Mathematical Modeling And Optimal Control Analysis of COVID-19 And TB Co-Dynamics

Abstract

COVID-19 and Tuberculosis (TB) are among the infectious diseases causing major public health problems and socioeconomic impacts. These diseases are spread globally having similar clinical symptoms which makes them difficult to be mitigated. Mathematical models are widely used for understanding the transmission mechanisms and control of infectious diseases. In this dissertation, we developed mathematical models for the transmission dynamics of COVID-19, and COVID-19 and TB co-infection with optimal control to establish mitigation strategies for their spread. An extended SEIR-type compartmental model for the COVID-19 pandemic is de veloped and analyzed both qualitatively and quantitatively. The threshold parameter usually referred to as the basic reproduction number which plays a key role in predicting disease persis tence or extinction is computed. It is shown that the disease-free equilibrium point is locally as well as globally asymptotically stable whenever the basic reproduction number is less than unity. On the other hand, the endemic equilibrium point exists, locally as well as globally asymptoti cally stable for the basic reproduction number greater than unity. The proposed model is fitted to the reported real data of COVID-19 confirmed cases from Ethiopia and its parameter values are estimated. Further, the model is extended to an optimal control problem to optimally manage the spread of COVID-19 disease. Various simulations of optimal control model were performed and it reveals that the spread of COVID-19 can be managed via minimizing the contact rate and increasing the quarantine of exposed individuals in addition to the isolation of symptomat ically infected individuals. We have then proposed a deterministic mathematical model to give insight into the co-infection of COVID-19 and TB. The biological meaningfulness of the model was proved and the stability analysis was performed by obtaining the basic reproduction num ber. Numerical simulations of the proposed co-infection model are carried out and it reveals that reducing the risk of COVID-19 infection by TB-infected individuals could greatly reduce the COVID-19 infection and the co-infection of both diseases in the community. We further formu lated and analyzed a mathematical model incorporating COVID-19 vaccination and exogenous reinfection for TB. The equilibria of the TB-COVID-19 model are locally asymptotically stable, but not globally, due to the occurrence of backward bifurcation. The incorporation of exogenous reinfection causes the occurrence of backward bifurcation for the basic reproduction number R0 < 1 and the exogenous reinfection rate greater than a threshold (η > η ∗ ) value. The study reveals that reducing R0 < 1 may not be sufficient to eliminate the disease from the community. Optimal control strategies were proposed to minimize the disease burden and related cost. The existence of optimal controls and their characterization is established using Pontryagin’s Min imum Principle. Moreover, different numerical simulations of the control-induced model were carried out and it shows that the optimal vaccination strategy significantly minimized the number of COVID-19 infected cases. Further, the combination of COVID-19 prevention and vaccination control strategy helps to mitigate new co-infection cases.