日本脳炎流行の理論モデル

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Japanese encephalitis (JE) is not considered essentially to be a disease of human beings in the sense that the cycle of JE can not be maintained without amplifying vertebrate animals. Pigs are probably the most important amplifying animal in Japan and some other countries, and the human epidemic of JE receives a direct influence of the pig epizootic. Therefore, a mathematical model for JE epizootic in pigs was developed. First of all, several assumptions are set up as follows: (1) There is no emigration nor immigration in the population of pigs and vector mosquitoes. (2) Age distribution of pigs is stable, births and slaughtereds being balanced. (3) Pigs when slaughtered are at a difinite age. (4) Probability of being bitten by a vector mosquito is the same in any pigs. (5) Vector mosquitoes feed only on pigs. (6) Probability of vector mosquito survival through one day is constant irrespective of age. (7) Vector mosquitoes which fed on a viremic pig are all infected, and all infected mosquioes which survived a certain duration become transmissible. (8) Immediately after infected mosquitoes became transmissible they feed on pigs. (9) All susceptible pigs which were fed on by the transmissible mosquito become viremic after a certain duration. (10) All infected pigs become immune after a viremic state. To develop the model the following conditions are given: (1) Population size or pigs is 1,000. (2) Pigs when slaughtered are at the age of 8 months. (3) Duration with maternal antibody in a pig is 2 months from the birth. (4) The number of pigs without maternal antibody is [numerical formula] (5) One cycle of infection is 0.5 month. (6) The number of pigs born or slaughtered in each infection cycle is [numerical formula] (7) Probability of mosquito survival through the incubation period of 10 days for the infected mosquito to become transmissible is 0.05. Under these assumptions and conditions, the mathematical model for JE epizootic of pigs is given as follows. At time t in terms of infection cycle, letting the number of vector mosquitoes biting one pig in one night be M(t), the number of infected pigs C(t), and the number of infected and immune pigs just after infection　A_1(t), then the number of susceptible pigs just after infection, S_1 (t), is 750-A_1(t). At time t+1, the number of transmissible mosquitoes, T(t+1), is shown as T(t+1)=M(t)×C(t)×0.05. The probability for one mosquito to be transmissible, R(t+1), is R(t+1)=T(t+1)/M(t+1)×1,000. Therefore, the probability for one pig to be infected by the bites of M(t+1) mosquitoes, I(t+1), is expressed by I(t+1)=1-Q(t+1)M(t+1) where Q(t+1) is the probability for one mosquito not to be transmissible, being equal to 1-R(t+1). Thus, the number of infected pigs, C(t+1), at time t+1 is given by C(t+1)=S_2(t+1)×(1-Q(t+1)M(t+1)) where S_2(t+1) is the number of susceptible pigs just before infection, obtained by subtraction of the number of immune pigs just before infection, A_2(t+1)=A_1(t)×(1-62.5/750),from 750. If the number of mosquitoes is constant in relation to time, M, then the equation is simplified as C(t+1)=S_2(t+1)×(1-(1-0.00005×C(t)^M). By giving the initial number of infected pigs at t=0, C(0), and the numbers of mosquitoes, M(t), t=0, 1, 2,…………, or M and by applying successively the equation for JE epizootic model, we have the sequence of infected pigs, C(t), t=1, 2,…………, as JE epizootic of pigs. Simulation studies with the model indicated that the number of transmissible mosquitoes, T(t), Which is considered to be proportional to the number of human cases if other conditions are the same, increases even at a higher rate than the number of mosquitoes feeding on pigs, M(t), increases (Table 1). In other words, it can be said that the density of vector mosquitoes and the scale of human epidemic are positively related each other. On the other hand, the increased number of mosquitoes, M(t), influences little the number of infected pigs, C(t), therefore the number of immune pigs (including infected pigs), A_1(t), or A_2(t), and also the rate of transmissible mosquitoes, R(t), unless the mosquito density is very low (Figs. 1, 2; Tables 2.1-2.3). The above indicates that it is rather difficult to predict the scale of the human epidemic by the appearance of pig epizootic. If the number of infected pigs at t=1, C(1),is smaller than the number at t=0, C(0),then the epizootic will not occur. Thus we can get the threshold density of vector mosquitoes for the occurrence of the JE epizootic as the value of M by giving the initial number of infected pigs, C (0), and solving the equation C(1)=S_2(1)×(1-(1-0.00005×C(0))^M) for M,where C(1)=C(0), and S_2(1)=750-C(0)×(1-62.5/750) (Table 3). Similarly, the threshold density of artificially immunized pigs, x, can be obtained by giving the number of mosquitoes, M, and initial number of infected pigs, C(0), and solving the same equation for x, where C(1)=C(0), and S_2(1)=750-(C(0)+x)×(1-62.5/750) (Table 4). The epidemic of JE is governed by many complicated factors, therefore it is thought that the simulation studies with the model can be a useful measures in understanding the epidemiology and also in attempting the control of the disease. If the results obtained with the model are much different from the observations in nature, then probably some of the assumptions set up and/or the conditions given in developing the model are not appropriate, and by modifying them we shall become to understand the natural events more rightly.
長崎大学熱帯医学研究所の論文

日本脳炎流行の理論モデル

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長崎大学熱帯医学研究所 | 論文

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