Stability and Bifurcation Analysis on Communicable and Non-communicable Diseases

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This dissertation provides mathematical guidance in developing effective control strategies for both communicable and non-communicable diseases through stability and bifurcation analysis. Our research employs mathematical tools such as Lyapunov functions, geometric approaches, and fluctuation lemma to perform the stability analysis. We also conduct bifurcation analysis, with a particular focus on backward and forward bifurcations. In our investigation, we formulated and analyzed two specific communicable and non-communicable disease models: one for low-level persistent transmission of Zika virus (ZIKV) and another for socially contagious overweight and obesity. Our work offers insight into the behavior of these diseases, which can aid public health policymakers in designing more effective control strategies. In our initial investigation, we focused on the low-level transmission of ZIKV, as observed in Thailand, which is a communicable disease that can be spread by mosquitoes and, to a lesser extent, through sexual contact. To model the sexual transmission route, we employed a Poisson point process that yielded an increasing and saturating contact rate. This nonlinear contact rate led to backward and Hopf bifurcations, which produced oscillations that demonstrated low-level persistent ZIKV transmission with sharp uniform outbreaks and extended silent periods. When we considered seasonal and stochastic variations in mosquito transmission, we observed varying amplitudes and outbreaks. We proposed criteria for disease elimination and conducted a sensitivity analysis, which revealed that the mosquito death rate had the most significant impact on the basic reproduction number, while human recovery rate was the most effective in reducing human host prevalence. In our second investigation, we examined the overweight and obesity epidemic, which, although not communicable, can be influenced by social factors. To understand the interpersonal dynamics involved in effective intervention programs, we analyzed the transmission of excess weight gain as if it were an infectious disease model. Through bifurcation analysis, we found evidence of a backward bifurcation in cases where the relative hazard of weight regain is high. This resulted in a bi-stable region where both stable obesity-free and obesity-endemic equilibria coexist. Using stability analysis, we developed strategies for obesity elimination and maintenance of a plateau. Our findings indicate that weight loss programs can be helpful in maintaining the plateau, but weight loss maintenance programs, such as follow-ups, must be promoted to eliminate the disease entirely.

Zika virus, Poisson point process, obesity, global stability, Hopf bifurcation, backward bifurcation