In this research, we developed a coinfection model of tuberculosis and COVID-19 with the effect of isolation and treatment. We obtained two equilibria, namely, disease-free equilibrium and endemic equilibrium. Disease-free equilibrium is a state in which no infection of tuberculosis and COVID-19 occurs. Endemic equilibrium is a state in which there occurs not only the infection of tuberculosis and COVID-19 but also the coinfection of tuberculosis and COVID-19. We assumed that the parameters follow the uniform distribution, and then, we took 1,000 samples of each parameter using Latin hypercube sampling (LHS). Next, the samples were sorted by ranking. Finally, we used the partial rank correlation coefficient (PRCC) to find the correlation between the parameters with compartments. We analyzed the PRCC for three compartments, namely, individuals infected with COVID-19, individuals infected with tuberculosis, and individuals coinfected with COVID-19 and tuberculosis. The most sensitive parameters are the recovery rate and the infection rate of each COVID-19 and tuberculosis. We performed the optimal control in the form of prevention for COVID-19 and tuberculosis. The numerical simulation shows that these controls effectively reduce the infected population. We also concluded that the effect of isolation has an immediate impact on reducing the number of COVID-19 infections, while the effect of treatment has an impact that tends to take a longer time.

We develop a new transmission model of COVID-19 in the presence of waning immunity with isolation and vaccination in humans. The spreading of the disease is determined by the basic reproduction number (ℜ0). Based on mathematical analysis, we found that the non-endemic equilibrium is locally asymptotically stable when ℜ0 < 1 whereas the endemic equilibrium is locally asymptotically stable when ℜ0 > 1. A numerical simulation is used to show the population dynamics. In conclusion, isolation, and vaccination (once it is available) are regarded as effective strategies for eliminating viruses, or at least to suppress the spread of the disease.