An age-structured SIR epidemic model with fixed incubation period of infection

Abstract

This paper is devoted to the study of an age-structured SIR epidemic system on the basis of the model of polycyclic population dynamics of susceptible, infected and recovered individuals. This model was considered as a nonlinear competitive system of three initial–boundary value problems for the nonlinear transport equations with non-local integral boundary conditions and discrete time delay (fixed incubation period of infection). It was obtained the explicit recurrent formulae for computing the traveling wave solution of such system provided the model parameters (coefficients of equations and initial values) satisfy the restrictions that guarantee the existence and uniqueness of continuous solution. Using some insignificant simplifications the age-structured SIR model was reduced to the nonlinear autonomous system of delay ODE. It was studied the dimensionless indicators and conditions of existence of trivial disease-free equilibrium, two non-trivial endemic equilibriums for the incurable and curable infection-induced diseases, respectively. By carrying out an analysis of the local asymptotical stability of the solution of such system we obtain the restrictions for the time delay that guarantee the stability of solutions in the neighborhood of equilibriums. The numerous simulations of dynamics of SIR population illustrate the results of theoretical analysis.