Bifurcation and Crossover Processes in the Dynamics of a Marburg Haemorrhagic Fever Model

Abstract

Just like Ebola disease, the Marburg virus disease (MVD) remains a constant threat to people’s live, not only in Africa, but also in the other parts of the globe. Recorded past outbreaks occurred in Europe, America and of course Africa, especially in places where inhabitants come into contact with wild animal products or live along-side wild animals such as apes, monkeys and fruit bats on a daily basis. Therefore, studying the dynamics of the Marburg virus and making some epidemiological assessments remain relevant and important in preventing future outbreaks. In this paper, we analyze a Marburg epidemic model in which the transmission is nonlinear. The well-posedness result is established as well as conditions for boundedness and dissipativity results. Then, we intensively analyze the stability of the Marburg model’s equilibria. The bifurcation dynamics indicate the existence of both transcritical (exchange of stability between the endemic equilibrium (EE) point and the disease free equilibrium (DFE)) and backward bifurcation where the classical epidemiological condition for the MVD to die out $(\mathcal{R}_0<1)$ is not sufficient anymore, but remains necessary. Lastly, some numerical simulations show convergence of the trajectories to both the EE and the DFE. Some adequate preventive measures are provided.