Mathematical analysis of a two-strain disease model with amplification

We investigate a two-strain disease model with amplification to simulate the prevalence of drug-susceptible (s) and drug-resistant (m) disease strains. We model the emergence of drug resistance as a consequence of inadequate treatment, i.e. amplification. We perform a dynamical analysis of the resulting system and find that the model contains three equilibrium points: a disease-free equilibrium; a mono-existent disease-endemic equilibrium with respect to the drug-resistant strain; and a co-existent disease-endemic equilibrium where both the drug-susceptible and drug-resistant strains persist. We found two basic reproduction numbers: one associated with the drug-susceptible strain $R_{0s}$; the other with the drug-resistant strain $R_{0m}$,and showed that at least one of the strains can spread in a population if ($R_{0s}$,$R_{0m}$) > 1 (epidemic).Furthermore, we also showed that if $R_{0m}$ > max($R_{0s}$,1), the drug-susceptible strain dies out but the drug-resistant strain persists in the population; however if $R_{0s}$ > max($R_{0m}$,1), then both the drug-susceptible and drug-resistant strains persist in the population. We conducted a local stability analysis of the system equilibrium points using the Routh-Hurwitz conditions and a global stability analysis using appropriate Lyapunov functions. Sensitivity analysis was used to identify the most important model parameters through the partial rank correlation coefficient (PRCC) method. We found that the contact rate of both strains had the largest influence on prevalence. We also investigated the impact of amplification and treatment rates of both strains on the equilibrium prevalence of infection; results suggest that poor quality treatment make coexistence more likely but increase the relative abundance of resistant infections.