SIRS epidemiological model with ratio‐dependent incidence: Influence of preventive vaccination and treatment control strategies on disease dynamics

Abstract

In this paper, we study an SIRS epidemic model with ratio‐dependent incidence rate function describing the mechanisms of infectious disease transmission. Impacts of vaccination and treatment on the transmission dynamics of the disease have been explored. The treatment rate is constant when the number of infected individuals is greater than the maximal capacity of treatment and proportional to the number of infected individuals when the number of infected individuals is less than the maximal capacity of treatment. Analysis shows that (1) the sufficiently large value of the preventive vaccination rate can control the spread of disease, and (2) a threshold level of the psychological (or inhibitory) effects in the incidence rate function is enough to decrease the infective population. It is also obtained that model undergoes transcritical and saddle‐node bifurcations with respect to disease contact rate. Moreover, in the presence of treatment strategy, the model has multiple endemic equilibria and undergoes a backward bifurcation. The maximal capacity of treatment plays important roles on the disease dynamics of the model. The number of infected individuals decreases with respect to the maximal capacity of treatment, and the disease completely dies out from the system for the large capacity of the treatment when constant treatment strategy is applied. Further, it is also found that the spread of disease can be suppressed by increasing treatment rate. From sensitivity analysis, we have observed that the transmission and treatment rates are most sensitive parameters. The effects of different parameters on the disease dynamics have also been investigated via numerical simulation.