Research Article Condition for Global Stability for a SEIR Model Incorporating Exogenous Reinfection and Primary Infection Mechanisms Isaac Mwangi Wangari Department of Mathematics and Actuarial Science, The Catholic University of Eastern Africa (CUEA), I Langata Main Campus I Bogani East Rd, Off Magadi Rd, P.O. Box 62157-00200 Nairobi, Kenya Correspondence should be addressed to Isaac Mwangi Wangari; mwangiisaac@aims.ac.za Received 14 June 2020; Revised 9 September 2020; Accepted 15 October 2020; Published 18 November 2020 Academic Editor: Markos G. Tsipouras Copyright © 2020 Isaac Mwangi Wangari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A mathematical model incorporating exogenous reinfection and primary progression infection processes is proposed. Global stability is examined using the geometric approach which involves the generalization of Poincare-Bendixson criterion for systems of n-ordinary differential equations. Analytical results show that for a Susceptible-Exposed-Infective-Recovered (SEIR) model incorporating exogenous reinfection and primary progression infection mechanisms, an additional condition is required to fulfill the Bendixson criterion for global stability. That is, the model is globally asymptotically stable whenever a parameter accounting for exogenous reinfection is less than the ratio of background mortality to effective contact rate. Numerical simulations are also presented to support theoretical findings. 1. Introduction Mathematical models, in particular, models tracking dynam- ics of infectious diseases, are of utmost importance due to their application in the assessment of public health policies by national and international agencies. In such models, one of the intriguing aspects that often occurs is when modellers need to know whether the disease will disappear or will per- manently remain in the population. This question is answered mainly through investigating the asymptotic stabil- ity of the disease-free equilibrium (DFE) as well the endemic equilibrium. It is already known that if the DFE is globally asymptotically stable, then the disease eradication is assured regardless of the initial number of infected individuals in the population [1]. An influx of infected cases may trigger an isolated epidemic outbreak, but they may not make the disease endemic in the population [2]. In contradiction to DFE, if the endemic equilibrium is globally asymptotically stable (GAS) and a few infected individuals are initially intro- duced, then, the disease will be permanently present in the population. In the sequel, there are two major methods used in the analysis of global stability of endemic equilibrium: Lyapunov direct method and geometric method. Although the Lyapu- nov direct method is often used in proving global stability of infectious diseases models, it is sometimes difficult to use because it requires an auxiliary function which is hard to con- struct. This is because there are no existing general methods for constructing such Lyapunov functions. Moreover, Lyapunov functions for models with parameter(s) that induce bistability phenomena may not even exist. The second method, sometimes referred to as geometric approach to global stability, is a generalization of the Poincare- Bendixson criterion for systems of n ordinary differential equations. This method was developed by Li and Muldowney [3, 4] in midnineties to address problems encountered with the Lyapunov direct method. The technique is extensively being used to analyze global properties of mathematical models emanating in mathematical epidemiology as well as in other biomathematical contexts. For instance, its applica- tions can be seen in toxicant-population interaction models, Lotka-Volterra models incorporating delay [5, 6], and Hindawi Computational and Mathematical Methods in Medicine Volume 2020, Article ID 9435819, 11 pages https://doi.org/10.1155/2020/9435819