In this paper, a mathematical model for assessing the impact of COVID-19 on tuberculosis disease is proposed and analysed. There are pieces of evidence that patients with Tuberculosis (TB) have more chances of developing the SARS-CoV-2 infection. The mathematical model is qualitatively and quantitatively analysed by using the theory of stability analysis. The dynamic system shows endemic equilibrium point which is stable when R 0 < 1 and unstable when R 0 > 1 . The global stability of the endemic point is analysed by constructing the Lyapunov function. The dynamic stability also exhibits bifurcation behaviour. The optimal control theory is used to find an optimal solution to the problem in the mathematical model. The sensitivity analysis is performed to clarify the effective parameters which affect the reproduction number the most. Numerical simulation is carried out to assess the effect of various biological parameters in the dynamic of both tuberculosis and COVID-19 classes. Our simulation results show that the COVID-19 and TB infections can be mitigated by controlling the transmission rate γ .

The present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number $\mathcal{R}_{0}$R 0 is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kutta of fourth order for simulating the proposed model with fractional-order derivative are presented. To validate the mathematical results, numerical simulations, sensitivity analysis, convergence analysis, and other important studies are given. The paper is finished with some conclusions and discussion.

In this paper, a Covid-19 dynamical transmission model of a coupled non-linear fractional differential equation in the Atangana-Baleanu Caputo sense is proposed. The basic dynamical transmission features of the proposed system are briefly discussed. The qualitative as well as quantitative results on the existence and uniqueness of the solutions are evaluated through the fixed point theorem. The Ulam-Hyers stability analysis of the suggested system is established. The two-step Adams-Bashforth-Moulton (ABM) numerical method is employed to find its numerical solution. The numerical simulation is performed to accesses the impact of various biological parameters on the dynamics of Covid-19 disease.