In this study, we propose new illustrative and effective modeling to point out the behaviors of the Hepatitis-B virus (Hepatitis-B). Not only do we consider the mathematical modeling, equilibria, stabilities, and existence–uniqueness analysis of the model, but also, we make numerical simulations by using the Adams–Bashforth numerical scheme. However, we apply the parameter estimation method to determine our model parameters and find the curve that best fits the model. Additionally, in this study, the stability analysis of the aforementioned model is considered, and also the sensitivity analysis of R₀ is examined. The results point out that the order of the fractional derivative has an essential effect on the dynamical process of the constructed model for Hepatitis-B.

A nonlinear mathematical model of differential equations with piecewise constant arguments is proposed. This model is analyzed by using the theory of both differential and difference equations to show the spread of HIV in a homogeneous population. Because of the solution of this differential equations being established in a certain subinterval, solutions will be analyzed as a system of difference equations. After that, results will be considered for differential equations as well. The population of the model is divided into three subclasses, which are the HIV negative class, the HIV positive class that do not know they are infected and the HIV positive class that know they are infected. As an application of the model we took the spread of HIV in India into consideration.

The aim of this article is to help predict the course of lung cancer patients. To make this prediction as close to reality as possible, we used data from lung cancer patients receiving treatment at Erciyes University Hospitals in Kayseri, Turkey. First, we developed a mathematical model considering the cells in the microenvironment of lung cancer tumors with the assistance of Caputo fractional derivatives. Subsequently, we identified the equilibrium points of the proposed mathematical model and examined the coexistence equilibrium point. In addition, we demonstrated the existence and uniqueness of the solutions through the fixed-point theorem. We also investigated the positivity and boundedness of the model’s solutions to show whether they are biologically meaningful. Using laboratory experimental results from cancer stem cells isolated from resected tumor tissues of lung cancer patients, we determined the most biologically realistic parameter values through the least squares curve fitting approach. Then, using these parameter values, we performed numerical simulations with the Adams-Bashforth-Moulton predictor-corrector method to validate the theoretical results. We considered different values of fractional derivatives to investigate how the model is affected by fractional derivatives. As a result, we obtained the dynamics and expectations of lung cancer and made predictions specific to individual patients. In our simulations based on the parameter values obtained from actual patient data, it has been observed that after a certain period, both tumor cells and cancer stem cells have been eliminated. Consequently, an increase in normal tissue cells and immune cells has been observed. This implies that the patient in question, and similar behaving patients, will recover and overcome cancer. The findings from this study provide insights into the dynamics and prognosis of lung cancer, opening up the possibility for more personalized and effective approaches to treatment.

This article explores and highlights the effect of stochasticity on the extinction behavior of a disease in a general epidemic model. Specifically, we consider a sophisticated dynamical model that combines logistic growth, quarantine strategy, media intrusion, and quadratic noise. The amalgamation of all these hypotheses makes our model more practical and realistic. By adopting new analytical techniques, we provide a sharp criterion for disease eradication. The theoretical results show that the extinction criterion of our general perturbed model is mainly determined by the parameters closely related to the linear and quadratic perturbations as well as other deterministic parameters of the system. In order to clearly show the strength of our new result in a practical way, we perform numerical examples using the case of herpes simplex virus (HSV) in the USA. We conclude that a great amount of quadratic noise minimizes the period of HSV and affects its eradication time.

This study aims to discuss the impact of discrete-time delay on the anti-tumor immune response against tumor growth, excess levels of estrogen, and the source rate of immune cells in a breast cancer model. The non-negativity and boundedness of the solutions of the model are discussed. The existence of equilibria and their stability are examined. It is found that if the estrogen level is normal and the source rate of immune cells is low, the stability of the model around the co-existing equilibrium switches to instability via a Hopf bifurcation as the time delay increases. To validate the theoretical findings, a few numerical examples have been presented. The main result of this study is that the growth of tumors can be controlled if the immune system quickly generates an anti-tumor immune response. However, if the immune system takes a longer time to generate anti-tumor immune responses, the tumor growth cannot be controlled, and the system becomes unstable, which may result in the further spread of the disease.