Asymptotic behavior of the final number of susceptible individuals in generalized epidemic processes
Many epidemic models describe the spread of infectious diseases by the so-called Susceptible-Infective-Removed scheme. In the literature, the standard model used to describe an epidemic is the so-called generalized epidemic process defined as follows: Consider a population of \(n\) susceptible individuals and let an epidemic start by introducing \(m\) infected individuals in the system. We suppose that each pair of individuals contact at the points of a homogeneous Poisson process of rate \(\beta\) and that the contacts are independent in different pairs. Every susceptible, if contacted with an infective, is assumed immediately infectious. Each infective \(i\), initial or subsequent, remains infectious for a time of random duration \(D_i\). The random times \(D_i\) are assumed independent and distributed identically with some random variable \(D\). After the period \(D_i\) the infective \(i\) dies or immunizes and becomes inessential for the further infection process; it is regarded as removed from the population. It is clear that with probability 1 the infection process of the epidemic terminates at some finite time when there are no infectives in the population. In the present article, we are interested in the final number of the susceptibles that survive the disease. Henceforth, we suppose that the initial number of susceptibles increases. In other words, we consider a sequence of generalized epidemic models with parameters \((n,m_n,\beta_n,D)\), \(n\to \infty \). Observe that the random variable \(D\) is assumed independent of \(n\), which is rather natural in practice. Let \(S_n (\infty)\), with law \({\mathcal L} [S_n (\infty)]\), be the final number of susceptibles. In our recent article Ann. Probab. 23, No. 3, 1139-1162 (1995; Zbl 0833.60023), we studied the Poisson approximation of the distribution \({\mathcal L} [S_n (\infty)]\) as \(n\to \infty\). Our purpose here is to supplement that study by examining the weak convergence of \(S_n (\infty)\) to an arbitrary distribution (nondegenerate at zero): \[ {\mathcal L} \bigl[ S_n (\infty) \bigr] \Rightarrow P_\xi \text{ as }n\to \infty, \tag{1} \] where \(P_\xi\) is the distribution of some nonnegative random variable \(\xi\) such that \(P(\xi> 0)>0\). First, we derive necessary and sufficient conditions on \({\mathcal L} (D)\) under which to every sequence \(\{m_n\}\) there is a sequence \(\{\beta_n\}\) such that (1) holds. Next, we consider a similar problem in the case when \(\{m_n\}\) is some fixed sequence. In particular, we demonstrate that if \(\{m_n\}\) is constant or varies in a certain way then the limit distribution is either a Poisson law or a mixed Poisson law with a random parameter having an asymmetric Cauchy stable law. To prove these results, we apply the convergence theorem for \(S_n(\infty)\) that was established in our Preprint No. 11, Université Libre de Bruxelles, IS-P (1995), for a wider class of epidemic models, namely, for the so-called collective epidemic models.
scientific article; zbMATH DE number 1023927