We formulate and theoretically analyze a mathematical model of COVID-19 transmission mechanism incorporating vital dynamics of the disease and two key therapeutic measures—vaccination of susceptible individuals and recovery/treatment of infected individuals. Both the disease-free and endemic equilibrium are globally asymptotically stable when the effective reproduction number R 0 v is, respectively, less or greater than unity. The derived critical vaccination threshold is dependent on the vaccine efficacy for disease eradication whenever R 0 v > 1 , even if vaccine coverage is high. Pontryagin’s maximum principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions to optimally mitigate the spread of the disease. The model is fitted with cumulative daily Senegal data, with a basic reproduction number R 0 = 1.31 at the onset of the epidemic. Simulation results suggest that despite the effectiveness of COVID-19 vaccination and treatment to mitigate the spread of COVID-19, when R 0 v > 1 , additional efforts such as nonpharmaceutical public health interventions should continue to be implemented. Using partial rank correlation coefficients and Latin hypercube sampling, sensitivity analysis is carried out to determine the relative importance of model parameters to disease transmission. Results shown graphically could help to inform the process of prioritizing public health intervention measures to be implemented and which model parameter to focus on in order to mitigate the spread of the disease. The effective contact rate b , the vaccine efficacy ε , the vaccination rate v , the fraction of exposed individuals who develop symptoms, and, respectively, the exit rates from the exposed and the asymptomatic classes σ and ϕ are the most impactful parameters.

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We propose and analyze an epidemiological model to evaluate the effectiveness of bed nets as a prophylactic measure in malaria-endemic areas. The main purpose in this work is the modeling of the aggressiveness of anopheles mosquitoes relative to the way humans use to protect themselves against bites of mosquitoes. This model is a system of several differential equations: the number of equations depends on the particular assumptions of the model. We compute the basic reproduction number 0  , and show that if 0 1  ≤ , the disease free equilibrium (DFE) is globally asymptotically stable on the non-negative orthant. If 0 > 1  , the system admits a unique endemic equilibrium (EE) that is globally and asymptotically stable. Numerical simulations are presented corresponding to scenarios typical of malaria-endemic areas, based on data collected in the literature. Finally, we discuss the relative effectiveness of different kinds of bed nets.

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COVID-19 is a respiratory illness caused by an RNA virus prone to mutations. In December 2020, variants with different characteristics that could affect transmissibility and death emerged around the world. To address this new dynamic of the disease, we formulate and analyze a mathematical model of a two-strain COVID-19 transmission dynamics with strain 1 vaccination. The model is theoretically analyzed and sufficient conditions for the stability of its equilibria are derived. In addition to the disease-free and endemic equilibria, the model also has single-strain 1 and strain 2 endemic equilibria. Using the center manifold theory, it is shown that the model does not exhibit the phenomenon of backward bifurcation, and global stability of the model equilibria when the basic reproduction number R 0 is either less or greater than unity as the case maybe are proved using various approaches. Simulations to support the model theoretical results are provided. We calculate the basic reproductive number for both strains R_1 and R_2 independently. Results indicate that - both strains will persist when both R 1 > 1 and R 2 > 1 - Stain 2 could establish itself as the dominant strain if R 1 < 1 and R 2 > 1, or when R 2 is at least two times R 1 . However, with the current knowledge of the epidemiology of the COVID-19 pandemic and the availability of treatment and an effective vaccine against strain 1, eventually, strain 2 will likely be eradicated in the population if the threshold parameter R 2 is controlled to remain below unity.