A diffusive age-structured SEIRS epidemic model

Abstract

A model of an epidemic disease is analyzed with the following fea- tures: (1) the susceptible and infective populations are diffusing in a spatial region and (2) infected individuals pass through an incubation period before becoming infectives throughout the spatial region. The model consists of a system of nonlin- ear partial differential equations in which the infective population is structured by age since first infected. It is proved that the infective population converges to zero and the susceptible population converges to a constant positive value throughout the region. The extinction of the infective population is due to the uniformly positive rate of their removal. We analyze a system of partial differential equations modeling the spatial spread of infectious disease through a population structured by age of infection. The population is confined to a bounded region in R n . We divide the population into susceptible, exposed, and infective subclasses, having density functions S(x, t), E(x, t), and I(x, t) respectively, for x ∈ and t ≥ 0. The total populations of the subclasses at time t are obtained by integrating the densities over . We envision the following scenario. The susceptible class consists of individuals capable of becoming infected, the exposed class is made up of individuals who have contracted the disease but who are not yet capable of transmitting it, and the infective class consists of individuals capable of transmitting the disease. We shall assume that at a rate proportional to the product of the susceptible and the infective populations, individuals leave the susceptible class and enter the exposed class. We stipulate that individuals, after remaining in the exposed class for a fixed incubation or latency period of length τ, enter the infective class where they remain for the finite length of time σ. Individuals are removed from both the latent and infective phases of the disease with a constant mortality rate λ and the probability of mortality is independent of the length of time the individual has been in either of these phases. At the conclusion of the infective period, individuals pass back into the susceptible class and do not enter a removed or immune class. Properly speaking, this process may be described as an SEIRS (susceptible, exposed, infective, removed, susceptible) disease model which describes the progression of a potentially fatal disease offering the possibility of recovery but no immunity. However, because the individuals comprising the removed class no longer affect the dynamics of the spread of the disease, we do not need to include the removed class in the modeling process.