Global Dynamics of a Two-Strain Disease Model with Amplification, Nonlinear Incidence and Treatment

In recent years, antibiotic resistance to the most effective treatments has emerged and spread. This has led to a decline in the efficacy of antibiotics used to treat patients, with drug-resistant strain experiencing much higher failure rates and serious side effects. In this study, we considered two-strains (e.g., drug-susceptible and drug-resistant) susceptible-infected-recovery disease model with amplification, nonlinear incidence and treatment. We assumed amplification develops mainly through the choice of naturally happening mutations in the presence of inappropriate treatment. We performed a rigorous analytical analysis of the model properties and solutions to predict late-time behavior of the disease dynamics and find that the model contains four equilibrium points: disease-free equilibrium, monoexistence endemic equilibrium 1 concerning drug-susceptible strain, monoexistence endemic equilibrium 2 concerning drug-resistant strain and coexistence equilibrium regarding to drug-susceptible as well as drug-resistant strains. Two basic reproduction numbers \({R}_{0\mathrm{s}}\) and \(R_{{0{\text{m}}}}\) are found, and we have presented that if both are less than one \(({\text{i}}.{\text{e}}.\max \left[ {R_{{0{\text{s}}}} ,{ }R_{{0{\text{m}}}} } \right] < 1)\), the disease fade-out, and if both greater than one \(({\text{i}}.{\text{e}}.\max \left[ {R_{{0{\text{s}}}} ,{ }R_{{0{\text{m}}}} } \right] > 1)\) the epidemic situation occurs. Moreover, epidemics occur regarding to any strain when the basic reproduction number remains above the value 1 and disease fade-out with regard to any strain when the basic reproduction number remains below the value 1. In all equilibrium points, the global stability analysis was determined with the help of appropriate Lyapunov functions. In addition, we also found that the drug-resistant strain prevalence increases when the drug-susceptible strain is treated due to the poor-quality treatment (i.e., amplification). We also performed the sensitivity analysis through evaluation of Partial Rank Correlation Coefficients (PRCC) to identify the most important model parameters and found that transmission rate of both strains had the maximum influence on disease outbreak. To support those analytical results, numerical simulations of the model were performed using ODE45 MATLAB routine.