Mathematical Models for Influenza a Virus and Pneumococcus: Within-Host and Between-Host Infection

Abstract

Infectious diseases have become problematic throughout the world, threatening individuals who come into contact with pathogens responsible for transmitting diseases. Pneumoccocal pneumonia, a secondary bacterial infection follows an influenza A infection, responsible for morbidity and mortality in children, elderly and immuno–comprised groups. The aims of this Thesis are to; develop a mathematical model for within–host co–infection of influenza A virus and pneumococcus, model between–host pneumococcal pneumonia in order to determine the effect of time delays due to latency and seeking medical care, and study the effect of antibiotic resistance awareness and saturated treatment in the control of pneumococcal pneumonia. Analysis of the stability of steady states of influenza A virus and pneumococcal co–infection, pnemococcal pneumonia with time delays and antibiotic resistance awareness is done. The graph theoretic method, combined linear and quadratic Lyapunov functions, Goh–Voltera Lyapunov function are used to get suitable Lyapunov functions for global stability of steady states. The results show that the endemic equilibrium of pneumococcal pneumonia is locally stable without delays and stable if the delays are under conditions. The results suggest that as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability and yields Hopf–bifurcation. The results of influenza A virus and pneumococcal co–infection show that, there exist a biologically important steady state where the two pathogens of unequal strength co–exist and replace each other in the epithelial cell population when the pathogen fitness for each infection exceeds unity. The impact of influenza A virus onto pneumococcus and vice–versa yields a bifurcation state. The results show that, the presence of antibiotic resistance awareness and treatment during the spread of pneumococcal pneumonia drastically reduces the basic reproduction number R0 to less than unity, hence the disease could be eradicated.