In this paper, we develop a mathematical model for the spread of COVID-19 outbreak, taking into account vaccination in susceptible and recovered populations. The model divides the population into eight classes, including susceptible, vaccinated in S class, exposed, infected asymptomatic, infected symptomatic, hospitalized, recovery, and vaccinated in recovered class. By applying a vaccine-distribution scenario, we investigate the impact of vaccines on the COVID-19 outbreak. After analyzing the equilibrium point and computing the basic reproduction number, we perform numerical simulation and sensitivity analysis to identify the most influential parameters and evaluate the impact of vaccine distribution on policies to control the spread of COVID-19. Our findings suggest that vaccine distribution can effectively suppress the spread of COVID-19, and increasing the $v$ parameter (vaccine distribution) and ${\alpha }_1$ parameter (acceleration of detection of undetected infected individuals who have recovered) can help control the outbreak. Moreover, decreasing the contact between vulnerable and infected individuals can lower the ${\beta }_1$ parameter, leading to $R_0<1$, which indicates a disease-free population. This study contributes to understanding the impact of vaccination on the spread of COVID-19 and provides insights for policymakers in developing control strategies.

This study presented a glioma growth model that accounts for drug-sensitive and drug-resistant cells in response to chemotherapy and anti-angiogenic therapy. Chemotherapy induces mutations in drug-sensitive cells, leading to the emergence of drug-resistant cells and highlighting the benefits of combined therapy. Anti-angiogenic therapy can mitigate mutations by inducing angiogenic dormancy. We have identified two reproduction numbers associated with the non-cell and disease-free states. Numerical sensitivity analysis has highlighted influential parameters that control glioma growth dynamics, emphasizing the interactions between drug-sensitive and drug-resistant cells. To reduce glioma endemicity among sensitive cases, it was recommended to decrease chemotherapy expenditure, increase angiogenic dormancy, and adjust chemotherapy infusion rates. In addition, to combat resistance to glioma endemicity, enhancing angiogenic dormancy is crucial.