Mathematical models of infectious diseases specific to COVID-19 and qualitative analysis of these models

This thesis investigates the effect of a partially protective vaccine on COVID-19 infection with original and mutant viruses using a compartmental deterministic mathematical model. The developed model consists of subcompartments S (susceptible), V (vaccinated), I_1 (infected with the original virus), I_2 (infected with the mutant virus), and R (recovered). The model considers the effects of both artificial active immunity (vaccinated) and natural active immunity (infected). The study considers the differences in the infectiousness, recovery, and mortality rates of the original and mutant viruses in COVID-19. First, the basic reproduction number was obtained using the next-generation matrix method. The local stability of the disease-free equilibrium point and the endemic equilibrium point of the model was analyzed using the Routh-Hurwitz criterion, and the global stability was analyzed using Lyapunov functions. Using the Castillo-Chavez and Song Bifurcation Theorem, a backward bifurcation was demonstrated, which occurs when the vaccine is insufficiently effective, leading to the simultaneous existence of both disease-free and endemic equilibrium points, even when the basic reproduction number is below 1. Three model parameters were estimated using parameter estimation, and model-sensitive parameters were determined using local sensitivity analysis. It was found that the parameter representing vaccine efficacy was the most sensitive to the basic reproduction number, and that increasing vaccine efficacy would reduce the average number of secondary cases. Three different simulations are presented to illustrate the fundamental mechanisms underlying the model's dynamics and to support the analytical findings, demonstrating a strong relationship between vaccine efficacy and the course of the epidemic. To reduce the average number of secondary cases and the likelihood of infected individuals remaining under the influence of the epidemic for extended periods, producing more effective vaccines and increasing the vaccination rate are necessary. Finally, an optimal control problem was posed to understand the conditions under which the epidemic could be controlled or mitigated. To reduce the number of infected individuals and minimize the cost of control measures, an objective function incorporating public health education, vaccination, treatment, and isolation control strategies was constructed and solved using Pontryagin's maximum principle. Numerical simulations demonstrate that the implementation of control strategies is effective in controlling the transmission dynamics of the COVID-19 pandemic, and that simultaneous implementation of four controls is more effective compared to the implementation of three, two, or a single control.