Modeling the Dynamics of the Marburg Virus Disease

Abstract

This work models the infection dynamics of the Marburg Virus Disease using the SEIQR model. It is a five-compartment model. We investigated the non-negativity of solutions, invariant region and boundedness, equilibrium points, basic reproduction number, sensitivity, equilibrium points’ stability, and bifurcation of the model. For all time  and positive initial conditions, we observed that the model has non-negative solutions. The invariant region of the model is established, revealing the population size is bounded. The sensitivity analysis shows that the contact rate, the probability of effective contact and the rate at which exposed individuals become infectious have positive sensitivity indices, while the quarantine rates for the exposed and infectious classes, the death rates, and the recovery rates for the exposed and infectious classes have negative sensitivity indices. We established that the non-negativity of the residual vector , resulting from the difference between the rate of change of infected individuals and the Metzler matrix, stops the existence of several endemic equilibria for . Thus, no critical value  exists, and there is an extension of local stability of the disease-free equilibrium to global stability on the entire domain. The bifurcation analysis reveals a forward bifurcation and the bifurcation plot confirms that for effective management and eradication the disease, the basic reproduction number  must satisfy , where  is a critical value. The mathematical analysis also reveals that the disease-free equilibrium is globally asymptotically stable. The effects of some sensitive parameters on the infectious class are shown via numerical simulations.