Abstract
A stochastic SIR epidemic dynamic model with distributed-time-delay, for a two-scale dynamic population is derived. The distributed time delay is the varying naturally acquired immunity period of the removal class of individuals who have recovered from the infection, and have acquired natural immunity to the disease. We investigate the stochastic asymptotic stability of the disease free equilibrium of the epidemic dynamic model, and verify the impact on the eradication of the disease.
Highlights
The recent advent of high technology in the area of communication, transportation and basic services, multilateral interactions have afforded efficient global mass flow of human beings, animals, goods, equipments and ideas on the earth’s multi-patches surface
The presented two-scale network delayed epidemic dynamic model with varying immunity period characterizes the dynamics of an SIR epidemic in a population with various scale levels created by the heterogeneities in the population
The disease dynamics is subject to random environmental perturbations at the disease transmission stage of the disease
Summary
The recent advent of high technology in the area of communication, transportation and basic services, multilateral interactions have afforded efficient global mass flow of human beings, animals, goods, equipments and ideas on the earth’s multi-patches surface. Epidemic dynamic processes in populations exhibiting varying time disease latency or immunity delay periods are represented by differential equation models with distributed time delays. A mathematical SIR (susceptible-infective-removal) epidemic dynamic model with distributed time delays representing the varying time temporal immunity period in the immune population class is studied by Blyuss and Kyrychko [19]. The global asymptotic stability of the disease free and endemic equilibria are shown by using Lyapunov functional technique They presented numerical simulation results for a special case SIR epidemic with temporal immunity. The temporary immunity period accounts for the time lag during which newly recovered individuals from the disease with conferred infection acquired or natural immunity lose the immunity and regain the susceptible state They utilized the Lyapunov energy function method to prove the global positive solution process existence, and defined a positively self invariant set.
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