DYNAMICAL MODELS FOR THE TRANSMISSION OF INFECTIOUS DISEASES AND RELEVANT CO-INFECTIONS

Abstract

In this thesis, some nonlinear mathematical models are developed and analyzed to study the transmission and control of typhoid as well as HIV-Typhoid co-infection. In general, the antimicrobialsensitive and antimicrobial-resistant strains of S Typhi bacteria are considered. After a brief introduction to the disease dynamics and literature survey, the model developed in the second chapter considers the treatment-induced acquired resistance and the dynamics of environmental bacteria. Rigorous qualitative analysis of the model shows that although the resistant strain has lower transmissibility than the sensitive strain, the resistant strain alone can persist in a community due to re-infection. Further analysis reveals that the treatment-induced acquired resistance enhances the load of resistant strain in a co-existence equilibrium state. The sensitivity analysis shows that the direct transmission rate of each strain and the consumption rate of bacteria in environmental transmission strongly impact the basic reproduction number (R0) of the model. The study suggests that access to safe drinking water combined with improved sanitation and hygiene practices can reduce the emergence and global spread of antimicrobial-resistant strain(s) of S Typhi. In the third chapter, a nonlinear deterministic model is proposed for assessing the influence of typhoid conjugate vaccine (TCV) on the emergence and global spread of antimicrobial-resistant typhoid infection. The qualitative analysis of the model with constant control variable shows that the model exhibits vaccine-induced backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium when the effective reproduction number (Re) is less than unity. Furthermore, numerical simulations have shown that the existence and stability of the endemic equilibria of the model depend on the sign of the difference between the effective reproduction numbers associated with each strain, Rs e and Rr e. When treated individuals develop treatment-induced acquired resistance, the resistant strain drives out the sensitive strain to extinction if Rr e > Rs e whereas the two strains coexist if Rr e ≤ Rs e. When all treated individuals are being cured, the co-existence equilibrium exists whenever Rs e = Rr e; otherwise, the strain with higher reproduction number drives out the other strain to extinction, showing the competitive exclusion of the two competing strains. Finally, the optimal control problem is studied, seeking to minimize the number of infectious individuals with antimicrobial-resistant strains and the total costs associated with the vaccination process. The results show that optimal vaccination can reduce the prevalence of antimicrobial-resistant typhoid infections. However, the number of cases did not approach zero, especially a significant number of antimicrobial-resistant chronic carriers still survive. Also, TCV vaccination has no significant effect on lowering the proportion of antimicrobial-resistant cases. This indicates that TCV vaccination alone is unlikely to control antimicrobial-resistant typhoid infections. The study suggests that more preventive measures v need to be implemented to control antimicrobial-resistant typhoid infection effectively. The fourth chapter devotes to co-dynamics of HIV and typhoid infections to suggest possible strategies for their effective control. The model has considered typhoid vaccine, vertical transmission of HIV, environmental transmission for typhoid, and treatment for both the diseases. The global stability of the disease-free equilibrium of the co-infection model is obtained when the co-infection reproduction number, R0, is less than unity. Further, the co-infection endemic state is locally asymptotically stable if it exists. The simulation results reveal that R0 can be reduced below unity by simultaneous preventive measures. The study suggests that both diseases need to be managed simultaneously through possible preventive measures to control HIV-Typhoid coinfection effectively. In chapter five, the model of chapter 2 is extended to a corresponding optimal control problem to predict the optimum level of measures required to control the two-strain typhoid infection. The treatment for symptomatic individuals in each strain and proper hygiene/sanitation practices are considered as control interventions. The simulation results show that comprehensive use of the control interventions highly influenced the number of symptomatic individuals and environmental bacteria in both the strain types. However, there are still a significant number of asymptomatic carriers in both types of the strain. This result shows that additional preventive measures need to be implemented to further reduce the population of asymptomatic carriers. Further, the impact of each control strategy on reducing the number of infected individuals and bacteria in both the strain types is investigated using efficiency analysis. The result shows that the combination of the three control interventions is the most effective strategy. Lastly, a nonlinear deterministic model for typhoid dynamics is developed to investigate the impact of control interventions such as vaccination, environmental sanitation, and limited treatment on the prevalence of typhoid. The model with constant control interventions is qualitatively analyzed. The results show that the model exhibits a backward bifurcation phenomenon. An optimal control problem is also studied to investigate the optimum intervention strategies by assessing their effects on typhoid prevalence and economic load. Numerical simulations are performed, and various combinations of intervention strategies are compared to assess their effects on typhoid prevalence and economic load. The results show that, in the absence of treatment, the combination of vaccination and environmental sanitation controls plays an important role in reducing the typhoid burden and economic load. Further, the comprehensive use of the three control interventions is found to be more effective and highly economical during typhoid outbreaks.