Stability Analysis of a Human–Phlebotomus papatasi–Rodent Epidemic Model

Cutaneous leishmaniasis (CL) represents a serious public health problem in Algeria. In the aim to understand the transmission dynamics of CL in the human–Phlebotomus papatasi–rodent cycle, and to improve the preventive strategies set up in Algeria, we developed a deterministic model for the transmission dynamics of the disease. The model includes an incidental host for human which acts only as a sink of infection, a primary reservoir host for rodent which acts as a source and a sink of infection, and a secondary reservoir host for P. papatasi which have a role in transmission by acting as the liaison between incidental host and primary reservoir. The global stability of the equilibria of the proposed model shows that the threshold conditions for disease persistence are completely determined by the reproduction number and do not explicitly include parameters relating to the dynamic transmission in the incidental hosts, which means that the disease becomes endemic if it persists endemically in the primary reservoir hosts, and therefore the control measures should be directed towards reservoir hosts. This is illustrated via numerical simulations of the model using parameters generated from data from M’Sila province in Algeria.