyclical Re-Emergence: Spotting Influenza Pandemics

To illustrate the use of R0A, we begin by applying it to the task of separating out influenza pandemic seasons from the normal run of influenza years over the course of the twentieth century. Three different spatial scales of data are used: France, a continental country (population 52 million in 1970); Iceland, an isolated island location (population 205,000 in 1970); and Cirencester, a small market town served by a single general practitioner, Edgar Hope-Simpson, for the second half of the century (population 15,000 in 1970). Then we combine the notion of different spatial scales within a single country by examining the potentially pandemic A/swine/H1N1 global outbreak of 2009 in the United States.

France: A Continental Country

As described in (Cliff, Smallman-Raynor, Haggett, et al. 2009, pp. 558–9), it is possible for the 113-year time window from 1887 to create two matrices giving the time series of monthly influenza mortality for two geographical frameworks: (i) n = 51 towns with populations over 30,000 (Figure 6.9) × 168 months, 1887–1900 and (ii) n " 90 départements × 1,188 months, 1901–99. These two matrices form the basis for the analysis reported as follows.

Figure 6.9
Towns of France, 1887. Geographical distribution of 51 towns with populations in excess of 30,000 used to estimate the spatial velocity of influenza waves in France, 1887–1905.

Figure 6.10 illustrates the temporal pattern of recorded deaths from influenza in France over the period. Figure 6.10A shows the seasonal distribution of mortality, with its winter peak characteristic of northern hemisphere countries. For this reason, in the analysis which follows, we have used as our temporal unit an influenza season running from 1 July through to 30 June of the following year, rather than calendar years which split months of high influenza mortality across year boundaries. Figure 6.10B charts the annual time series. In Europe, as described by (Dowdle 1999), pandemics (marked) occurred as follows within this 113-year window:

1. (i) 1889. First deaths were reported in France in November 1889; the peak month was January 1890. The pandemic occurred in three successive waves in most parts of the world, producing mortality and morbidity greater than had been seen in decades. H2N? was the likely strain.

2. (ii) 1900. (Dowdle 1999) has queried whether this was truly a pandemic. Excess mortality was reported in North America and in England and Wales but not globally. H3N? was the likely strain.

3. (iii) 1918–19. This occurred in three waves as follows – Wave I: April–July 1918; Wave II: August–November 1918; Wave III: March 1919 (Patterson and Pyle, 1991). It produced mortality unequalled in recorded history – up to 50 millions or more worldwide. H1N1 was the likely strain.

4. (iv) 1957. This pandemic occurred in two main waves – Wave I: February–December 1957; Wave II: October 1957–January 1958 (Cliff, et al., 2004). H2N2 was the causative virus.

5. (v) 1968–69. Wave I: 1968–69 (30 percent of deaths); Wave II: 1969–70 (70 percent of deaths). Wave I primarily affected North America and to a lesser extent Europe. In Wave II the situation was reversed and was associated with significant drift in the N surface antigen (Viboud, et al., 2005). H3N2 was the causative strain.

Figure 6.10
Influenza mortality in France, 1887–1999. (A) Radar chart showing the proportion of deaths reported in each month over the period (dark shading). The average monthly proportion is marked by light shading. (B) Time series of annual deaths from influenza reported in French records. For 1901–5, the national series of deaths by cause was recorded for towns whose populations exceeded 5,000 rather than 30,000.

Figure 6.10B shows that, as in most countries of the world, the 1918–19 pandemic stood alone in France in terms of mortality caused. While the long-term trend in mortality is steadily downwards, nevertheless it is the case that pandemic years generally experienced heightened mortality as compared with adjacent years.

The time series illustrated in Figure 6.10B reinforces the need for robust methods of analysis. Influenza mortality appears to have run at a much higher level between 1887 and 1900, than in the rest of the time series. However, this is an artefact of the recording method; from 1887–1900, influenza mortality in France was estimated from excess pneumonia deaths, whereas in the rest of the series, it was recorded directly.

Figure 6.11A plots the results for the spatial basic reproduction number, R0A, and the two edge parameters, VLE, and VFE, along with the linear regression trend lines. It shows that the long term trend in R0A was slowly downwards over the last century; i.e. the rate of geographical propagation of influenza epidemics gradually diminished, probably reflecting less intense epidemics arising from (i) improved standards of living and healthcare, and (ii) no shift of virus strain since 1969. This century-long declining trend in R0A is reflected in a similar decline in the leading edge (LE) velocity parameter, and a rising trend in the following edge (FE) parameter implying that epidemics have arrived later and ended sooner in each influenza season; influenza seasons have become of shorter duration.

Figure 6.11
Swash model parameters for influenza in France, 1887–1999. (A) Time series plots of calculated values for the spatial basic reproductive number, R0A, and for the edge velocity parameters, VLE and VFE. (B) Time series of the infected (I), susceptible (S) and recovered (R) integrals. Linear regression trend lines are also shown.

Within these long term trends, however, the pandemic seasons of 1900, 1918–19, 1957 and 1968–69 showed a locally raised R0A. In the case of 1889, this is not evident, while the rise to a higher level in 1969 persisted for four seasons. In the latter case, this is consistent with the description by (Viboud, et al. 2005) of the 1968–69 pandemic as a “smouldering pandemic”. From the late 1980s, R0A oscillated wildly, suggestive of locally intense epidemics in each influenza season involving only a few départements rather than the entire country.

Table 6.1 highlights the difference between pandemic and non-pandemic seasons in France. The 103 seasons for which data were available have been classified into three groups: (a) pandemic-affected (11), (b) high-intensity inter-pandemic (54), with death rates greater than the lowest pandemic season, and (c) low-intensity inter-pandemic (38), with death rates lower than the lowest pandemic season. The average velocity of the leading edge ($t − L E ,$ equation A1 in Appendix 6.1) for the three groups is (a) = 2.39 months, (b) = 3.59 months and (c) = 4.93 months. The results for VLE in Table 6.1 confirm that pandemic seasons had higher velocities than inter-pandemic years. This higher velocity was maintained even compared with inter-pandemic influenza seasons of similar intensity levels as pandemic seasons. Moreover, pandemic seasons appeared to be of greater spatial intensity (larger R0A) and were slower to clear (lower following edge velocities).

Table 6.1 French influenza seasons, 1887–1999. Values of swash model parameters for 103 influenza waves classified by intensity of mortality

Type of influenza wave

Mortality (mean/100,000 population per season)

Start LE (mean in months)

Velocity (VLE)

End LE (mean in months)

Velocity (VFE)

R0A (mean)

Pandemic (a) (n = 11)

27.29

2.39

0.80

11.45

0.05

0.93

Interpandemic:

High intensity (b) (n = 54)

14.91

3.59

0.70

11.20

0.07

0.84

Low intensity (c) (n = 38)

3.14

4.93

0.59

10.12

0.16

0.80

Notes:

(a) Pandemic seasons of 1889–91, 1900, 1918–19, 1957–58 and 1968–69.

(b) High intensity = Deaths per 100,000 population greater than the least-intense pandemic season.

(c) Low intensity = Deaths per 100,000 population less than the least intense pandemic season.

Data unavailable for seasons 1920–24 and 1936–39.

The plots of the susceptible and recovered integrals in Figure 6.11B show rising trends over the twentieth century, and this is consistent with the generally declining trend in the infective integral. As noted in Figure 6.10B, these trends imply declining intensity of influenza epidemics over the century-long study period. Again the pandemic years bucked the trend with raised infective integral values. The main other variations in the long term fall in the infectives integral occurred following the arrival of Asian influenza in 1957, when the infected integral remained above the trend (shaded in Figure 10.6B) for the next quarter of a century, and then, from the mid-1980s, when the integral remained resolutely below the trend line. The extended period of higher values for the infectives integral post-1957 may be attributed to the combined action of three effects: (i) the long interval since the last major strain shift (40 years since 1918); (ii) the shift from H2N2 to H3N2 in 1968–69; and (iii) the re-emergence in 1976 of Russian influenza (strain of H1N1) which has been co-circulating with the Hong Kong strain ever since. Together these effects meant that a greater proportion of the French population was likely to be susceptible to one or other of the mix of circulating strains than if just a single strain had been present over the period. After a generation, with no new major strains emerging, herd immunity appears to have caught up from the mid-1980s, leading to a general collapse of nationwide epidemics.

Iceland: An Island Location

This subsection is based upon (Cliff, Smallman-Raynor, Haggett, et al. 2009, pp. 566–72). Since 1895, Iceland has required direct notification of influenza cases by physicians. This concern for data collection stems from the island’s early history which was marked by disastrous externally-introduced epidemics, including the 1843 influenza outbreak which, although lasting only two months, doubled the expected death rate for the year (Schleisner, 1851). Annual totals and other summary data have been published in that country’s annual public health reports (Heilbrigðisskýrslur) since that date with, for influenza, national monthly morbidity time series available from 1913 (Figure 6.12A). For the 61-year period spanning the middle of the twentieth century from 1915 (Figure 6.12B), monthly data are broken down to a local level for some 50 medical districts (Figure 6.12C). This allows application of the swash model to estimate the changing spatial and temporal velocity of influenza epidemics over the period.

Figure 6.12
Reported influenza morbidity in Iceland. (A) Monthly records for total reported influenza cases from January 1913 to December 1976. (B) Monthly record of the number of Icelandic medical districts reporting one or more cases of influenza, January 1915 to December 1976. (C) Map of boundaries of Icelandic medical districts for the year 1945. (D) Cumulative total of reported monthly influenza cases for each Icelandic medical district in the period January 1915 to December 1976. Circles are drawn proportional to the cumulative number of cases but note the special position of the capital city, Reykjavík.

Source: All influenza data based on Heilbrigðisskýrslur (Public Health in Iceland)

Over the whole 61 years studied, Iceland’s doctors reported 530,276 cases of influenza, half of them from Reykjavík and immediately surrounding areas (Figure 6.12D). Although reported cases are likely to be under-estimates, the broad shape of outbreaks in both space and time is readily discernible. The distribution throughout the year shows clear peaks in March–April with the low periods in August–September, a pattern which, as we have seen for France in Figure 6.10A is typical of many northern latitude countries. However, the sub-Arctic climate of Iceland pushes the peak influenza months slightly later in the influenza season than in France, and so we use for Iceland a slightly different definition of season, running from September 1 through to August 31 of the following year.

The swash–backwash model was applied to Iceland as a whole. The time series of both edges is shown in Figure 6.13A. Despite marked year to year variation, the average trend shown by the linear regression line for the leading edge is distinctly upward implying that waves have speeded up over time, i.e. influenza waves moved around the island faster at the end than at the beginning of the study period. By contrast the position of the following edge when influenza incidence ceased in any influenza season has remained essentially unchanged. This implies that the duration of reported influenza incidence grew slowly longer, from around 2.5 months in 1915–16 to nearly 4.0 months in 1975–76 (Figure 10.8B), a finding which differs from that for France.

Figure 6.13
Velocity of epidemic waves in Iceland for the influenza seasons 1915–16 to 1975–76. (A) Velocities of the leading edge (LE, solid lines) and following edge (FE, pecked lines) are shown as the velocity ratios, VLE and VFE, defined in Appendix 6.1 equation (A2). V is in the range, 0 ≤ V ≤ 1. The larger the value of V, the faster the edge. (B) The widening time gap between the two edges illustrated in (A) is confirmed in this graph which plots the average duration of each district’s epidemic wave in months. In both (A) and (B) the trend lines were determined by OLS linear regression.

Source: All influenza data based on Heilbrigðisskýrslur (Public Health in Iceland)

Three of the 61 seasons studied were associated with pandemics of influenza A (the Spanish, Asian and Hong Kong pandemics). Figure 6.14 uses data on the spatial extent of influenza in each season to plot the position of the pandemic front with reference to the three seasons which immediately preceded or followed it. In the Spanish and Asian pandemics the front stands out clearly but in the third (Figure 10.9C) the Hong Kong front appears, as in France, to have been spread over two seasons.

Figure 6.14
Pandemic seasons in the Icelandic records. (A)–(C) Profiles showing the cumulative number (scaled to unity) of medical districts reporting influenza cases by month as a measure of the spatial extent of influenza in Iceland for the three shift seasons of 1918–19, 1957–58 and 1968–69. In each case the profile for the shift season is compared with the profiles of the three preceding and three following non-shift seasons. (D) Value of the spatial version of the basic reproduction number, R0A, for Iceland for the influenza seasons 1915–16 to 1975–76. Peaks are associated with the three viral shift years for influenza A (1918–19, 1957–58 and 1968–69) and for three other non-shift seasons (1927–28, 1943–44, and 1972–73).

Source: All influenza data based on Heilbrigðisskýrslur (Public Health in Iceland)

Although pandemic years had large numbers of influenza cases, they were not the largest recorded over the period. The 1937–38 season had the largest number of cases (21,977) and the highest monthly rate and, as Figure 6.12A shows, monthly case numbers in several inter-pandemic years exceeded those with pandemics. For Iceland in Table 6.2, as for France in Table 6.1, we have divided the 61 seasons into three groups: (a) pandemic (3); (b) high-intensity inter-pandemic (24), with case rates greater than the lowest pandemic season; and (c) low-intensity inter-pandemic (34), with case rates lower than the lowest pandemic season. The average velocity of the leading edges (LE) for the three groups is (a) = 2.83 months, (b) = 5.53 months and (c) = 6.03 months. Echoing the results for France, this suggests that pandemic seasons have higher velocities than inter-pandemic years, and that this higher velocity is maintained even compared with inter-pandemic influenza seasons of similar intensity levels as pandemic seasons. Table 6.2 also shows that values for R0A, when calculated for the three categories of influenza season, (a), (b) and (c), defined above produced the same differentials as the leading and following edge parameters.

Table 6.2 Icelandic influenza seasons, 1915–16 to 1975–76. Swash model parameters for Iceland’s 61 influenza waves

Type of influenza wave

Morbidity (mean cases/season)

Start (t̄–LE) (mean in months)

Velocity (VLE)

End (t̄–LE) (mean in months)

Velocity (VFE)

Duration (mean in months)

R0A (mean index)

Pandemic waves (a)(n = 3)

11,027

2.83

0.76

7.03

0.41

3.82

1.48

Interpandemic waves:

High intensity (b)(n = 24)

10,113

5.53

0.54

9.14

0.24

3.56

0.84

Low intensity (c)(n = 34)

2,302

6.03

0.50

9.04

0.25

3.10

0.80

(a) Three pandemic seasons of 1918–19, 1957–58 and 1968–69.

(b) High intensity = Cases per 100,000 population greater than the least-intense pandemic wave.

(c) Low intensity = Cases per 100,000 population less than the least intense pandemic wave.

Thus application of the swash–backwash model to Iceland’s influenza morbidity records has shown that (i) the onset of waves speeded up over the period 1914 to 1975 and (ii) waves in three viral shift (pandemic) seasons spread significantly faster and were of longer duration than other equally large waves in non-shift (inter-pandemic) seasons. These are identical to the results reported for France. It suggests that both the swash model and the findings are robust across the transfer of geographical scales from a large country (France) to a small country (Iceland), and from a continental to an island setting. To test transferability across geographical scales further, we now apply the model at the smallest of our spatial scales by looking at influenza waves between 1947–48 and 1975–76 in the English market town of Cirencester.

Cirencester: A Small English Town

Cirencester is a small market town lying between Gloucester and Swindon in the English Cotswolds. It had a population of about 12,000 in 1957, rising to 15,000 by 1970 and 18,000 in 2011 (Figure 6.15B). Based in the town, R.E. Hope-Simpson ran a general practice covering an area of about 210 km2 from a centrally located surgery. In this practice, individual patients were identified and influenza was diagnosed and studied by a single doctor and his partner over a 30-year period following the end of the Second World War. The panel consisted of between three and four thousand patients and Hope-Simpson kept very detailed records of the incidence of several infectious diseases, but especially influenza, for each patient. Working with the help of his wife and later with the support of the UK’s Medical Research Council and the Public Health Laboratory Service he set up an Epidemiological Research Unit in Cirencester. By converting cottage rooms at his surgery in Dyer Street, he established a laboratory to permit identification of the viruses isolated from his patients. As a result, this unique practice became internationally known and it provides an unrivalled window into the behaviour of influenza epidemics at the micro-scale.

Figure 6.15
Influenza waves in an English GP practice. The country practice of a local doctor, Edgar Hope-Simpson, near Cirencester, England. Hope-Simpson made a special study of influenza in the area for some 50 years. (A) Boundaries of the 22 geographical units used by Hope-Simpson for data collection. (B) Location map of Cirencester. (C) Block diagram of cases of clinically-diagnosed influenza A weekly, 1946–74. The earlier appearance of influenza in the pandemic seasons of 1957–58 and 1968–69 is striking, as is the greater number of cases reported.

Source: Data from Cliff et al, Spatial Aspects of Influenza Epidemics, Figures 3.1 and 3.12, p. 48 and 65, Pion, London, UK, Copyright © 1986.

Hope-Simpson recorded every case of influenza identified in the practice, along with the causative strain of the virus and the geo-coordinates of the patient. Figure 6.15A maps the geographical framework used by Hope-Simpson for data collection, along with a block diagram (Figure 6.15C) of the cases reported in each influenza season, 1946–74. The earlier appearance of influenza in the town in the pandemic seasons of 1957–58 and 1968–69 is striking. This appearance of an early peak at times of antigenic shift in the causative virus can be related to the larger stock of susceptibles available for immediate infection on such occasions.

In Figure 6.16, the results of applying the swash–backwash model to the Hope-Simpson data from 1963–75 are summarised. They echo those already obtained for France and Iceland, with a sharp upturn in R0A and an increase in the leading edge velocity in the second season of Hong Kong influenza (1969). Thereafter R0A fell slowly to reach pre-1968 levels by 1973.

Figure 6.16
Cirencester, England: the 1968–69 pandemic of influenza in the Hope-Simpson general practice. Values of the reproductive number, and leading and following edge velocities for the swash–backwash model, 1963–75. As in France and Iceland, influenza moved more rapidly and struck earlier in 1968 than in non-pandemic seasons.

United States: The Swine Flu Outbreak of 2009

As noted in Section 6.2, the first cases of influenza caused by a novel influenza virus, subsequently typed 2009 swine A/H1N1 influenza, were reported in the state of Veracruz, Mexico, in March 2009, although the evidence suggests that there had been an ongoing epidemic for months before it was officially recognised. The Mexican government closed most of Mexico City’s public and private facilities in a futile attempt to contain the spread of the virus. The first cases in the United States were reported from California in April 2009. Rapid global diffusion of the new strain continued from the Mexican epicentre so that, on 11 June 2009, WHO declared the world to be at stage six of its six-phase pandemic alert scale. The crisis continued through into 2010, although the new strain had largely run its course by the turn of the year. The US Public Health Emergency for 2009 H1N1 influenza expired on June 23 2010 and, on 10 August 2010, WHO declared an end to the 2009 H1N1 pandemic globally. The new strain has subsequently behaved like normal seasonal influenza and is likely to be around for many years. In the US, the CDC estimates that between 43 million and 89 million cases (mid-range 61 million) of 2009 H1N1 occurred between April 2009 and April 10 2010; that there were between about 195,000 and 403,000 (mid-range 274,000) H1N1-related hospitalisations; and that there were between 8,870 and 18,300 (mid-range 12,470) 2009 H1N1-related deaths. US cases surged when schools reopened for the autumn term. The global impact of the pandemic is illustrated in Figure 6.17 and for the United States in Figures 6.18 and 6.19.

Figure 6.17
Pandemic (A/swine/H1N1) influenza, 2009. Countries, territories and areas with laboratory confirmed cases and cumulative number of deaths reported to the World Health Organization, Geneva, from the start of the pandemic. (A) 27 April 2009. (B) 22 July 2009. (C) 25 October 2009. (D) 21 March 2010.

Source: World Health Organization Map Production: Public Health Information and Geographic Information System (GIS). (Map A) Reproduced from Pandemic (H1N1) 2009: Countries, territories and areas with lab confirmed cases and numbers of deaths reported to WHO, Status as of 27 April 2009, available from http://gamapserver.who.int/mapLibrary/Files/Maps/GlobalSubnationalMasterGradcolour_weekly_20090427.png; (Map B) Reproduced from Pandemic (H1N1) 2009: Countries, territories and areas with lab confirmed cases and numbers of deaths reported to WHO, Status as of 22 July 2009, available from http://gamapserver.who.int/mapLibrary/Files/Maps/GlobalSubnationalMasterGradcolour_20090722_weekly.png; (Map C) Reproduced from Pandemic (H1N1) 2009: Countries, territories and areas with lab confirmed cases and numbers of deaths reported to WHO, Status as of 25 October 2009, available from http://gamapserver.who.int/mapLibrary/Files/Maps/GlobalSubnationalMasterGradcolour_20091029_weekly.png and (Map D) Reproduced from Pandemic (H1N1) 2009: Countries, territories and areas with lab confirmed cases and numbers of deaths reported to WHO, Status as of 21 March 2010, available from http://gamapserver.who.int/mapLibrary/Files/Maps/GlobalSubnationalMasterGradcolour_20100321_weekly.png with permission from the World Health Organization.

Figure 6.18
Geographical development of the swine flu pandemic in the United States, 2009–10 influenza season. Each histogram shows the number of positive influenza virus isolates by Department of Health and Human Services regions reported by WHO/NREVSS Collaborating Laboratories. The preponderance of swine flu A(H1N1), shown in black, as opposed to normal seasonal strains (grey) is evident in all regions.

Source: Data from US Centers for Diseases Control, Influenza Division, available from http://www.cdc.gov/h1n1flu/update.htm and http://www.cdc.gov/flu/weekly/

Figure 6.19
Pneumonia and influenza mortality in the United States, 2007–11. (A) The percentage of all deaths due to pneumonia and influenza in 122 US cities is plotted against the US national seasonal influenza baseline and the 95% confidence band above which epidemics (shaded) are declared. While the 2009–10 season in the round does not appear exceptional compared with adjacent seasons, especially 2008, the epidemic of swine flu in the fall of 2009 stands out as the largest autumn influenza event in this time series. (B) Number of influenza-related paediatric deaths. The high infant mortality during the swine flu season as compared with normal seasonal influenza in adjacent seasons is evident.

Source: Data from US Centers for Diseases Control, Influenza Division, available from http://www.cdc.gov/flu/weekly/

Figure 6.20 plots the values of the spatial basic reproduction number, R0A, and the leading edge parameter, , of the swash model for weeks 27–39 (second week of July–first week of October of the five normal influenza seasons, 2004–5 to 2008–9, and for the first swine flu influenza season, 2009–10). Note the substantially higher velocity through the US population for the new swine flu strain as compared with normal seasonal influenza at all three spatial scales as evidenced by the larger values of R0A and smaller values of in the 2009–10 season. The time window plotted covers the start of each influenza season and the reopening of schools, colleges and universities in the Fall.

Figure 6.20
Swine flu, United States, 2009–10. Values of the swash model parameters, R0A and t–LE, for the first 39 weeks (July–October) of each of the influenza seasons, 2004–5 to 2008–9, and for the first swine flu season, 2009–10 at three geographical scales. (A) 122 cities comprising the ILINet (influenza-like illnesses) PI (pneumonia–influenza) mortality dataset. (B) Fifty states. (C) Ten United States Department of Health and Human Services regions.

Source: Data from Centers for Disease Control, available from http://www.cdc.gov/h1n1flu/update.htm and http://www.cdc.gov/flu/weekly/

Discussion

In this section, a spatial version of R0A has been developed in the context of what we have called a swash–backwash model. This has been applied to records of influenza from France (1887–1999), Iceland (1916–75), Cirencester (1963–75) and the United States in 2009, locations of vastly different geographical scales and settings. But common threads have been found: pandemic influenza at all spatial scales appeared (i) earlier than normal in the influenza season and (ii) spread spatially more rapidly than normal. If our findings are confirmed elsewhere, this will have wider implications for public health measures. It would suggest that any new influenza pandemic in the twenty-first century, whether emerging from highly pathogenic avian influenza, swine influenza, or other sources, will give little lead time if the public health is to be protected. This conclusion underscores the role of surveillance and virus watch systems, issues which we considered in Chapter 2 and for which we now provide a practical example.

Pandemic Detection

As recent WHO reports have stressed, among the critical needs of any epidemic forecasting model if it is to be useful in informing control strategies is the ability to predict an epidemic’s likely spatial spread at some point relatively early in its unfolding history. Here we test the capability of the swash model for this purpose. We use the French influenza data. A positive outcome would imply that the swash–backwash model has a potential use as part of a disease early-warning system.

The following experiment was devised:

1. (i) In month 2 (August) of a given influenza season, the swash model parameters, R0A, VLE and VFE, defined in Appendix 6.1 in equations (A2) and (A7), were calculated and plotted.

2. (ii) In each successive month through to the end of the influenza season, the data set was updated as new data arrived. The model parameters were recalculated and plotted.

3. (iii) The experiment was applied to the following groups of influenza seasons, each centred around a pandemic year(s): (a) 1887–92; (b) 1896–1902; (c) 1914–20; (d) 1955–59; (e) 1966–72.

Figure 6.21 plots the results for R0A when the approach was tested for the group (c) seasons, centred on the 1918 pandemic. This pandemic spread over three influenza seasons (Wave I in the spring of 1918 in the 1917–18 season, Wave II in the autumn of 1918 in the 1918–19 season, and Wave III in the spring of 1919, affecting both the 1918–19 and 1919–20 seasons; cf. Figure 6.2). The plots for R0A show that, for the main part of the pandemic (Waves II and III), R0A ran at a consistently higher level from September than in immediately preceding and succeeding seasons. From the viewpoint of online monitoring, public health officers would have had early warning from the beginning of the 1918–19 season that the impending influenza experience was likely to be unlike anything they had encountered in the war years.

Figure 6.21
Real-time detection of the 1918–20 influenza pandemic in France. Time series plot of the values of the spatial basic reproduction number, R0A, for the swash–backwash model applied to continually updated influenza mortality reports in France for the 1914–20 influenza seasons.

Figure 6.22 summarises the results of the experiment for all the pandemic seasonal groups, (a)–(e). Each chart plots the median values of the model parameters obtained in each seasonal group for (i) the virus shift season(s) and (ii) the non-shift seasons. From October in the influenza seasons studied, this diagram shows that the spatial basic reproductive number, R0A, was consistently greater in virus shift seasons than in other seasons. For the leading edge parameter, VLE, velocity of spread was also greater in shift seasons than in other seasons except for 1889. For the following edge parameter, VFE, no consistent picture emerges from Figure 6.22.

Figure 6.22
Real-time detection of influenza pandemics in France, 1887–1999. Median values of (A) the spatial basic reproduction number, R0A, (B) the leading edge parameter, VLE, and (C) the following edge parameter, VFE, for the swash–backwash model applied to continually updated influenza mortality reports in France for the following groups of influenza seasons: (a) 1887–92; (b) 1896–1902; (c) 1914–20; (d) 1955–59; (e) 1966–72. In each group, influenza seasons were classified as shift if they were affected by a new strain of influenza virus and rest otherwise. Months in which the parameter values for shift seasons exceeded those for rest are shaded.

When a new strain of human influenza A appears, the population will have at best limited natural resistance, resulting in a much larger stock of susceptibles for infection than normal. Under these circumstances, pandemic influenza may occur, with the new strain running rapidly through the population. Ceteris paribus, characteristic features of virus shift seasons are likely to be (i) high levels of morbidity and mortality compared with ‘normal’ influenza seasons, manifested in larger values of R0A, and (ii) rapid geographical spread (large values for VLE). The expected behaviour of the following edge parameter, VFE, is not clear. In shift seasons, long spatial occupancy would lead to a low value for VFE, whereas rapid passage through and clearing of an area, which is feasible with a new virus strain, would yield a small value for VFE. Indeed, the ambiguous behaviour of VFE is confirmed by a correlation of 0.07 between R0A and VFE for the French data.

The results presented in this subsection broadly match expectation. Figure 6.23 shows that, in shift years, epidemics occurred earlier in the season and that they were, for the 1918, 1957 and 1968 shifts, more sharply peaked than in adjacent seasons. Figure 6.22A displays a long-term decline in the reproduction number for influenza which was linked to declining velocity of spread (the correlation coefficient between R0A and VLE is 0.77). The localised peaks in R0A and in epidemic velocity set against the century-long falling trend all occurred in virus shift seasons and their immediate aftermath.

Figure 6.23
Block diagram of influenza seasons in France, 1887–1999. The recorded monthly mortality from influenza in France is plotted by season. Viral shift years have been highlighted. Outbreaks in shift years generally start earlier and commonly had larger mortality than non-shift seasons. The block has been drawn from two different viewpoints to highlight these features.

Swash Model and Other Infectious Diseases

Poliomyelitis in England and Wales, 1919–71

The historical geography of poliomyelitis as a twentieth-century emerging disease is described in (Smallman-Raynor, et al. 2006). Figure 6.24 plots the monthly time series of reported cases of poliomyelitis per 100,000 population in England and Wales, 1919–71. The upsurge of epidemic poliomyelitis after the Second World War until the impact of mass vaccination against the disease kicked in from the 1960s is dramatic (see Smallman-Raynor, et al., 2006, pp. 317–51 and pp. 482–6). In this epidemic sequence, that of 1947 was by far the largest and most severe. From an apparent onset in the late spring of 1947, the epidemic spread across the entire country, attacking both urban and rural populations with force. At the height of the epidemic, in late August and early September, more than 600 cases of poliomyelitis were recorded each week. All told, the epidemic was associated with some 7,655 civilian and 735 military cases – a toll far in excess of the 1,600 or so notifications associated with the previous epidemic high of 1938. As (Gale 1948) observes, it was not simply that the local intensity of the 1947 epidemic was greater than ever before; the geographical reach of the epidemic, too, exceeded all previous experience of the disease.

Figure 6.24
Poliomyelitis in England and Wales, 1919–71. Monthly number of notified cases of poliomyelitis per 100,000 population.

Source: Data from General Register Office: Registrar General's Weely Returns, 1919–71.

Origin and course of the epidemic

The events of the summer and autumn of 1947 were not wholly unanticipated by the medical community of England and Wales, although some purveyed complacency. As early as 28 June – a full month before the usual seasonal upturn in poliomyelitis – The Medical Officer observed that notifications of poliomyelitis in the week ending 14 June were “higher than usual for the time of year,” adding that “an upward trend is expected” (Anonymous, 1947a, p. 260). A few weeks later, and with the count of poliomyelitis notifications still rising, readers of The Lancet were alerted to a Ministry of Health memorandum that warned of “an unprecedented prevalence” of poliomyelitis in the months ahead (Anonymous, 1947b, p. 155). The deepening fears of the medical community were underscored by an editorial comment in The Medical Officer. “The brisk rise in notification of poliomyelitis in the late spring of the present year,” the editorial noted, “gives us much uneasiness…Nobody can foretell what poliomyelitis is going to do, but an epidemic on an extensive scale appears to be due in this country” (Anonymous, 1947c, p. 23). “Unfortunately,” the editorial added, “we can do nothing about it, for there is no proof that any measures to control the disease have any value.”

Figure 6.25 confirms the pandemic-like qualities of the 1947 outbreak as compared with the previous three poliomyelitis upswings and the one which followed in 1948. It attacked the geographical mesh of boroughs and districts more completely, more rapidly (R0A, LE), and persisted longer (time between LE and FE) than any of the other four outbreaks. Table 6.3 gives the swash parameters defined in Appendix 6.1 for the five polio seasons between 1944 and 1948. As with influenza, the swash model picks out the extraordinary poliomyelitis epidemic of 1947 from its neighbouring, less intense, polio years.

Figure 6.25
Poliomyelitis in England and Wales, 1944–8: leading and following edges of poliomyelitis activity in England and Wales. Graphs plot the cumulative percentage of administrative units (county, metropolitan and municipal boroughs, urban districts and rural districts) that first notified (leading edge, LE; graph A) and last notified (following edge, FE; graph B) cases of poliomyelitis in the epidemic year of 1947 (heavy line trace) and in the corresponding weeks of the four non-epidemic years of 1944–46 and 1948.

Table 6.3 Poliomyelitis in England and Wales, 1944–48. Values of the swash model parameters for the poliomyelitis outbreaks in each year at the county borough and district scale

N of infected units

t LE

t FE

V LE

V FE

S A

I A

R A

R 0A

1944

All units

256

13.64

15.30

0.48

0.41

0.49

0.10

0.41

0.87

1945

All units

321

13.94

17.72

0.46

0.32

0.50

0.18

0.32

0.74

1946

All units

283

13.34

16.31

0.49

0.37

0.47

0.15

0.37

0.84

1947

All units

1113

10.29

18.50

0.60

0.29

0.36

0.35

0.29

0.90

1948

All units

532

13.66

18.41

0.47

0.29

0.49

0.22

0.29

0.72

6.4 Forecasting for Intervention

Any intervention against the spread of a communicable disease is likely to be more effective, to be more likely to succeed, and to make the best use of resources if the intervention can be targeted. Targeted interventions become feasible if we can forecast where to and how rapidly a communicable disease is spreading through a set of locations. In this section we look at possible ways to forecast spread from the point(s) of introduction of a disease (the disease impulse) to other areas (the response). To provide a common geographical framework for the discussion, the country of Iceland is used as an epidemiological laboratory. Our discussion focuses upon approaches to the disease forecasting problem rather than upon mathematical details of model fitting for which appropriate references are provided.

Iceland as an Epidemiological Laboratory

A number of reasons make Iceland particularly valuable as an arena within which to test geographical models of communicable disease spread:

1. Quality of Data

Iceland’s public health and epidemiological records are among the highest quality in the world in terms of their completeness, geographical granularity and temporal resolution. For many human communicable diseases, mortality and morbidity counts are available on a monthly basis for around 125 years in some 50 geographical areas; for some diseases the record goes back on a similar basis to the middle of the eighteenth century. Figure 6.26 summarises the general demographic history of Iceland from that date, along with the occurrence of the major epidemics of communicable diseases until the last quarter of the twentieth century by which time mass vaccination had all but eliminated them. Population size was stable at around 40–60,000 for the period 1751–1900, when a remarkable growth set in; the population approximately quadrupled between 1900 and 1970 to about 200,000. Up to 1950, this growth was concentrated mainly in the capital, Reykjavík. Since then, Reykjavík and the rest of Iceland have grown in population at about the same rate. The decline in infant mortality since 1850 and the higher birth-rates of the twentieth century (especially since 1940) are the main reasons for Iceland’s population growth.

Figure 6.26
Iceland demography and communicable disease history, 1751–1970. Annual population total, births, deaths and major epidemics of communicable diseases.

Source: Reproduced from Cliff et al, Spatial Diffusion: An Historical Geography of Epidemics in an Island Community, Figures 3.8 and 3.9, pp. 47-48, Cambridge University Press, Cambridge, UK, Copyright © 1981, by permission of the authors.

Throughout its history, neither the island as a whole nor any individual settlement approached the critical community size required to maintain endemically the spectrum of human infectious diseases (Section 4.3). As a result, epidemics of infectious diseases in Iceland have been spatially and temporally separated, with inter-epidemic intervals during which either no or only isolated cases occurred. Figure 6.27 illustrates the point by plotting Iceland’s twentieth-century time series for three diseases – measles, rubella and pertussis. Thus for a significant outbreak of one of these diseases to occur on an Icelandic farmstead, the disease agent had to cross several hundred miles of sea, travel from a seaport or airport to the rural community, and eventually to the farm itself. Once the susceptible population was exhausted, the agent disappeared from the island until the susceptible population grew again by births to a sufficient size to sustain a new outbreak. This temporal and spatial separation of outbreaks facilitates both the development and testing of forecasting models.

Figure 6.27
Epidemic diseases in Iceland, 1900–90. Monthly time series of reported cases (grey, log scale) and rate per thousand population (black, arithmetic scale) of measles, rubella and pertussis.

Source: Reproduced from Cliff et al, Island Epidemics, Figure 6.3, p. 245, Oxford University Press, Oxford, UK, Copyright © 2000 by permission of Oxford University Press.

2. Population Distribution

Iceland is a large island (about the size of southern England or the state of Indiana) located just south of the Arctic Circle. Its 1990 population was somewhat over 250,000. It is the least-densely settled country in Europe and the harsh environment of the interior plateau has restricted population to the peripheral lowlands (diagonal shading in Figure 6.28). This diagram maps the geographical distribution of Iceland’s population by medical districts using proportional circles and squares. The percentage change in population that has taken place over the last century is also shown. A pattern familiar across Western Europe – of rural depopulation on a large scale – is evident. The biggest single settlement is the capital, Reykjavík, which has been growing both absolutely and in its share of Iceland’s total population. In 1901 its 6,700 inhabitants accounted for less than a tenth of the island’s 78,000; by 1990, Reykjavík had grown to 100,000 out of a total of 256,000. The overall impression is of population distributed around the coast much like the beads on a necklace. The degree of isolation of settlements around the coast means that it is often possible to treat such communities as separate epidemiological cells from a modelling point of view, giving a spatial equivalent to the discrete epidemic waves in time evident in Figure 6.27.

Figure 6.28
Iceland population distribution. Settled areas of Iceland with 1990 medical district populations shown by proportional circles and squares. Medical district boundaries plotted as pecked lines. Percentage population change, 1900–90 indicated numerically.

Source: Reproduced from Cliff et al, Island Epidemics, Figure 6.2, p. 243, Oxford University Press, Oxford, UK, Copyright © 2000 by permission of Oxford University Press.

3. Physician Reports

Historically, Icelandic legislation required the principal physician of each of the country’s c. 50 medical districts to submit an annual report to the Chief Medical Officer of Health in Reykjavík. The reports contain contact tracing information for each epidemic of an infectious disease to have affected the district that year. The reports give, in varying detail, information on the origin of the outbreakand its local spread. They are much fuller prior to 1945 than thereafter. Figure 6.29 maps the results of this contact tracing for all epidemics of measles, rubella and pertussis which have affected Iceland since 1900. The last map, D (All), pools the results for the three diseases. The principal vectors fan out from the capital, Reykjavíkur (population 95,000 in 1990), to the three main regional towns in Iceland of Akureyrar (population 15,000), Ísafjarðar (population 3,600) and Seyðisfjarðar (population 1,000) – cascade or hierarchical diffusion. Local spread from these centres into their hinterlands (contagious diffusion) is illustrated by the vectors within the diagonally shaded areas on each map, and is most clearly seen on map All around Akureyrar.

Figure 6.29
Iceland: internal epidemic pathways. Observed transmission routes between Icelandic settlements for measles, rubella and pertussis, 1900–90. Vectors show the number of times the diseases were recorded in Heilbrigðisskýrslur (Public Health in Iceland) as passing between linked medical districts.

Source: Reproduced from Cliff et al, Island Epidemics, Figure 6.4, p. 247, Oxford University Press, Oxford, UK, Copyright © 2000 by permission of Oxford University Press.

Lag Maps

The number of steps in the particular pathway followed as an epidemic moves from one geographical area to another will clearly affect its time of arrival. Using lag maps, this subsection shows the impact of Iceland’s internal pathways upon spatial timing. Measuring epidemic velocity has attracted theoretical attention because of its importance for possible preventive measures; the spread of slow-moving waves may be simpler to check than that of rapidly moving waves. Basic references are (Mollison 1991) and (van den Bosch, et al. 1990). If we are dealing with a simple spatial process where the epidemic spreads with a well-defined wave front (as in the case of the studies by (Mollison 1977)), then the physical concept of distance travelled over time may be appropriate. However, where the wave front is not a well-defined line, and where the susceptible population through which the epidemic moves is both discontinuous in space and has sharp variations in density, then alternative definitions of velocity must be sought.

Calculation of Lag Maps

For epidemic , code the first month in which a disease was reported anywhere in Iceland as month 1 and, for medical district i, note the month in which the disease was reported in that medical district as month 2, or 3, or 4, etc. Denote this month as. The desired quantities are then (6.1) $t − i = ( 1 / n ) ∑ l t i l , l = 1 , 2 , ... , n ,$ where i is subscripted over the 47 medical districts.

Figure 6.30 plots the ${ t ¯ }$ for three of the Icelandic diseases separately (measles, rubella and pertussis) and for an average map, where the average has been calculated from the results for the three separate diseases and influenza; data for 1900–90 have been used. Circle sizes are proportional to the 1990 populations of medical districts. Circles have been coloured black on the map in which they first appear and have been left as open circles thereafter. A similar pattern is seen across the diseases. Reykjavík and the Reykjavík region are reached early; the principal regional capitals of Akureyri, Ísafjörður and Seyðisfjörður are attacked at the second stage; and epidemic decay occurs in the smaller and more remote settlements of the fjord areas of eastern and northwest Iceland. This is consistent with the pathway diagrams in Figure 6.29, and yields a schematic space–time model like Figure 6.31.

Figure 6.30
Time lag maps for infectious diseases in Iceland, 1900–90. Average time in months from commencement of epidemics to reach medical districts in different parts of Iceland. On each map, medical districts reached for the first time are shown as solid circles. Districts appearing on earlier maps appear as open circles on later maps. Circle sizes are proportional to 1990 medical district populations. (A) Measles. (B) Rubella. (C) Pertussis. (D) Average across all three diseases and influenza.

Source: Reproduced from Cliff et al, Island Epidemics, Figure 6.12, p. 264, Oxford University Press, Oxford, UK, Copyright © 2000 by permission of Oxford University Press.

Figure 6.31
Communicable disease diffusion in Iceland. Hypothetical sketch of the spread of an infectious disease agent from an international source down to a farm family in rural Iceland. The arrows indicate possible contacts. At levels 1 and 2, the movements are likely to be young adults; at 3 and 4, children. The synchrony of Iceland with Europe and the United States as international sources which seed infectious diseases in Iceland is discussed in Weinberger, et al. (2012).

Source: Reproduced from Cliff et al, Spatial Diffusion: An Historical Geography of Epidemics in an Island Community, Figure 3.14, p. 54, Cambridge University Press, Cambridge, UK, Copyright © 1981, by permission of the authors.

With the spatial and temporal structure of epidemic spread in Iceland over the twentieth century in mind, we can now turn to modelling such space–time patterns and we use two approaches:

1. 1. The first is to try to capture the process and build a spatial SIR model. Here the particular difficulties are (i) obtaining reliable estimates of the susceptible population and (ii) constructing a model in which the mixing parameter, β, between I and S in Figure 1.17 can vary over space and time. The homogeneous mixing assumption of Figure 1.17 is clearly not tenable when both cascade and contagious disease diffusion occurs. Such spatial inhomogeneities in the spread process are usually handled via a compartment model in which homogeneous mixing is allowed within the compartments or boxes, but different interaction rates are estimated between the compartments. See, for example, (Baroyan, et al. 1969, 1971, 1977). The mixing parameters themselves may also change over time as intervention to control spread occurs.

2. 2. The second is to use time series methods in which the main emphasis is upon identifying structural regularities in the data and projecting these forward spatially and temporally. These methods tend to be less data hungry than SIR models but do not handle non-stationary processes in time and space very easily. They accordingly run the risk that an unexpected change in the process over time will cause serious errors of forecast.

We now give examples of the results which may be obtained with each approach. The data used are monthly reported measles cases by medical district in Iceland, 1945–74. See (Cliff, et al. 1981) for details of the data.

SIR Models

(Cliff and Ord 1995, pp. 153–64) have adapted the basic SIR model of Section 1.4 to examine epidemic return times for measles outbreaks in Iceland’s medical districts between 1945 and 1974. In addition to the transitions between infected, susceptible and recovered states shown in Figure 1.17, the model also allowed for migration of infectives into each medical district to approximate the priming of epidemics between geographical areas as shown in Figure 1.18. Estimation of likely epidemic return times to different geographical areas, especially if they are disease-free, will become increasingly important in the context of the UNICEF/World Health Organization’s Global Immunization Vision and Strategy (GIVS) programme (Sections 4.4 and 6.5). It will be important to link the model to intervention strategies. Planned vaccination campaigns, for example, feedback to affect epidemic return times.

Figures 6.32 and 6.33 show the results obtained for the eight medical districts which comprise Iceland’s northwest fjords region. The time periods used for model fitting and for ex-post forecasting of the time gap in months between the penultimate and ultimate epidemics in the measles time series of each district are marked on Figure 6.32. The bar graphs in Figure 6.33 show that the observed return times lay within the upper and lower ten percent points of the estimated distribution for return times in all eight districts, but that the correspondence between individual observed and 50th percentile forecast values was poor in two districts (⇒ingeyrar and Reykhóla).

Figure 6.32
Iceland, northwest fjords, 1945–74: time series of reported measles cases in eight medical districts. Model calibration and forecast periods with inter-epidemic intervals in months are given.

Source: Reproduced from Cliff, A.D., and Ord J.K., Estimating epidemic return times, Figure 8.6, p. 155, in Cliff A.D. et al (eds), Diffusing Geography, Institute of British Geographers Special Publications Series, 31, Blackwell, Oxford, UK, Copyright © 1995, by permission of the authors.

Figure 6.33
Iceland, northwest fjords, 1945–74: measles epidemic return times. Observed and projected epidemic return times in eight medical districts. Median, 10th and 90th percentiles for 50 simulations are plotted.

Source: Reproduced from Cliff, A.D., and Ord J.K., Estimating epidemic return times, Figure 8.7, p. 157, in Cliff A.D. et al (eds), Diffusing Geography, Institute of British Geographers Special Publications Series, 31, Blackwell, Oxford, UK, Copyright © 1995, by permission of the authors.

Observed epidemic return times for measles across Iceland in the post-1945 period have been generally 36–60 months (3–5 years), often a little shorter in the capital, Reykjavík, and sometimes longer in very small settlements. This is expected from Figure 1.18. Table 6.4 gives the results for the return times model if the analysis is extended from northwest Iceland to the whole country. In the table, settlements have been classified according to their time lag in months behind Reykjavík to be reached by measles outbreaks (cf. Figure 6.30A) after virus arrival in Iceland: group 1 settlements, six months or less; group 2, 6–9 months; group 3, over nine months. The classification reflects the geographical structure for disease propagation shown in Figure 6.29. The entry point into Iceland for most communicable diseases is Reykjavík, from which spread to regional centres – Akureyri (group 1), Ísafjörður (group 2) and Seyðisfjörður (group 3) – occurs, with localised spread from thence into regional hinterlands.

Table 6.4 Iceland: inter-epidemic waiting times for measles, 1945–74. Observed and ex-post forecasts for groups of settlements

Inter-epidemic interval (months)

Settlement group ( n )

Parameter

Observed

Model percentile

50

10

90

Group 1 (9)

Median

36

29.9

13.5

65.3

Mean

62.9

34.9

17.4

71.6

Minimum

34.0

20.2

6.7

52.3

Maximum

167.0

57.4

34.4

100.6

Corr (obs, 50th percentile)

0.76

Group 2 (29)

Median

46

41.4

20.1

85.7

Mean

58

43.9

22.3

88.7

Minimum

24

24.4

8.1

63.1

Maximum

168

89.1

57.7

146.0

Corr (obs, 50th percentile)

0.26

Group 3 (6)

Median

44

39.5

18.2

85.4

Mean

52

38.6

17.6

83.8

Minimum

26

32.5

13.2

75.8

Maximum

92

42.6

20.4

89.0

Corr (obs, 50th percentile)

0.03

The return times model was fitted to each settlement group separately. Parameter estimates were calculated using all the epidemic intervals except the last in the study period, and these estimates were then used to generate ex-post forecasts of the waiting time in months to the last measles epidemic in the time window. Reykjavík was excluded from the analysis because of its disproportionate population size compared with any other settlement (eight times the next biggest town in the mid-1970s), and because this city generally led the diffusion process. The other settlements ranged down in size from c. 15,000 population. As Table 6.4 shows, despite low correlations between the point estimates, the average 50th percentile point from the model shows reasonable correspondence with the average for each settlement group. Both observed and model means reflect the 3–5-year cycle for measles epidemics seen historically in Iceland. In all but eight of the 44 settlements included in the analysis, observed return times fell within the 10th and 90th percentage points of the model estimates.

Spatial Time Series Approaches

(Cliff, et al. 1993, pp. 360–410) have tested a variety of time series models on Icelandic measles data. The models used, including SIR for comparison, and the advantages and disadvantages of each are summarised in Table 6.5. As Figure 6.32 shows, the time series of Icelandic measles epidemics comprises two main states: epidemic and no epidemic. A persistent difficulty from a modelling viewpoint with such two-state series, even with different models for each state, is devising a rule to switch models from the no epidemic to the epidemic state in the correct time period. And modelling the epidemic state is made problematic by the non-stationarity of the build up and fade out phases of an epidemic. The impact of these issues is seen in Table 6.5. Forecasts produced by different models often miss the start of epidemics by a month and then echo the shape of the epidemic curve one month in arrears – the so-called one month lag effect in Table 6.5. Then the non-stationarity of the build up and fade out phases can cause models to over-estimate epidemic size when the reported cases decline sharply after the epidemic peak.

Table 6.5 Time series forecasting models. Characteristics with Icelandic measles data, 1945–70

Model

Application format

Main data inputs

Temporal parameter structure

Single region

Multiple region

S

I

Fixed

Variable

SIR

×

×

×

×

Good at forecasting epidemic recurrence years ahead; average to poor at estimating epidemic size.

Autoregressive

Reykjavík

×

×

Initial one-month lag effect; adapts to changing phase characteristics; overestimates epidemic size.

GLIM (Generalised linear integrated model)

Northwest Iceland

×

×

×

Use of main town as epidemic lead indicator produces good estimates of epidemic starts in other medical districts; poor estimates of epidemic size.

Kalman filter

Reykjavík

×

×

Initial lag effect; locks on to epidemic course but overestimates epidemic size.

Bayesian entropy

Reykjavík

×

×

×

Separate models for epidemic and no epidemic states. One-month lag effect in predicting epidemic curve; probability forecasts of epidemic/no epidemic good; model switches states in correct month.

Simultaneous equation

Multi-region chains

×

×

Areas studied as causal chains; phase characteristics guaranteed by model formulation.

Logit

Multi-region chains

×

×

Good probability forecasts of epidemic/no epidemic states; slow state switching.

A further modelling complication with communicable diseases is the frequent non-congruence of data recording intervals and the serial interval of the disease; the serial interval is defined as the duration of time between the onset of symptoms of an index individual and the onset of symptoms in a second case directly infected by the first (Fine, 2003). A great deal of surveillance data for communicable diseases is reported for comparatively long discrete time intervals (e.g. weekly, monthly, quarterly), whereas the serial intervals for many communicable diseases of humans are often much shorter (e.g. a few days for influenza, about 14 days for measles). The spatial effect of this lack of congruence is that geographically widely separated cases, where the second is caused by the first, can appear as simultaneous events when they are in fact associated but separated temporally by a serial interval which is shorter than the data recording interval. Thus the spatial development of an epidemic on a monthly map series may appear as bursts of growth when there is in fact a causal chain between cases which would be observable if the data recording interval were of finer temporal granularity than the serial interval.

From the various time series models in Table 6.5, we have selected simultaneous equation models to illustrate how this issue may be tackled. Figure 6.34 takes reported monthly cases of measles for seven Icelandic medical districts (Keflavíkur, Reykjavíkur, Akureyrar, Hofsós, Siglufjarðar, Ísafjarðar, Bolungarvíkur), 1945–70, and divides the data into a model calibration period (1945–57) and a forecast period (1958–70). The maps use vectors to show the measles diffusion chains modelled, reflecting the internal pathways identified in Figure 6.29A. Both time lags and spatial disease impulses between medical districts are incorporated. Mathematical details are given in (Cliff, et al. 1993, pp. 383–88). The graphs show that the probability estimates of the likelihood of an epidemic are a better indication of observed epidemics than are the estimated number of cases. In addition, the size of epidemics as measured by number of reported cases is better modelled in larger than in smaller communities. This is unsurprising. As implied by Figure 1.18, the initiation of an epidemic in a small community is as likely to depend upon the chance arrival of an infective (e.g. a tourist; not allowed for in the model) as the systematic arrival of infection through the population size hierarchy of settlements.

Figure 6.34
Icelandic measles diffusion chains: simultaneous equation models. The maps show (A) spread from Keflavíkur to Reykjavíkur, on to Akureyrar, and finally from Akureyrar into the local medical districts of Hofsós and (B) Siglufjarðar. (C) Spread from Keflavíkur to Reykjavíkur across to Ísafjarðar, with local spread (neighbourhood) from thence to Bolungarvíkur. The identifiers, t, t–1 and t–2, reflect the monthly time steps in the causal chains. Arrows run from impulse districts to the response districts. The graphs plot the district monthly reported cases of measles (grey bars) against simultaneous equation estimates of cases (solid black lines) and epidemic probabilities (pecked lines). The model calibration period is 1945–57 and the forecast period is 1958–70.

Source: Adapted from Cliff et al, Measles: An Historical Geography of a Major Human Viral Disease from Global Expansion to Local Retreat, 1840–1990, Figure 14.2 and 15.6, p. 384 and 404, Blackwell, Oxford, UK, Copyright © 1993, by permission of the authors.

Model Assessment

Forecasting the return times and sizes of epidemics of communicable diseases is still an art form and much remains to be done. In this, epidemic forecasting is no different from forecasting in other fields like economics, public health, or indeed the weather in the UK. Our broad conclusions after experimenting with many models over a spectrum of diseases and geographical areas are as follows:

1. (i) We have yet to find the Rosetta stone of a model which produces accurate projections of both epidemic recurrence times and epidemic size. All too often, if a model is devised which will forecast recurrence acceptably, epidemic size is over-estimated. To forecast inter-epidemic times accurately, it is generally necessary to tune the model to be sensitive to changes signalling the approach of an epidemic, with the result that it overshoots when the epidemic is in progress.

2. (ii) Models which are based only on the size of the infective population in previous time periods consistently fail to detect the approach of an epidemic. Instead, they provide reasonable estimates of cases reported, but lagged in time.

3. (iii) Models with parameters fixed through time have a tendency to smooth through epidemic highs and lows because they are unable to adapt to the changes between the build-up and fade-out phases. Time-varying parameter models are better at avoiding this problem.

4. (iv) Epidemic recurrences can be reasonably anticipated only by incorporating information on the size of the susceptible population and/or properly identifying the lead–lag structure for disease transmission among geographical areas. Addition of spatial interaction information markedly improves our ability to forecast recurrences in lagging areas. Information on susceptible population levels also serves to prime a model to the possibility of a recurrence. Models based on susceptible populations, but which are single- rather than multi-region, tend to miss the start of epidemics but rapidly lock on to the course of an epidemic thereafter. Models which are dominated by spatial transmission information, at the expense of information on the level of the susceptible population in the study region, produce estimates of epidemic size which reflect the course of the epidemic in the triggering regions rather than in the study region.

5. (v) Stochastic process models enhance our understanding of disease transmission across geographical space and increase the chances of devising time series models appropriate to the task of forecasting.

These conclusions highlight the fact that naïve models produce poor results. However, Table 6.5 also indicates the gains to be made for each extra element of complexity added to our models, namely:

1. (i) time-varying parameters to handle the non-stationary nature of within-epidemic structure, particularly the fundamentally different character of the build-up and fade-out phases;

2. (ii) separate models for epidemic and inter-epidemic episodes to recognise the different character of these periods;

3. (iii) spatial lead–lag information to improve our ability to forecast epidemic recurrences and to understand the transmission of disease between areas;

4. (iv) incorporation of data upon the susceptible population level to improve estimates both of epidemic size and likely recurrence intervals.

And it is important to be able to identify the gains obtained by increasing model complexity, since it is all too easy to specify sophisticated models which are either insoluble or contain multiplicative structures that magnify errors when applied to data of variable quality.

6.5 Intervention, Value-for-Money and Demography

All health interventions cost money, and the associated opportunity costs of the other uses to which the money used for health interventions could be put. Against the costs must be weighed the gains in lives saved and extended, the improvements in the quality of life which interventions can yield, and potential economic and social benefits in the labour market from a healthier population. In the end, resolution of the tensions between costs and benefits are resolved by societal debate and political choices. Sometimes the politicians buffer themselves from the difficult choices. In the United Kingdom, for example, the independent National Institute for Health and Clinical Excellence (NICE) was set up in 1999 to reduce variation in the geographical availability and quality of the UK’s National Health Service (NHS) treatments and care. NICE’s evidence-based guidance is designed to assist in resolving uncertainty about which medicines, treatments, procedures and devices represent the best quality care and which offer the best value for money for the NHS. Inevitably, NICE’s value-for-money guidance leads to making uncomfortable decisions to restrict or deny certain drugs on cost grounds to the NHS as opposed to the benefits gained (e.g. prolonging life for terminally-ill patients by a few months). No-one enjoys making these choices but they are the reality of the current world (cf. Figure 6.1).

In this section, we look at ways of assessing the impact of interventions to prevent the geographical spread of communicable diseases, costs, benefits and demographic outcomes. We begin by examining ways of measuring intervention impacts.

Measuring Intervention Impacts

Substantial resources are devoted to reducing the incidence, duration and severity of major diseases that cause morbidity and mortality, and to reducing their impact on people’s lives. It is important to capture both fatal and non-fatal health outcomes in summary measures of average levels of population health. For mortality, one of the earliest and still most commonly used ways of measuring intervention impact is to assess the extension of life produced by intervention as compared with no-intervention scenarios.

Years of Potential Life Lost (YPLL)

Agencies such as the World Health Organization and the US Centers for Disease Control and Prevention have been experimenting with statistics which measure years of potential life lost (YPLL). This takes into account the year at which a death occurs as compared with their life expectancy at, say, birth, as given in the life tables published by most national population agencies. If an individual dies before their normal life expectancy, the difference between date of death and life expectancy in years represents one measure of YPLL. More complicated versions of this simple model exist because life expectancy itself changes with age. Thus, in England in 2000, a 20-year-old male could expect on average to live for another 55 years (60 for a female) whereas an 80-year-old male could expect to live on average for another seven years. So under YPLL, the two deaths would be weighted differently, the younger death having a weight much higher than the older.

The effect of measures of this kind is to change the order of some leading causes of death, as compared with the list order based on death rates alone, all tending to give greater weight to the infectious diseases (which especially affect children) and to injuries (which especially affect young people); where these conditions cause mortality, it is most commonly in the first two decades of life. YPLL also reduces the relative importance of heart disease and cancers (which tend to affect the old).

Disability-Adjusted Life Years (DALYs) and Healthy Life Expectancy (HALE)

With illness and disability, rather than mortality, similar arguments may be used, giving special weight to diseases which cause a lifetime of disability (so-called disability-adjusted life years, DALYs) rather than a brief illness. The World Health Organization makes frequent use of healthy life expectancy or HALE. HALE at birth sums expectation of life for different health states, adjusted for severity distribution, making it sensitive to changes over time or differences between countries in the severity distribution of health states. It gives the average number of years that a person can expect to live in ‘full health’ by taking into account years lived in less than full health due to disease and/or injury. A full account of the many different measures which have been proposed to quantify what is meant by ‘full health’ is given in (Murray, et al. 2002). Yet other weighting systems attempt to weight morbidity data by quality of life measures. But the difficulty with all these approaches is defining unambiguous and generally accepted sets of weights.

Figure 6.35 shows the YPLL, DALY and HALE maps for the countries of the European Union for infectious and parasitic diseases in the first decade of the new millennium. All show essentially the same pattern of higher loss of life and poorer health primarily in southern Europe and the new republics of Eastern Europe which were originally part of the former USSR. The YPLL map focuses on death before age 65, so the figures give an indication of the premature loss of economically productive life. In the EU-27, the average YPLL is 116 per 100,000 men, as compared with 47 YPLL per 100,000 women. The increased resilience of women against infectious and parasitic diseases is a well known biological fact. The highest losses of potential life due to infectious diseases among men were found in Portugal (538 YPLL per 100,000 men) and in Romania (444 per 100,000 men). The variance is high in the highest quintile, showing the patchy nature of local epidemics in people under 65: in Lisboa the number of YPLL increased to over 1,000 among men. The lowest figures were found in the Czech Republic (23.1 YPLL). The same geographical pattern is found among women, with the highest YPLL again found in Romania (190) and Portugal (157, with 300 in Lisboa). The lowest figures were returned by Malta (only 1.5 years, which may be an artefact of Malta’s small population) and the Czech Republic (10.3 YPLL).

Figure 6.35
YPLL, DALY and HALE maps for the countries of the European Union. (A) Male YPLL. (B) Female YPLL. The YPLL is estimated against an age of death of 65. (C) Age-standardised DALYs per 100,000 population. (D) HALE. The YPLL and DALY maps are for infectious and parasitic diseases at the beginning of the new millennium. The HALE map gives the estimated healthy life expectancy for all persons at birth. In the British Isles, although the total burden of deaths from infectious and parasitic diseases was small at the millennium, the map shows that the greatest YPLL and lowest expectations of good health were in Scotland and the London area. The conurbations in South and West Yorkshire, Greater Manchester, and in the West Midlands also experienced comparatively high risks of premature mortality from infectious and parasitic diseases as did the Welsh Marches.

Sources: Figures (A) and (B) reproduced from Eurostat Statistical Books, Health Statistics – Atlas on Mortality in the European Union, Figures 6.3 and 6.4, pp. 46–47, Office for Official Publications of the European Communities, Luxembourg, Copyright © European Communities, 2009, with permission from the European Union; (C) Data from WHO Global Burden of Disease data by member countries, 2004, available from http://www.who.int/healthinfo/global_burden_disease/estimates_country/en/index.html and (D) Data from World Health Organization, World Health Report, World Health Organization, Geneva, Switzerland, Copyright © 2004.

Global Immunization Vision and Strategy: Intervention Costs and Benefits

As discussed in Section 4.4, the Expanded Programme on Immunization (EPI) was established in 1974 through a World Health Assembly resolution (WHA27.57) to build on the success of the global smallpox eradication programme and to ensure that all children in all countries benefited from life-saving vaccines. The operations of the EPI are currently set within the framework of the Global Immunization Vision and Strategy (GIVS), launched as a joint initiative of the WHO and the United Nations Children’s Fund (UNICEF) in 2006.

GIVS was developed in the context of increasing resources for immunisation. As described in Section 4.4 and in (Wolfson, et al. 2008), in 2000 a public–private partnership, the Global Alliance for Vaccines and Immunisation (GAVI) was initiated to provide financial support for immunisation in the world’s poorest countries. By the end of 2005, government and private sources had pledged a total of US$3.3 billion to the GAVI Alliance, enabling it to provide support to 73 of 75 eligible countries. Between 2000 and 2005, total GAVI Alliance disbursements were US$760.5 million. GAVI Alliance’s resource outlook over the next decade has improved with the launch of two innovative funding mechanisms: the International Finance Facility for Immunisation (IFFIm), which could provide up to US$4 billion over the next 10 years, and the Pneumo Advance Market Commitment (AMC), which will provide US$1.5 billion to support low-income countries for the purchase of new vaccines against Streptococcus pneumoniae, a leading cause of childhood meningitis and pneumonia mortality.

In 2005, WHO and UNICEF undertook, as a companion to the GIVS document, to estimate the costs to reach immunisation goals which had been established, and (Wolfson, et al. 2008) report on the methods and results of that exercise. A model was developed to estimate the total cost of reaching GIVS goals by 2015 in 117 low- and lower-middle-income countries. Current spending was estimated by analysing data from country planning documents, and scale-up costs were estimated using a bottom-up, ingredients-based approach. Financial costs were estimated by country and year for reaching 90% coverage with all existing vaccines; introducing a discrete set of new vaccines (rotavirus, conjugate pneumococcal, conjugate meningococcal A and Japanese encephalitis); and conducting immunisation campaigns to protect at-risk populations against poliomyelitis, tetanus, measles, yellow fever and meningococcal meningitis (Figure 6.36).

Figure 6.36
GIVS costing blocks. Block A: maintenance of current routine system (baseline cost) Current levels of investment in immunisation were estimated using available data from 40 Financial Sustainability Plans (Block A1), and extrapolated for the period 2006–2015 by accounting for the impact of inflation and population increases (Block A3). They assume no change in vaccination schedules and no improvement in immunisation coverage levels (Block A2). This does not include campaigns or vaccine costs. Block B: vaccine costs Vaccine costs were estimated by using coverage targets, population projections and applying the most recent available data on unit prices of different vaccine presentations. The estimates account for wastage rates and the need for buffer stock. The cost of safe injection equipment is bundled in the vaccine cost estimates. The element below the line represents the vaccine costs to continue immunisation at 2005 levels, and above the line is the vaccine portion of scaling-up. Block C: scaling-up of routine system This is estimated using an ingredients-based approach. Block D: campaigns A schedule of needed campaigns was generated based on a combination of the projections of vaccine coverage and the epidemiological coverage required to reduce rapidly the burden of disease. Campaign costs include both operational costs and vaccine costs.

Source: Reproduced from Wolfson et al, Estimating the costs of achieving the WHO–UNICEF Global Immunization Vision and Strategy, 2006–2015, Bulletin of the World Health Organization, Volume 86, Figure 1, p. 31, Copyright © 2008, with permission from the World Health Organization.

The 72 poorest countries of the world spent US$2.5 (range: US$1.8–4.2) billion on immunisation in 2005, an increase from US$1.1 (range: US$0.9–1.6) billion in 2000. By 2015 annual immunisation costs will on average increase to about US$4.0 (range US$2.9–6.7) billion. Total immunisation costs for 2006–2015 are estimated at US$35 (range US$13–40) billion; of this, US$16.2 billion are incremental costs, comprising US$5.6 billion for system scale-up and US$8.7 billion for vaccines; US$19.3 billion is required to maintain immunisation programmes at 2005 levels. In all 117 low- and lower-middle-income countries, total costs for 2006–2015 are estimated at US$76 (range: US$23–110) billion, with US$49 billion for maintaining current systems and$27 billion for scaling-up. These projected costs are staggeringly large, particularly in the context of the state of the global debt and sovereign debt crises existing at the end of the first decade of the new millennium. In the 72 poorest countries, US$11–15 billion (30%–40%) of the overall resource needs are unmet if the GIVS goals are to be reached. While the authors acknowledge that the methods they used are approximate estimates with limitations, they do provide some idea of the financing gaps that need to be filled to scale up immunisation by 2015 if the world’s communicable disease control aspirations are to be converted into practical reality. What are the benefits? If the immunisation goals are realised, (Wolfson, et al. 2008, pp. 35–6) estimate that, by 2015, more than 70 million children in the world’s 72 poorest countries can be protected annually against 14 major childhood diseases if an additional US$1 billion per year can be invested towards immunisation, saving some 10 million more lives in the decade, 2006–15. The cost equates to an additional US$0.5 per capita per year above current levels (< US$1 per capita) of investment in immunisation. Wolfson and colleagues argue that, at such modest per capita costs and high benefits, immunisation continues to be one of the best values for public health investment today. They see other benefits too over and above saving lives: (i) in impoverished countries, lives saved boost economies, potentially yielding a rate of return of up to 18%; (ii) immunisation can serve as a platform to strengthen health systems and deliver other life-saving interventions such as those against non-communicable conditions like malnutrition, as well as other communicable infections like malaria and intestinal worms.

Failure to control the spread of communicable diseases has consequences other than demographic; it also throws sudden burdens upon healthcare systems, potentially leading to degradation of other services. For example, Figure 6.37 shows the country by country impact of the 2009 swine flu scare upon healthcare services for two months of the pandemic: the first week of November and the third week of December, 2009. Note the shift from an Americas focus of high impact to Europe as the pandemic spread from its hearth continent to another geographical region.

Figure 6.37
Pandemic (A/Swine/H1N1) influenza, 2009: global impact upon healthcare services. Impact refers to the degree of disruption of healthcare services from increased levels of acute respiratory diseases attributed to the pandemic, and is mapped at four levels of intensity for WHO member countries. (A) Early October 2009; (B) mid December 2009.

Source: World Health Organization Map Production: Public Health Information and Geographic Information System (GIS). (Map A) reproduced from Impact on health care services, Status as of Week 41, 05 Oct-11 Oct 2009, available from http://gamapserver.who.int/mapLibrary/Files/Maps/Global_Impact_week41_20091105.png with permission from the Word Health Association and (Map B) reproduced from Impact on health care services, Status as of Week 51, 14 Dec-20 Dec 2009, available from http://gamapserver.who.int/mapLibrary/Files/Maps/Global_Impact_week51_20100114.png with permission from the Word Health Association.

6.6 Conclusion

This chapter concludes our analysis of the control of the geographical spread of human infectious diseases. We have tried to show the ways in which modelling approaches, linked to real-time surveillance, can be used to guide intervention strategies using quarantine, isolation and vaccination to interrupt spatial spread. These interventions all have the capacity to reduce the global burden of transmissible infectious diseases which ruin lives, increase mortality rates, and drain public health resources. There is little sign that, in the foreseeable future, the spread of re-emerging old infections and the emergence of new infections will cease. As we noted in the Preface to this book, if pestilence, prayer and appeals to saints to control the spatial spread of communicable diseases are thought of solely as quaint relicts of medieval life, the late twentieth century came as a nasty shock. Since 1981, 25 millions have died from the AIDS pandemic, and Africa alone has 12 million AIDS orphans. The worldwide toll of HIV infection today has reached 35 million and in some African countries, one-quarter of the adult population is infected. Worries over new H5N1 and H1N1 strains of human influenza rekindle memories of the devastating Spanish influenza pandemic of 1918–19. The brief outbreak of SARS in 2003 showed how rapidly viruses can now race through the global urban network. Indeed, as this conclusion is being written, China has just reported (31 December 2011) its first death from H5N1 bird flu for over a year in the southern city of Shenzhen.

Study of the temporal history of human disease shows that ‘new’ or ‘renewed’ infectious diseases are a constant theme rather than a recent episode. Very old diseases such as measles or cholera undergo periodic eruptions and invasions which share many of the features now associated with emerging diseases. This longer-run perspective helps to set the context within which current events are viewed. It enables the exceptional conditions which are forcing the present rapid pace of disease change to be more readily isolated. In the twenty-first century, the world is still working at control, eradication and elimination of the spatial spread of infectious diseases, just as the Italian states and principalities, along with their saints, did seven centuries ago when confronted by plague epidemics. And it reinforces our need to intervene to control the spread of communicable diseases in timely and cost-effective ways.

Appendix 6.1: Swash–Backwash Model Equations

Let the first month of an influenza season (July, say) be coded as t = 1. The subsequent months of the season are then coded serially as t = 2, t = 3,…, t = T, where T is the number of monthly periods from the beginning to the end of the season (i.e. T = 12 in our yearly cycle). We call the month in which an influenza case is first recorded in a given geographical unit the leading edge (LE) of the outbreak in that unit, and the last month of record as the following edge (FE). As described in (Cliff and Haggett 2006), standard statistical analysis of the distribution of the two edges enables us to define a time-weighted arithmetic mean, $t ¯ L E and t ¯ F E$ for each edge. For the leading edge, the equation is (A1) $t − L E = 1 N ∑ t = 1 T t n t$ where nt is the number of units whose leading edge occurred in month t and $N = ∑ n t$. The time-weighted mean is a useful measure of the velocity of the wave in terms of average time to unit infection. A similar equation can be written for FE, and higher order moments can also be specified. To allow comparison between diseases with different wave characteristics, we convert these time-weighted means to a velocity ratio, V,(0≤V≤1), (A2) $V L E = 1 − t − L E D$ where D is the duration of the wave. A similar equation can be written for VFE.

The conventional basic reproduction number (or rate or ratio), R0, is defined as the ratio between an infection rate (β) and a recovery rate (μ): (A3) $R 0 = β u .$

In the spatial domain, A, the spatial reproduction number, R0A, is the average number of secondary infected geographical units produced from one infected unit in a virgin area. In a given study area, the integral SA (the proportion of the study area at risk of infection) is given by: (A4) $S A = ( t − L E − 1 ) T$ while the proportion of the area which is infected (the infected area integral) is (A5) $I A = t − L E T − S A$

The recovered area integral, RA, is (A6) $R A = 1 − ( S A + I A ) .$

All three integrals are dimensionless numbers with values in the range [0, 1]. S parallels the reciprocal of β; a small value of SA indicates a very rapid spread. The integral RA parallels μ in that a large value suggests very rapid recovery. Because SA is inversely related to its power we suggest that complements might be substituted in estimating a spatial version of R0, namely (A7) $R 0 A = 1 − S A 1 − R A .$

Other formulations of R0A are possible. For example, (Smallman-Raynor and Cliff 2013) have used R0A = (1 – SA)/RA equally successfully.

Such a spatial basic reproduction number would measure the propensity of an infected geographical unit to spawn other infected units in later time periods. In effect it provides an indicator of the tendency of an infected unit to produce secondaries. Values of R0A calibrate the velocity of such spread (the larger the value, the greater the rate of spread). It is important not to over-stretch the analogy between R0 and R0A. R0 is defined for the hypothetical situation when a new case is introduced into a wholly susceptible population. While R0A is defined for a virgin region, it is calculated using spatial data for the entire span of the outbreak. As a result, it is contaminated with data from the later phases of the outbreak when many spatial sub-areas are no longer virgin. This may account for the frequent small calculated values. Normal R0 is useful because it distinguishes between situations where an epidemic can take off (R0 > 1) and those where it cannot (R0 < 1), and this is arguably the most important attribute of R0 as a summary parameter in epidemiology. R0’s spatial cousin, R0A, does not share this property – for example in Tables 6.1–6.3 there are analyses of real epidemics that had sustained spread from one district to another in which R0A < 1. But it does allow large numbers of spatial settings to be examined and the relative velocity at different stages of outbreaks to be assessed and compared. Finally, it should be noted that the parameters R0A, LE and FE are correlated, but each gives slightly different insights into the progress of an outbreak through a geographical area.