We proposed a mathematical model of the coronavirus disease 2019 to investigate the transmission and control mechanism of the disease in Nigeria. Using stability theory of differential equations, the qualitative behavior of model is studied. The pandemic indicator represented by basic reproductive number R 0 is obtained from the largest eigenvalue of the next-generation matrix. Local as well as global asymptotic stability conditions for the disease-free equilibrium is obtained which determines the conditions to stabilize the exponential spread of the disease. Further, we examined this model by using Atangana-Baleanu fractional derivative operator and existence criteria of solution for the operator is established. We consider the data of reported infection cases from April 1, 2020, till April 30, 2020, and parameterized the model. We have used one of the reliable and efficient method known as iterative Laplace transform to obtain numerical simulations. The impacts of various biological parameters on transmission dynamics of COVID-19 is examined. These results are based on different values of the fractional parameter and serve as a control parameter to identify the significant strategies for the control of the disease. In the end, the obtained results are demonstrated graphically to justify our theoretical findings.

An SEIR epidemic model is investigated to ascertain the impact of exposed individuals at latent period (individuals who are infected but not yet infectious) on the transmission dynamics of measles. Mathematical analysis is carried out that completely determine the dynamics of the model. The impact of exposed individuals at latent period are discussed through the stability analysis and numerical simulation.

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Summary A new mathematical model of tuberculosis (TB) featuring exogenous re‐infection and incomplete treatment is presented and analyzed. The model divides total population into susceptible, latently infected, actively infected (uninformed and enlightened), and treatment classes. The model with or without incomplete treatment exhibits backward bifurcation phenomenon, which is caused by the presence of exogenous re‐infection. However, further investigation reveals that the absence of incomplete treatment has the potential to reduce the backward bifurcation range significantly. The global dynamics of the TB model without exogenous re‐infection is completely determined by the basic reproduction number under certain modifications on parameters. Furthermore, the model is extended to include three time‐dependent control functions, such as public awareness campaign, treatment effort, and preventive control against incomplete treatment. The existence of the optimal control for the nonautonomous model is proven and the three controls are characterized using Pontryagin's maximum principle. Numerical simulations are performed to show the significance of singular implementation of each of the controls and combination of the three controls in minimizing the TB burden in the population.

Protecting children from vaccines-preventable diseases such as measles, is among primary goals of health administrators worldwide. Since vaccination turned out to be the most effective strategy against childhood disease, developing a framework that would predict an optimal vaccine coverage level needed to control the spread of these disease is crucial. In this paper, we used a compartmental mathematical model to study the transmission dynamics of measles. The effect of vaccination on transmission dynamics of measles were study. The stability of the disease-free equilibrium is established. Numerical simulations are carried out. We discussed in details the implications of our analytic and numerical solutions.