Abstract
Sudden environmental perturbations may affect the positivity of the solution of the susceptible-infected-recovered-susceptible (SIRS) model. Most of the SIRS epidemic models have no analytical solution. Thus, in order to find the appropriate solution, the numerical technique becomes more essential for us to solve the dynamic behavior of epidemics. In this paper, we are concerned with the positivity of the numerical solution of a stochastic SIRS epidemic model. A new numerical method that is the balanced implicit method (BIM) is set, which preserves the positivity under given conditions. The BIM method can maintain positive numerical solution. An illustrative numerical instance is presented for the numerical BIM of the stochastic SIRS model.
Highlights
Mathematical models have been an important tool in analyzing the dynamic behaviors of infectious diseases, such as susceptible-infected-recovered (SIR) models, susceptible-infected-vaccinated-susceptible (SIVS) epidemic models, and SIRS models
SIRS mathematical models attract the attention of many scholars. e positivity is a basic characteristic in plenty of areas, such as economy and ecology areas
The susceptible number, infected number, and recovered number are positive in SIRS models, it means to find the positive numerical solution of SIRS model
Summary
Mathematical models have been an important tool in analyzing the dynamic behaviors of infectious diseases, such as susceptible-infected-recovered (SIR) models, susceptible-infected-vaccinated-susceptible (SIVS) epidemic models, and SIRS models (see, e.g., [1,2,3,4,5,6,7]). By equation (3), we can see that the EM approximation cannot guarantee the positivity of the SIRS model. (i) We constructed a new BIM numerical method for the stochastic SIRS model with local Lipschitz condition, and the numerical scheme will keep positive solution for the stochastic system. E outline of the paper is arrayed as follows: in Section 2, we have presented some assumptions and preparations for studying the numerical method for stochastic SIRS equation (1). We have proof of the BIM (17) numerical solution of SIRS model convergences to the true solution of system (1).
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