A two dimensional stochastic differential equation is suggested as a stochastic model for the Kermack-McKendrick epidemics. Its strong (weak) existence and uniqueness and absorption properties are investigated. The examples presented in Section 5 are meant to illustrate possible different asymptotics of a solution to the equation.
This paper proposes a stochastic diffusion model for the spread of a susceptible-infective- removed Kermack–McKendric epidemic (M1) in a population which size is a martingale Nt that solves the Engelbert–Schmidt stochastic differential equation (2). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coeffients depend on the size Nt. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer simulations performed.
This paper proposes a deterministic model for the spread of an epidemic. We extend the classical Kermack-McKendrick model, so that a more general contact rate is chosen and a vaccination added. The model is governed by a differential equation (DE) for the time dynamics of the susceptibles, infectives and removals subpopulation. We present some conditions on the existence and uniqueness of a solution to the nonlinear DE. The existence of limits and uniqueness of maximum of infected individuals are also discussed. In the final part, simulations, numerical results and comparisons of the different vaccination strategies are presented.