We consider an SIR model with nonlinear contact and vaccination rates influenced by media information. In the theoretical analysis, we include the well-posedness of the system, computation of the basic reproduction number, and local stability of disease-free and disease-persistent fixed points. Further, we construct a nonstandard finite difference scheme (NSFD) for numerically establishing the results. The NSFD system is shown to maintain positivity for all positive initial values and preserves the conservation law. We also find that the NSFD method shares the same fixed points and stability criteria as the corresponding continuous model. We also compare the method with existing finite difference schemes to demonstrate its superior performance.

This study discretizes a continuous-time smoking model using the standard forward Euler and non-standard finite difference (NSFD) schemes. The dynamical properties of both systems are compared to the original continuous model and it is found that the NSFD system remains positive for all positive initial values and has the same fixed points and local stability properties as the continuous model. In contrast, the dynamics of the Euler system is contingent on the step size and may allow numerical instabilities. In the end, numerical investigations are carried out to support the theoretical findings and highlight the benefits of the NSFD scheme compared to the conventional Euler method.

Many people are unaware of the symptoms of tuberculosis (TB) or how the disease is spread. In some areas, availability of healthcare is limited and people may not have easy access to diagnostic tests or treatment for TB within a timely manner. These factors can lead to delays in seeking medical attention and receiving a diagnosis. In this paper, a TB model incorporating a delay in the detection and treatment of the uninformed actively infected population is presented and analyzed. The delay under consideration here is the total delay, which takes into account both health system delay and patient delay. Local stability analysis of the model is performed. Additionally, the critical value for the delay that determines the local stability of the disease-free equilibrium (DFE) is derived. The presence of a Hopf-bifurcation, along with its direction as supercritical or subcritical, is discussed both analytically and numerically. According to the analysis, when the delay exceeds a critical threshold, a Hopf-bifurcation occurs, resulting in instability of the unique endemic equilibrium point (EEP). A suitable Lyapunov function in the presence of delay is constructed in order to examine the global stability of the unique EEP. Further, the occurrence of stable closed periodic orbits, also known as limit cycles, is observed. In addition to this, the presence of Hopf-bifurcation for the non-delayed scenario is also confirmed. These findings are validated numerically under various parameter settings. The analysis highlights that both the presence and absence of delay contribute to the diverse and complex nature of the model.

The information shared by users on social media, including symptoms and the current status of their illness potentially aids in identifying and forecasting disease outbreaks. In this work, we introduce and analyze a deterministic compartmental model for tuberculosis (TB) which considers exogenous reinfection, relapse or treatment failure, and also incorporates the impact of media. The threshold quantity R0T and the equilibria for the model are obtained. Stability analysis of the disease-free equilibrium (DFE) is performed. The direction of supercritical (forward) and subcritical (backward) bifurcations is also obtained at R0T=1. The existence of multiple endemic equilibrium point (EEP) in the absence of exogenous reinfection (p=0) is examined. The existence of Hopf-bifurcation is discussed for the model analytically. Numerical simulations are conducted to support and broaden our analysis. Our numerical simulation reveals the presence of a subcritical bifurcation even in the absence of exogenous reinfection. The occurrence and disappearance of periodic oscillations through Hopf-bifurcation are observed under various parameter settings. More specifically, we observe Hopf-bifurcation by selecting the following parameters as bifurcation parameters: the disease transmission coefficient (β), the coefficient that decides how effectively TB information can influence the transmission rate (a), the rate at which the individuals leave the treated class (ρ), and the modification parameter for relative infectiousness of treated people (δ). Furthermore, we observe a stability switch phenomenon of the unique EEP with changes in a, giving rise to an interesting structure known as an endemic bubble. The presence of a limit cycle (closed periodic curve) around an unstable EEP is noticed. Thus, due to the significant nonlinearity of the model, it revels complex and diverse dynamics.

In this current study, we formulate a new Zika virus model in light of information-induced behavioural change. Firstly, we investigate the model without control and basic mathematical results are obtained. Non-negativity, boundedness, basic reproduction number, sensitivity analysis, and the stability of different equilibrium quantities are discussed. It is observed that disease-free equilibrium (DFE) point is globally asymptotically stable (GAS) if the aware individuals do not participate in the disease progression. Secondly, a nonstandard finite difference (NSFD) scheme is developed for the proposed model. It is observed that, unlike traditional numerical methods such as the Euler and fourth-order Runge–Kutta (RK4) methods, which can fail with larger step sizes, our proposed NSFD method consistently preserves the equilibrium quantities and their stability characteristics. Thereafter, we introduce time dependent controls into the model in order to investigate the optimal effects of information-induced behavioural change, use of condoms, use of insecticide-treated nets, treatment, and indoor residual spraying. Using Pontryagin’s Maximum Principle (PMP), the proposed control system is examined and the optimal control profiles for the implemented controls are acquired. Afterward, different combinations of control strategies are applied and compared numerically. It is found that though each combination has its usefulness, the combined effect of all control measures is observed to be highly effective in reducing the spread of the disease and the overall cost.

Tuberculosis poses a worldwide threat, particularly to low and middle-income nations. The timely screening, identification, and management of TB become problematic in these regions due to resource constraints, inadequate education and awareness, stigma and discrimination, weak healthcare infrastructure, and concurrent health conditions. This delay may lead to postponed diagnosis and treatment. This work presents and examines a mathematical model for tuberculosis that incorporates exogenous reinfection, slow-fast progression, and delay in detection and treatment. More precisely, we examine a scenario where a certain proportion of people with active tuberculosis receive prompt diagnosis and treatment, while the rest experience a delay in detection and appropriate treatment. First, we establish the non-negativity and boundedness of the solutions, and discuss the criteria for the existence of different equilibria. Then, the local stability of the equilibria is analyzed. The threshold value of the time delay which decides the local stability of the disease-free equilibrium (DFE) is obtained. We observe that the model exhibits a Hopf-bifurcation when the time delay crosses a certain threshold value, τ 0 * . This reveals that the EEP is stable for τ < τ 0 * . However, as τ > τ 0 * , the EEP loses its stability via Hopf-bifurcation, leading to sustained periodic oscillations. Further, by applying center manifold theory, we analytically determine the direction and stability of the Hopf-bifurcation and validate the results numerically. It is observed that the Hopf-bifurcation is supercritical and the bifurcating periodic oscillations are stable. Global stability of the endemic equilibrium point (EEP) under certain conditions is established by choosing a suitable Lyapunov function. Additionally, the system also exhibits a Hopf-bifurcation in the absence of delay and numerical simulations are performed to determine the critical parameter values at which the bifurcation occurs. The impact of treatment delay is significant, as it not only contributes to the emergence of complex dynamical behavior but also significantly influences the stability properties of the system. This delay can lead to critical transitions, such as changes in equilibrium stability and the onset of oscillatory dynamics, depending on the interaction of system parameters.

To manage risks and minimize the transmission of contagious diseases, individuals may reduce their contact with each other and take other precautions as much as possible in their daily lives and workplaces. As a result, the transmission of the infection reduces due to the behavioral changes. These behavioral changes are incorporated into models by introducing saturation in disease incidence. In this article, we propose and analyze a tuberculosis model that incorporates saturated exogenous reinfection and treatment. The stability analysis of the model’s steady states is rigorously examined. We observe that the disease-free equilibrium point and the endemic equilibrium point (EEP) are globally asymptotically stable if the basic reproduction number ( R 0 ) is less than 1 and greater than 1, respectively, only when exogenous reinfection is not present ( p = 0 ) and when treatment is available for all ( ω = 0 ). However, even when R 0 is less than 1, tuberculosis may persist at a specific level in the presence of exogenous reinfection and treatment saturation, leading to a backward bifurcation in the system. The existence and direction of Hopf-bifurcations are also discussed. Furthermore, we numerically validate our analytical results using different parameter sets. In the numerical examples, we study Hopf-bifurcations for parameters such as β , p , α , and ω . In one example, we observe that increasing β leads to the loss of stability of the unique EEP through a forward Hopf-bifurcation. If β is further increased, the unique EEP restores its stability, and the bifurcation diagram exhibits an interesting structure known as an endemic bubble. The existence of an endemic bubble for the saturation constant ω is also observed.