Global Analysis of an SEIRS Model for COVID-19 Capturing Saturated Incidence with Treatment Response

Abstract

Sequel to [10], who studied the dynamics of COVID-19 using an SEIRUS model. We consider an SEIRS model capturing saturated incidence with treatment response. In this theoretical model, we assumed that the treatment response is proportional to the number of infected as long as the incidence cases are within the capacity of the healthcare system, after which the value becomes constant, when the number of confirmed cases exceed the carrying capacity of the available medical facilities. Thus, we obtain the reproduction number stating that when , the disease free equilibrium is globally asymptotically stable. Also, we studied the existence of the local and global stability of the disease free and endemic equilibria and found that the kind of treatment response and inhibitory measures deployed in tackling the COVID-19 pandemic determines whether the disease will die out or become endemic.

Backward and Hopf bifurcation analysis of an SEIRS COVID-19 epidemic model with saturated incidence and saturated treatment response

“…We should remark, however, that the models in [15] and [16] did not include saturation effects in the treatment rate. Our results for model (1) show that the limitation of medical resources alone can be a cause for bistability behaviour and backward bifurcation, similarly to what was studied in [5] and [14]. Moreover, we considered a saturation effect in the incidence rate that was not included in the above mentioned literature.…”

“…We can see that model (1) is similar to [17]. Here, we will extend the research done in [1] and [17] by studying the backward and Hopf bifurcation dynamics of model (1).…”

“…It follows that the region Ω is positively invariant with respect to the system (1), which implies that the ω-limit sets of all solutions of the model in R 4 + are contained in Ω for all time and those outside Ω are eventually attracted to that region. Hence, the system (1) is epidemiologically well posed.…”

“…Thus, having establish the threshold quantity R 0 in the system (1), which measures the average number of infections generated by a single infected individual in a completely susceptible population, Theorem 2 states that when R 0 < 1, the introduction of a small number of infected individuals into the community would not lead to large outbreak and the disease dies out in time but when R 0 > 1, the disease persists. However, the study of backward bifurcation shows that the disease may persist even when the reproduction number is less than unity (R 0 < 1), as we will study in Section 6.…”

“…Local stability analysisIn this section, we will study the local stability of the disease-free and endemic equilibria of model(1).…”