MATHEMATICAL MODELLING OF INFLUENZA EPIDEMICS

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1 British Medical Bulletin (1979) Vol. 35, No. 1, pp MATHEMATICAL MDELLING F INFLUENZA EPIDEMICS THE MATHEMATICAL MDELLING F INFLUENZA EPIDEMICS C C SPICER MRCS Division of Medical Computing Clinical Research Centre, Harrow 1 Methods 2 Studies on data for England and Wales 3 Results 4 Discussion References For some years workers in the USSR have used a mathematical technique for predicting the course of influenza epidemics in the Soviet Union. The methods used have been recently described in a book (in Russian) by Baroyan el al. (1977) which gives details of the mathematical model used and the accuracy with which it fits the data. They also give extensive references to their own and other work in this field. A report in English is given by Baroyan et al. (1971). The model used was derived by Baroyan and his co-workers from the mathematical theory of transfer processes in continuous media, but is in fact identical with one proposed about 60 years ago by A G McKendrick and subsequently developed by Kermack & McKendrick (1927). A very good account of existing theory and results in this field is given by Bailey (1975), who devotes a chapter to the Russian methods. The Russian system covers two processes: firstly the spread of the influenza epidemics between major cities of the Soviet Union and second the behaviour of the separate epidemic waves of influenza in individual cities. All these are described by a single large set of differential equations incorporating the rates of migration between the major cities, which are known from travel statistics. The most striking and unexpected feature of the model is that the parameter on which the spread of an epidemic t within a city depends is the same for every city in any one epidemic within the USSR. This parameter itself is a product of two unknown quantities related to the proportion of persons in the population who are susceptible to the epidemic virus and the transmissibility of the virus. The latter is a complex entity and includes the degree of mixing in the population and the survival of the virus in the environment. It could depend on meteorological and sociological conditions and, in fact, on many factors other than the immune state of the population, which is defined by the proportion of susceptible individuals at risk. Another interesting point about the Russian model is that it uses a single empirically determined function to describe the falling off in the infectivity of influenza cases, and this again seems to apply quite generally throughout the USSR. The applicability of the Russian results has not been tested outside the USSR and Eastern Europe and the following account is concerned with some preliminary work on the data available in England and Wales. 1 Methods The basis of all the applications of mathematics to the spread of infectious disease has been the assumption that the progress of an epidemic is analogous to that of a second-order chemical reaction whose rate is proportional to the product of the two reacting substances. In this case these "substances" are the susceptible and the infectious members of the population so that where R is the incidence rate of new cases of the disease, S and I are the concentrations of susceptible and immune individuals, and X is the average number of new cases produced by a single infectious individual. Considering the complexity of the underlying process it seems incredible that such a simple law could apply, but there is now a good deal of evidence from family and community studies that it is often a very good approximation. Given that this is so the progress of an epidemic can be described as shown in fig. 1. It can be seen from the figure that the number of infectious people in the population at a given moment is the sum of those infected at each previous epoch multiplied by the probability of surviving as an infectious case till that moment. The infectivity at each stage of the disease is assumed, in the present model, to be a constant, X. The total of infectious survivors multiplied by X FIG. 1. Diagram illustrating the epidemic model used in this paper Number still infectious at each stage of illness Susceptible individuals (x) New cases (y) Yo X w w \V X X y 2 y, 4*,, y 0 4*, N X X '3 12 T 0 71 T 1 '0 T 2 *0 ^o Yo = "i "o - Yo - Vt = X 2 ^o Yo Yi VJ ^ X3 y, =Xx oy o v }' 0 Y2 = x i Wi ^0 + Yo ^1) y 3 =Xx 2 (y 2 >I' 0 + y,>j' 1 +y 0 4' 2 ) KEY susceptible individuals at time t y< new cases at time t 4* : proportion of cases still infectious j intervals after being infected X: transmissibility parameter 23 Vol. 35 No. 1

MATHEMATICAL MDELLING F INFLUENZA EPIDEMICS and the number of susceptible individuals gives the number of new cases arising, and the number of susceptible individuals is correspondingly reduced by

2 MATHEMATICAL MDELLING F INFLUENZA EPIDEMICS and the number of susceptible individuals gives the number of new cases arising, and the number of susceptible individuals is correspondingly reduced by this amount It is probable that the infectivity of an infected person is not constant throughout the course of the disease, but in the case of influenza this assumption is not critical. The actual calculations in the above model are laborious but quite simple and can be done easily on a desk calculator and even more easily on an electronic computer. Mathematically the process is really a continuous one in time and much theoretical work has been done on the resulting differential equations (see Bailey, 1975); but in practice the available data are always in discrete time units. Application therefore takes the form of fig. 1. Algebraically the basic equations can be written in the following form using the notation of fig. 1: The factor y t is the probability that a patient is still infectious at time t after falling ill. In other words it describes the progress of a patient as indicated by the arrows in fig. 1. When applying these equations to an actual epidemic it is necessary to have estimates of the initial numbers of infectious (yo) and susceptible persons (x^). In small communities these may be known, but in larger ones they have to be ascertained in some way from epidemiological observations, as does the value of the parameter A. With the equations in the form given above this gives rise to a basic indeterminacy since, if x,, is multiplied by a positive constant c, and X by 1/c, the course of the subsequent simulated epidemic is unaltered for a given y,,. The system used by the USSR estimates a single parameter, equivalent to tafl, and y 0 is found by adjusting the beginning of the simulated epidemic to coincide with that observed. A mathematical technique of fitting Xxj has been developed by Baroyan et al. (1977) based on the theoretical properties of the epidemic process. It can be shown from the first two steps of the epidemic that, if XJC 0 is fixed, y 0 is determined for all identical epidemics. The function y T can be derived from the probabilities that a patient infected at time t will still be infectious at time t + 1. Baroyan et al. (1977) give the following values of these probabilities as having been obtained from observation: p, T= It will be seen that this implies that no patient is infectious for longer than six days. The data available in the USSR are the daily notifications of new cases of influenza, and a further set of parameters is used to describe the variations in frequency of notification on different days of the week, there being a much smaller rate during and immediately before weekends. 2 Studies on Data for England and Wales As mentioned above, the Russian system is designed to cover the spread of influenza over the major cities of the whole of the USSR. For this purpose it includes data, of a kind not available 6+ 0 in the UK, on interchange of population between these cities. However it seemed worth while to make some attempt to see whether the form of epidemic curve applicable within these cities could be used on the data for England and Wales and Greater London published by the ffice of Population Censuses and Surveys. It was felt originally that this was not likely to be very fruitful, but the results have turned out to be sufficiently interesting to be worth reporting. There are some important differences between the English and the Russian statistics. In the USSR the notifications are made daily from polyclinics during epidemic periods. This would correspond roughly to daily notifications of new cases of acute respiratory infections by general practitioners in England and Wales, but these do not exist What is available are the deaths registered as due to influenza and influenzal pneumonia weekly for England and Wales and, also, separately for Greater London. A consideration of the underlying equations suggested that if the proportion of deaths to clinical cases is approximately constant throughout any given epidemic the model should apply to the incidence of deaths allowing for their distribution in time after infection. The weekly nature of the notifications does not seriously affect the model owing to the short infectious period of influenza, and it introduces a useful smoothing effect The curve of variation of infectiousness with time given by Baroyan et al. (1977) was assumed applicable. This assumes that no patient is infectious for more than six days, and a value of 0.64 weeks was derived from it for the average duration of infectiousness. The incidence of death after infection presented some difficulty owing to the paucity of actual data, and the calculations were based on observations reported by Stuart-Harris et al. (1950). It was found, however, that variations in these figures made little difference to the final fitting of the epidemic curve. The use of numbers of deaths instead of notifications of infections makes it necessary to add one more equation to the set given above and we now have t = x t~ v t+i t+i Z, +1 = K y t +i-e<f>6 e-o where the new quantities are defined as z t = number of registered deaths at time t to t + 1, K = probability that an infected person will die, (j>e = probability of dying 8 time units after infection, the time unit being one week, and once again all epidemics will be identical when (x 0, y 0, X, K) = (cx 0, cy 0, Vc, K/C), where c is a positive constant The technical method of fitting was to use a standard computing procedure (Powell, 1964) to find a set of parameters which minimized the sum of squares of the differences between the observed and calculated weekly deaths (W), i.e., to minimize where z, is the number of calculated deaths in week t 24 Br. Med. Bull. 1979

3 MATHEMATICAL MDELLING F INFLUENZA EPIDEMICS FIG. 2. Weekly deaths from influenza and influenzal pneumonia in (a) England and Wales, and (b) Greater London H 200 1M m- 14M ; a A\ / ' A A. A \ 100- u* M#- M V o A weighted version of this function was also minimized for each epidemic, the weights being the observed z t 's. This has the effect of fitting the epidemic more closely near the peak and less closely in the tails. To the order of accuracy of the whole process this weighting made little difference to the conclusions and the discussion below is confined to the unweighted calculations. It is hoped that a full discussion of the statistical and mathematical details will be published elsewhere. a \ KEY: observed numbers calculated numbers n the abscissae, each division represents a period of one week in the later stages of the epidemic. Weighted fits tend to reverse this effect. The values of the parameter a = Xx 0 are given in Table I, and in fig. 3 the values for Greater London are plotted against those TABLE I. Values of o = Xx 0 in England and Wales and Greater London, England and Wales Greater London 3 Results The model was fitted to all influenza epidemics recorded in England and Wales during the period , except those whose curves showed obvious evidence of being bimodal. A selection of the curves for England and Wales and Greater London is given in fig. 2. It can be seen that the fit is reasonably good and of the same order as that in the graphs published by Baroyan et al. (1977) for Russian cities. Generally speaking the fitted curve is too high at the beginning of the epidemic and too low at the end. This is not unexpected, since it is known that there is a tendency to under-notify at first and then over-notify Years which were clearly unsuitable because of small numbers or bimodality are omitted 25 VoL 35 No. 1

4 MATHEMATICAL MDELLING F INFLUENZA EPIDEMICS FIG. 3. Relationship between parameter o = Xx 0 in England and Wales and Greater London (correlation coefficient is 0.67) "a S. 6 0 Greater London o for the same years in England and Wales. There is a marked correlation of about 0.7 between a for London and England and Wales, but no very obvious interpretation of its values in terms of the arrival of new strains of virus. The large value for follows closely the appearance of the Hong Kong A strain in The earlier values from 1958 to 1965 show a general downward trend which may indicate an increasing mass immunity to the Asian (H2N2) influenza A virus and its antigenic variants. The actual values of a imply that in the early stages of the epidemic one infected person transmits infection to about five others which seems to be an implausibly high number. The value of K was fixed for all calculations at 2 x \0~*, which is intended as a rough estimate of the influenza death rate. At least one parameter can always be fixed as only the ratios between the parameters can be estimated, as explained above. If the death rate is in fact reasonably constant, A. will give some indication of the transmissibility of the disease. The values of X/K plotted for England and Wales and Greater London infig.4 suggest that transmissibility was low in the early stages of the introduction of the new Asian (H2N2) virus subtype in 1957 and the Hong Kong (H3N2) virus subtype in and high for some years after. This is biologically plausible if the virus is adapting to new conditions of spread. This effect is most clearly demonstrated in the data for Greater London. If K, the death rate, is a constant proportion of the rate of recovery from the disease, then mathematical theory (see Bailey, 1975) suggests that the final size of the epidemic should be inversely proportional to K/X. This is not apparent in fig. 5, which shows clearly that epidemic size as judged by influenza deaths increases almost linearly with K/X. This may arise because variations in the death rate swamp the expected theoretical relationship. The fitting of the model depends on an arbitrary empirical judgement as to when the epidemic begins and ends, and some numerical studies were done on the effect of this factor. It was found that none of the estimates of the parameters was altered by more than about 5-10% by truncations of 2-3 weeks at the start and end of the epidemic. Further numerical studies showed that the goodness of fit FIG. 4. Values of X/x in (a) England and Wales, winter to winter , and (b) Greater London, winter to winter , 2D0 so 19M '59 '60 'S1 '62 '63 '64 '65 '66 '67 '6» '69 '70 '71 '72 '73 Year 195J '5» 'S '61 '62 '63 '64 ts '66 '67 "St '69 '70 '71 '72 '73 Year 26 Br.Med.Bull 1979

5 MATHEMATICAL MDELLING F INFLUENZA EPIDEMICS FIG. 5. Relationship betweenx/xand total influenza death* in (a) England and Wales, and (b) Greater London Total deaths depended much more on x^ and X than on y 0 and K. In other words, y 0 and K could vary considerably from their optimal values without altering the goodness of fit, while variations of 10-20% in X and x 0 seriously affected it. It was also clear that the position of the optimum fit was determined more by X/x^ than by the individual values of X and x,,. 4 Discussion It seems from the results given above that the simple mathematical model used is capable of fitting a variety of influenza epidemics to a reasonably good approximation. It also appears not to be seriously affected by assumptions about the distribution of the infectious period and the time of death after onset. It should be said, however, that the fitting process may break down for some epidemics and result in negative numbers of cases in the tails of the epidemic. Apart from this, a few numerical studies have indicated that the best fit is usually unique when one of the parameters has been fixed and is not afflicted by local stationary points, as can be the case when fitting complex mathematical models. The biological significance of the parameters is difficult to assess as only their ratios can be determined and their absolute values are unknown. It is not possible to relate them to those occurring in the usual theoretical development. To be of practical use the model depends on obtaining, as early as possible, a good estimate of the combined parameter a = XJC. This is likely to be difficult unless it is found to be as stable over other large geographical areas as it has been found to be in the USSR. If a can be well established early, estimates of the time when the peak will be reached, and the over-all size of the epidemic, can be made. It would be worth while to do some K/X «o- so M M0 t M Total deaths retrospective studies on this possibility using statistical techniques of the kind described by Box & Jenkins (1962) for process control. The most satisfactory approach to the problem, though this would be expensive and time consuming, would be to investigate the relationships between the parameters and the infectiousness of the disease and the number of susceptible individuals in the population. In principle, both these could be measured in field studies: the former in family and small community epidemics, the latter by studies on antibody levels in the population. Such studies would be best conducted in relatively isolated communities of reasonable size and based on a reliable system of notification. It seems likely from the results obtained by Hammond & Tyrrell (1971) in Tristan da Cunha that epidemics in such communities would follow the present model quite closely though a considerably larger population would be more desirable. Studies on a number of epidemics would be necessary. Scientific considerations apart it is not altogether clear whether the expense of such a programme of field studies would be justified in terms of strict practical benefit. To predict the course of an epidemic from its behaviour in the first few weeks would not allow much to be usefully done, except to ensure the provision of more hospital beds and medical supplies at the peak and later. n the other hand, if several months' notice could be obtained by estimating x<, and X from the behaviour of strains isolated in some other part of the world, more effective immunological and chemotherapeutic preventive measures could be put into operation. Antibody surveys using the new strain might indicate the value of Xg, but an estimate of X would have to be made in the region of origin of the virus and this might not be practicable. 27 Vol. 35 No. 1

6 MATHEMATICAL MDELLING F INFLUENZA EPIDEMICS Even if it is practicable the estimated value for X might not be applicable elsewhere. However, a direct estimate of x 0 might be of considerable help in fitting the early stages of the epidemic to the model. ACKNWLEDGEMENTS This work arose out of my visit to the USSR under the terms of an Anglo-Soviet collaboration agreement I am most grateful to Dr Yu G Ivannikov and Dr L A Rvachev for their patience in explaining to me the details of the USSR system for predicting influenza epidemics, as I have no knowledge of Russian. Most of the calculations for this paper were done by Mr Robert lesen as part of an MSc thesis at the Mathematics Department, Imperial College of Science and Technology, London, and I would like to thank him for his help and advice. Thanks are also due to Professor D R Cox who encouraged this collaboration and to Dr Valerie Isham for her help. REFERENCES B«iley N T J (1975) Mathematical theory of infectious diseases and its applications. Gnffin, London Baroyan V, Rvachev L A, Basfleviky U V, Ermakov V V, Frank K D, Rvachev M A & Shashkov V A (\97\)Adv. Appl. Probab. 3, Baroyan V, Rvachev L A & Ivannikov Yu G (1977) Modelling and prediction of influenza epidemics in the USSR. (In Russian)* Box G E P & Jenkins G M (1962)/. R. Stal. Soc. B, 24, Hammond B J & Tyrrell D A J (1971)/. Hyg. (Cambridge) 69, Kermack W & McKendrick A G (1927) Proc. R. Soc. A, 113, Powell M J D (1964) Computer J. 7, Stuart-Harris C H, Franki Z & Tyrrell D A Jl (1950) Br. Med. J. 1, Tim work wai produced by the N F Gnmatd Innhute of Epidemiology and Microbiology, Moscow The Quarterly Journal of Medicine fficial rgan of the Association for Physicians of Great Britain and Ireland ne of the leading medical journals published in the U.K. (Established 1907, New Series 1932). It covers the whole field of medicine but gives some emphasis to internal medicine. Its aim is to report advances in both diagnosis and treatment, and many of its papers are concerned with the development of growing points of current interest. The basic orientation is clinical, but the journal welcomes contributions with a patho-physiological approach. In the face of the present tendency to fragment the subject, The Quarterly Journal of Medicine attempts to provide general physicians with a synoptic view. Many of the contributions provide a desirable link between clinical and basic science. In addition to original papers, the journal also publishes occasional requested review articles on subjects of special interest. Quarterly, 1979 prices: (UK 17.50, US?44.00)p.a. Single issues 6.00 (? 13.50). xford University Press Journals Press Road, Neasden, London NW10 DD 28 Br. Med. Bull. 1979

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