Unidentified diseases are becoming more prevalent among humans due to various climatic factors, and some of these diseases originate in animals before spreading to humans. One virus that has been of particular concern is the avian influenza virus, which primarily infects bird and can subsequently transmit to humans. This article presents a mathematical model describing the spatio-temporal reaction-diffusion process involved in the transmission of avian flu in human population. The paper begins by studying the proposed model’s well-posedness and the calculated basic reproduction number, which provides valuable insights into the dynamics of virus transmission. The paper also provides stability analysis for the disease-free steady state of the model. All theoretical studies are validated using computational results.

This article examines the convergence of the two-step backward difference formula (BDF2)-discontinuous Galerkin (DG) scheme in the context of a nonlocal reaction–diffusion equation. Our methodology integrates BDF2 for temporal discretization with a DG finite element method for spatial discretization, enabling a comprehensive analysis of the nonlocal diffusion equation. We establish optimal-order convergence rates for the fully discrete system. To assess the effectiveness and precision of the proposed scheme, we provide numerical simulations alongside convergence results.

Understanding the dynamics of the African swine fever virus during periods of intense replication is critical for effective combatting of the rapid spread. In our research, we have developed a fractional-order SVEIR model using the Caputo derivatives to investigate this behaviour. We have established the existence and uniqueness of the solution through fixed point theory and determined the basic reproduction number using the next-generation matrix method. Our study also involves an examination of the local and global stability of disease-free equilibrium points. Additionally, we have conducted optimal control analysis with two control variables to increase the number of recovered pigs while reducing the number of those infected and exposed. We have supported our findings with numerical simulations to demonstrate the effectiveness of the control strategy.

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results.

Humans have been affected by various epidemic diseases, mostly are airborne and exhibit high transmission rates. Given these nature properties, quarantine measures are essential to control the spread of the diseases effectively. Motivated by this fact, and due to the successful use of mathematical modeling, we investigate a SIR model with quarantine and vaccination compartments. This model uses a system of fractional differential equations (SIQVR-based) with specific parameters to track the dynamics of model variables. We examine the well-posedness and boundedness results via standard tools. An effective threshold parameter ℛ_0 is determined using a generation matrix and equilibrium points of the model are obtained. To effectively manage the transmission of infection within the outlined model, we employ the strategy of optimal control. This approach involves implementing control measures and interventions guided by mathematical optimization techniques to minimize the spread of disease. These control strategies may encompass vaccination campaigns, quarantine protocols, social distancing measures and other preventive actions. Further, to evaluate the effectiveness of proposed model and the applied optimal control strategy, we conduct a series of numerical simulations. Computational results involve running the model under different scenarios, considering a range of parameters and meticulously analyzing the resulting outcomes.

This study introduces a novel fractional-order model to investigate the interplay between cancer and obesity and their treatment. Initially, we examine the solution’s existence and uniqueness for the proposed model. Additionally, we establish the boundedness of these solutions. Subsequently, we identify some potential equilibrium points of the cancer–obesity model and investigate their stability. To address the considered model, we propose fractional Euler’s and Adam’s methods. Theoretical and numerical analyses are conducted to assess the error estimates and performance of both methods with varying fractional-order derivatives. Moreover, we formulate an optimal control problem concerning cancer density and drug concentration. We delve into the existence of control and explore the first-order optimality conditions. We validate the analytical findings through numerical computations, demonstrating that administering drugs with control variables enhance immunity levels and reduce the burden of cancer.

We introduce an epidemic disease reaction–diffusion model to study the transmission of the varicella-zoster virus in both space and time. More precisely, we present a system of partial differential equations with the Neumann boundary conditions (NBC) concerned to model the evolution of the virus. Firstly, the wellposedness results of the model are studied using the semigroup theory. Then, the boundedness of the solutions is also derived. Further, the basic reproduction number (BRN) for the proposed model is determined using the eigenvalue problem. Moreover, asymptotic profiles of the equilibrium points of the susceptible and infected compartments of the model are investigated. Finally, the advantage of the spatiotemporal model and the above theoretical results are validated with numerical experiments.

A system of partial differential equations modeling the transmission dynamics of tuberculosis is considered to represent the density of susceptible, vaccinated, latent stage infected, active stage infected, and treated individuals.We studied the optimal control problem of the coupled nonlinear system with nonlocal diffusion. First, an optimal solution for the proposed model is established and we derive the optimality system. Then, solutions for the direct and the adjoint problem are proved. Numerical simulations are provided to validate the theoretical results.