Two-Population Model Accounting for the Different Patterns Observed in the Log-Log Plot of the Cumulative Numbers of Those Infec

Jpn. J. Infect. Dis., 62, 2009

Laboratory and Epidemiology Communications Two-Population Model Accounting for the Different Patterns Observed in the Log-Log Plot of the Cumulative Numbers of Those Infected and Killed in the Early Phase of the 2009 H1N1 Pandemic in Contrast to the One-Population Model Accounting for the 1918 - 1919 Pandemic in San Francisco Hiroshi Yoshikura* National Institute of Infectious Diseases, Tokyo 162-8640, Japan Communicated by Ichiro Kurane (Accepted October 30, 2009)

The log-log plot of the cumulative numbers of those infected throughout the epidemic (i.e., independent of X), k has to be 1. and killed in the early phase of the 2009 H1N1 pandemic (WHO Fig. 1 shows two typical log-log plots, one for Mexico data at http://idsc.nih.go.jp/index-j.html [in Japanese]) fol- (Mexico-DT) and the other for the USA (USA-DT) (the lowed straight lines expressed by the relation logY = klogX shapes of the plots for countries/regions other than Mexico k – klogN0 (equivalent to Y = (X/N0) ), and was characteristic are essentially similar to that of the USA) (1). The value of k to countries or regions. Here, X and Y respectively denote was not 1 but was about 0.65 for Mexico and about 1.8 for the cumulative numbers of infected and killed persons, N0 the USA; the value of N0 was 4-5 for Mexico, and 2,000- (determined by extrapolation of the line to the horizontal axis) 3,000 for the USA. This means that m continued to change denotes the number of those infected that accumulated be- from its value at the time when the first fatal case was reported fore the first pandemic-related death, and the coefficient k (20-25% for Mexico and 0.03-0.05% for the USA). It was denotes the slope of the straight line. The case-fatality rate m as if the epidemic in Mexico that started with a relatively k is expressed by m = (X/N0) /X. If m should remain the same high case-fatality rate followed a process of continuous de- crease of the viral virulence (i.e., lowering of case-fatality *Corresponding author: Mailing address: Department of Food Safety, rate), while the epidemic in the USA that started with no fatal Ministry of Health, Labour and Welfare, 1-2-2 Kasumigaseki, cases became an epidemic of an increasingly virulent virus Chiyoda-ku, Tokyo 100-8916, Japan. Tel: +81-3-3595-2142, Fax: (higher case-fatality rate) when the number of the infected +81-3-3503-7965, E-mail: [email protected] persons reached 2,000-3,000. A changing case-fatality rate

482 Fig. 1. Log-log plot of the cumulative numbers of those infected and killed. Mexico-DT and USA-DT, plots obtained by using WHO data for Mexico and the USA; Mexico-SML and USA-SML, plots obtained by simulation using Table 1; 1919 Pandemic, plots obtained by 1918-1919 pandemic in San Francisco (2). Horizontal axis, cumulative number of the infected; vertical axis, cumulative number of the killed. is rarely encountered in infection epidemiology. This paper of infections reached 1,000 (assumption 6), and (iii) in the deals with how to interpret this curious phenomenon. subpopulation of vulnerable people, the virus spread at an By trial and error, I arrived at a model that can explain the amplification rate twice as high as that in the normal population above observations reasonably well. Namely, the influenza (assumption 4) and the case-fatality rate became as high as virus that caused the initial epidemic in Mexico consisted of 25% (assumption 3). two virus strains, one with high virulence and the other with The simulated plots (Mexico-SML and USA-SML) agreed low virulence, and only the latter was exported to other coun- well with the real ones (Mexico-DT and USA-DT) (Fig. 1), tries, where only vulnerable people succumbed to it. I call the meaning that events quite close to the model constructed us- model thus constructed a “two-population” model, because ing the above six assumptions (shown in Table 1) were likely it postulates two populations. Further assumptions used for to have happened. One concern with this model is that no the modeling were as follows: laboratory data so far published have indicated the presence ・Assumption 1: The case-fatality rate does not change of virulent mutant(s) in the Mexican epidemic. However, during the epidemic so far as the combination of the virus without postulating more and less virulent strains, I was strain and the host population remains the same. unable to make a model that fits the data. Likewise, omission ・Assumption 2: The case-fatality rate is higher for the of any one of the assumptions resulted in failure of the model- virus strain of higher virulence if the susceptibility of ing as I performed it. the target population remains the same. After the above analysis, I realized that it was rather evident ・Assumption 3: The case-fatality rate is higher in the vul- from the beginning that whenever the virus changes its niche nerable population if the virus is the same. or its virulence, parameters like basic reproduction number ・Assumption 4: In the vulnerable population, the virus (R0, the average number of other individuals each infected spreads more quickly than in the normal population. individual will infect in a population that has no immunity to ・Assumption 5: In the normal population, if the trans- the disease), susceptible population (S, proportion of the popu- missibility is the same, the virus with higher virulence lation who are susceptible to the disease), and case-fatality spreads more slowly than that with lower virulence. This rate also change during the course of the epidemic. Monitor- assumption may contradict the general belief. However, ing of the case-fatality rate using the log-log plot of the as patients infected with a virulent strain are confined to cumulative numbers of those infected and killed may be use- bed while those infected with the low-virulence one freely ful for detecting possible changes in the virulence or the niche circulate in the community, the chance of transmission of the virus. is much higher for the low-virulence virus. It should be noted that, if the same reporting system is main- ・Assumption 6: When the virus with low virulence is in- tained throughout the epidemic, k will not be affected by the troduced into a country, the virus will not immediately sensitivity of the surveillance. This is because the original enter the subpopulation of vulnerable people but will do equation can be transformed to k = logY/log(X/N0), where so by chance after long circulation in the normal popula- numerators and denominators of Y (denominator is 1 repre- tion. senting the first killed case) and X/N0 are equally affected by Based on these six assumptions, Table 1 was produced. the sensitivity. Therefore, while the determination of N0 re- For Mexico, it was postulated that (i) the virus strain with quires the reporting of all cases, surveillance using properly high virulence (case-fatality rate 25% in the general public) selected sentinel points is sufficient for estimating k. and the strain with low virulence (case-fatality rate 0.1% in Finally, I tried to determine whether the log-log plot analy- the general public) started to circulate simultaneously, and sis is applicable to the past data. The data used are the ones (ii) the virulent strain propagated at an amplification rate half appearing in Arseny K. Hrenoff’s report “The Influenza 1918- that of the less virulent one (assumption 5). For the USA, it 1919 in San Francisco” in Military Surgeon, vol. 89 (Novem- was postulated that (i) only the virus of low virulence entered ber 1941), p. 807, which was cited by Alfred W. Crosby in this country, (ii) the virus could not enter the subpopulation the book “America’s Forgotten Pandemic: the Influenza of of the vulnerable population until the cumulative number 1918” (2). The plot is shown also in Fig. 1. It followed a

483 Table 1. Table for simulation Mexico simulation Normal population Normal population Normal population Time (ti) Low-virulence virus (LVV) High-virulence virus (HVV) LVV + HVV Infected (x) Killed (0.1%)(y) Infected (x´) Killed (25%) (y´) Infected (X=x+x´) Killed (Y=y+y´) t1 2 0 2 0 4 0 t2 8 0 4 1 12 1 t3 32 0 8 2 40 2 t4 128 0 16 4 144 4 t5 512 0 32 8 544 8 t6 2,048 2 64 16 2,112 18 t7 8,192 8 128 32 8,320 40 t8 32,768 32 256 64 33,536 96

USA simulation Normal population (NP) Vulnerable population (VP) NP + VP Time (ti) Low-virulence virus (LLV) Low-virulence virus (LLV) LLV Infected (x) Killed (0.1%) (y) Infected (x´) Killed (25%) (y´) Infected (X=x+x´) Killed (Y=y+y´) t1 128 0 0 0 128 0 t2 256 0 0 0 256 0 t3 512 0 0 0 512 0 t4 1,024 1 2 0 1,026 1 t5 2,048 2 8 2 2,056 4 t6 4,096 4 32 8 4,128 12 t7 8,192 8 128 32 8,230 40 t8 16,384 16 512 128 16,896 144 Table used to produce plots Mexico-SML and USA-SML in Fig. 1. The upper panel is for simulation of Mexico epidemic and the lower panel for that of the USA epidemic. Numbers indicate cumulative numbers of those infected or killed. Time (ti) indicates the time when the numbers in its right side panels were attained; as epidemic may not progress at the same speed, time from t1 to t2, time from t2 to t3, etc., are not necessarily the same. straight line, but with a break between the fourth and the fifth it may be reasonable to use the latter part of the data for the plots. The former part before the break is for October 13- analysis. November 9, 1918, and the later for November 10, 1918- January 25, 1919. The value of k was nearly 2 for the former I thank Mr. Makoto Tanaka, Department of Food Safety, and 1 for the latter, and N0 was about 300 for the former and Ministry of Health, Labour and Welfare and Dr. Hiromu about 10 for the latter. If I assume that the data after the break Yoshida, Department of Virology II, National Institute of In- better reflect what happened (vide infra), the case-fatality rate fectious Diseases for stimulating discussion. was continuously 10%. In other words, a one-population model applies for the 1918-1919 pandemic in San Francisco. REFERENCES The reason why there was a break is not known. It should be 1. Yoshikura, H. (2009): Two parameters characterizing 2009 H1N1 swine kept in mind, however, that the influenza virus was discov- influenza epidemic in different countries/regions of the world. Jpn. J. ered only in 1931, more than a decade later, by Richard Shope, Infect. Dis., 62, 411-412. and the diagnosis could be understandably very inaccurate in 2. Crosby, A.W. (1989): America’s Forgotten Pandemic: the Influenza of 1918-1919. Inaccuracy in this case, underreporting affects 1918. Cambridge University Press. p. 114 (translated into Japanese by the data particularly when the number of cases is low. Thus, H. Nishimura).

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