On Trends of AIDS and an Estimate for the Number of HIV
Infected in Japan

Hisashi INABA

In this paper we first observe the growth trends of number of HIV/AIDS cases in Japan based on the AIDS surveillance data. We can observe that the cumulated AIDS incidence in Japan has been growing exponentially since the AIDS surveillance system was established in May 1989. Let C(t) be the cumulated AIDS incidence by time t(years since May 1989). Then we have

　　　　　(1)

where R² =0.998. Our observations strongly support that the number of newly infected is also growing exponentially. That is, AIDS in Japan is in the exponential phase as European countries in the late 80th.
Next we develop a method to estimate the number of HIV infected in the exponential phase. Let () be the force of developing AIDS at disease-age (duration since infection) , let () = exp( - ) be the survival function and let be the proportion of newly infected who eventually develop AIDS. Then the incubation distribution for infecteds is given by F() : = (1-()). Let I(t) be the number of HIV infecteds at time t and let B(t) be the number of newly infecteds per unit yime at time t. Then we have

　　　　　(2)

　　　　　(3)

If we assume that the epidemic is in the exponential phase, that is, B(t)=B(0)e^rt, and the infected population is closed, it is easily shown that

　　　　　(4)

where denotes the crude rate of developing AIDS given by

　　　　　(5)

In order to estimate the number of HIV infected in Japan, we assumed that () is given by some Weibul distributions with different average incubation period (8.5 years [model I ] , 9 years
[model II] and 10 years [model III]) or by a log-logistic distribution (model IV, its average incubation period is 11 years) . Using the above estimation formula (4) and assumed survival functions, we conclude that the number of infected individuals is about from 10 times (model I) to 17 times (model IV) as much as the size of cumulated AIDS incidence in Japan. This means that the number of HIV infected (excluding about 1800 cases infected by contaminated blood products) at the end of June 1994 in Japan is estimated to be about from 3200 to 5200. However even if the report of AIDS incidence is complete, this estimate would be an underestimate, since it does not take into account infected short-term stayers who emigrate to home countries before developing AIDS in Japan and the average incubation period could be longer.
Thirdly if we assume that infected individuals are diagnosed as HIV positives with a constant rate, it is proved that the disease-age structure of reported HIV infected is ageing in compare with that of total HIV infected. Then the crude rate of developing AIDS for the reported HIV infected individuals would be very different from the crude rate of developing AIDS in all of infected. As a result, it follows that the proportion of reports of HIV infection to the unknown total HIV infection cannot be estimated as the proportion of the reported AIDS cases who develop AIDS after diagnosed as HIV positives to the total AIDS cases.
Finally we formulate the Lotka's integral equation for HIV infected and calculate the basic reproduction ratio R₀ for the HIV infection. We show that if the basic reproduction ratio of the reported HIV infected becomes zero, the subcritical condition R₀< I could be attained by making the rate of diagnosis larger than the Malthusian parameter of the HIV infecteds.

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