Annals of Communications in Mathematics

Abstract:In this paper, we analyze a vector-host epidemic model with a piecewise-smooth treatment rate. The use of piecewise-smooth treatment depicts the limited medical resource situation in the community. The treatment increases linearly with infective population until the treatment capacity is reached, after which constant treatment (i.e., maximum treatment) is applied. The analysis indicates that there exists a critical value \( I_{h0}^c = \frac{b_h}{\mu_h} \) for the infective human population level \( I_{h0} \) at which the health care system reaches its capacity. We derive that when \( I_{h0} \geq I_{h0}^c \), the dynamics of the model is completely determined by the basic reproduction number \( R_0 \). When \( I_{h0} < I_{h0}^c \), the model exhibits multiple endemic equilibria.

Abstract:In our study, we propose a vector-host epidemic model with fixed-time pulse interventions guided by population size of susceptible humans and susceptible vectors. This approach aims to reflect the situation where public health interventions, namely, vaccination campaigns, mass treatments, insecticide spraying, or sterile mosquito releases, are generally implemented periodically rather than continuously. A saturated incidence function is used to fit the situation where the number of infective cases is getting larger with the limited medical facilities. The model incorporates a general rule of implementation of control measures such us vaccination, treatment and mosquitoes elimination. We theoretically investigate the dynamic behavior with the appearance of (k + m)T - periodic solutions. Firstly, we derive the existence and stability of the order-1 (k + 1)T - disease-free periodic solution with nonnegative integer k and the order-m (1 + m)T -disease-free periodic solution with m being any positive integer. Furthermore, we study the existence and stability of the disease-free (1 + 2)T -periodic solution. In addition, numerical simulations are performed to support the theoretical results. Our results indicate that there exists a critical monitoring period or vaccination rate such that the intervention strategy can successfully control the disease.