Narrow Results

In this paper, an optimal control theory was applied to the tuberculosis (TB) model governed by system of nonlinear ordinary differential equations. The aim is to investigate the impact of treatment failure on the TB epidemic. An optimal control strategy is proposed to minimize the disease effect and cost incurred due to treatment failure. The existence and uniqueness of optimal controls are proved. The characterization of optimal paths is analytically derived using Pontryagin’s Minimum Principle. The control-induced model is then fitted using TB infected cases reported from the year 2010–2019 in East Shewa zone Oromia regional state, Ethiopia. Different simulation cases were performed to compare with analytical results. The simulation results show that the combined effect of awareness via various mass media and continuous supervision during the treatment period helps to reduce treatment failure and hence reduced the TB epidemic in the community.

In this paper, the general nonlinear multi-strain Tuberculosis model is controlled using the merits of Jacobi spectral collocation method. In order to have a variety of accurate results to simulate the reality, a fractional order model of multi-strain Tuberculosis with its control is introduced, where the derivatives are adopted from Caputo’s definition. The shifted Jacobi polynomials are used to approximate the optimality system. Subsequently, Newton’s iterative method will be used to solve the resultant nonlinear algebraic equations. A comparative study of the values of the objective functional, between both the generalized Euler method and the proposed technique is presented. We can claim that the proposed technique reveals better results when compared to the generalized Euler method.

This paper formulates a diffusive tuberculosis (TB) model with early and late latent infections, vaccination and treatment that may more properly describe the slow and fast dynamics of TB transmission. We develop a new concise approach to determine the combination coefficients in the Lyapunov function candidate for the model and its time derivative in the case that both are the linear combinations of several Volterra-type functions, which highly simplifies the computations in global dynamical analysis for the nonlinear high-dimensional model. Based on the TB case data reported in China, the parameter values of the model are estimated. We further predict the TB prevalence trend in China. Sensitivity analysis for the control reproduction number and endemic equilibrium is conducted to seek some effective interventions that can significantly reduce initial TB transmission and lower TB prevalence levels in China. In the end, numerical simulations show that the bigger diffusive rates pick up the speeds of convergence to the equilibria of the model.

China is one of the countries in the world carrying a heavy burden of tuberculosis. Due to the unbalanced economic development, the number of people working in other parts of country is huge, and the mobility of personnel has exacerbated the increase in tuberculosis cases. Most patients affected by this are in their middle and young ages. It is having a great impact among the family and society. Therefore, research on how to control this disease is absolutely necessary. The population is divided into two categories such as local population and the immigrant population. A pulmonary tuberculosis dynamic model with population heterogeneity is established. We calculate the basic reproductive number and the controlled reproductive number, and discuss the two types of population under the constraints given by the amount of vaccine and the optimal immunization ratio obtained is $(0.118,0.107)$, which can reduce the effective reproduction number from 5.85 to 0.227. It is understood that immunizing the local population will control the spread of the epidemic to a large extent, and we simulate the final scale of infection after immunization under the optimal immunization ratio. It can take a minimum of at least 10 years to reduce the spread of this disease, but to eliminate it forever, it needs at least a minimum of 100 years.