Dynamics of Salmonella Infection

In this chapter, we propose a mathematical epidemic model, with integer and fractional order to describe the dynamics of Salmonella infection in animal herds. We investigate the qualitative behaviors of such model and find the conditions that guarantee the asymptotic stability of disease‐free and endemic steady states. To assess the severity of the outbreak, as well as the strength of the medical and/or behavioral interventions necessary for control, we estimate basic reproduction number R0. This threshold parameter specifies the average number of secondary infections caused by one infected individual during his/her entire infectious period at the start of an outbreak. We also provide an unconditionally stable implicit scheme for the fractional‐order epidemic model. The theoretical and computational results give insight into the modelers and infectious disease specialists

Numerical Modeling of Fractional-Order Biological Systems

We provide a class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4+ T cells. Stability and nonstability conditions for disease-free equilibrium and positive equilibria are obtained in terms of a threshold parameter (minimum infection parameter) for each model. We provide unconditionally stable method, using the Caputo fractional derivative of order and implicit Euler’s approximation, to find a numerical solution of the resulting systems. The numerical simulations confirm the advantages of the numerical technique and using fractional-order differential models in biological systems over the differential equations with integer order. The results may give insight to infectious disease specialists

Hopf Bifurcation and Stability of Periodic Solutions for Delay Differential Model of HIV Infection of CD4 +

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of CD4+ T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results

Dynamics of a Fractional-Order Delayed Model of COVID-19 with Vaccination Efficacy

In this study, we provide a fractional-order mathematical model that considers the effect of vaccination on COVID-19 spread dynamics. The model accounts for the latent period of intervention strategies by incorporating a time delay τ. A basic reproduction number, R0, is determined for the model, and prerequisites for endemic equilibrium are discussed. The model’s endemic equilibrium point also exhibits local asymptotic stability (under certain conditions), and a Hopf bifurcation condition is established. Different scenarios of vaccination efficacy are simulated. As a result of the vaccination efforts, the number of deaths and those affected have decreased. COVID-19 may not be effectively controlled by vaccination alone. To control infections, several non-pharmacological interventions are necessary. Based on numerical simulations and fitting to real observations, the theoretical results are proven to be effective

Stability and Bifurcation Analysis of the Caputo Fractional-Order Asymptomatic COVID-19 Model with Multiple Time-Delays

Throughout the last few decades, fractional-order models have been used in many fields of science and engineering, applied mathematics, and biotechnology. Fractional-order differential equations are beneficial for incorporating memory and hereditary properties into systems. Our paper proposes an asymptomatic COVID-19 model with three delay terms τ1,τ2,τ3 and fractional-order α. Multiple constant time delays are included in the model to account for the latency of infection in a vector. We study the necessary and sufficient criteria for stability of steady states and Hopf bifurcations based on the three constant time-delays, τ1, τ2, and τ3. Hopf bifurcation occurs in the addressed model at the estimated bifurcation points τ10, τ20, τ30, and τ10*. The numerical simulations fit to real observations proving the effectiveness of the theoretical results. Fractional-order and time-delays successfully enhance the dynamics and strengthen the stability condition of the asymptomatic COVID-19 model

Delay Differential Model for Tumour-Immune System With Chemo-therapy and Optimal Control

In this paper, we present a mathematical model of tumour-immune interactions in presence of chemotherapy treatment. The model is governed by a system of delay differential equations with optimal control variables. The control variables are included to justify the best strategy of treatments with minimum side effects, by reducing the production of new tumour cells and keeping the number of normal cells above the average of its carrying capacity. Existence of optimality and optimality conditions are also proved. The numerical simulations show that the optimal treatment strategy reduces the load of tumour cells and increases the effector cells after few days of therapy

Numerical Modelling of Biological Systems with Memory using Delay Differential Equations

This is a review article, to show the consistency of delay differential equations with biological systems with memory, in which we present a class of mathematical models with time-lags in immunology, physiology, epidemiology and cell growth. We also incorporate optimal control parameters into a delay model to describe the interactions of the tumour cells and immune response cells with external therapy. We then study parameter estimations and sensitivity analysis with delay differential equations. Sensitivity analysis is an important tool for understanding a particular model, which is considered as an issue of stability with respect to structural perturbations in the model. We introduce a variational method to evaluate sensitivity of the state variables to small perturbations in the initial conditions and parameters appear in the model. The presented numerical simulations show the consistency of delay differential equations with biological systems with memory. The displayed results may bridge the gap between the mathematics reserach and its applications in biology and medicine

Four-Dimensional Data Assimilation and Numerical Weather Prediction Abstract

All forecast models, whether they represent the state of the weather, the spread of a disease, or levels of economic activity, contain unknown parameters. These parameters may be the model’s initial conditions, its boundary conditions, or other tunable parameters which have to be determined. Four dimensional variational data assimilation (4D-Var) is a method of estimating this set of parameters by optimizing the fit between the solution of the model and a set of observations which the model is meant to predict. The four dimensional nature of 4D-Var reflects the fact that the observation set spans not only three dimensional space, but also a time domain. Although the method of 4D-Var described in this report is not restricted to any particular system, the application described here has a Numerical Weather Prediction (NWP) model at its core, and the parameters to be determined are the initial conditions of the model. The purpose of this report is to give a survey covering assimilation of Doppler radar wind data into a high-resolution NWP model. Some associated problems, such as sensitivity to small variations in the initial conditions or due to small changes i

Dynamics of Tumor-Immune System with Random Noise

With deterministic differential equations, we can understand the dynamics of tumor-immune interactions. Cancer-immune interactions can, however, be greatly disrupted by random factors, such as physiological rhythms, environmental factors, and cell-to-cell communication. The present study introduces a stochastic differential model in infectious diseases and immunology of the dynamics of a tumor-immune system with random noise. Stationary ergodic distribution of positive solutions to the system is investigated in which the solution fluctuates around the equilibrium of the deterministic case and causes the disease to persist stochastically. In some conditions, it may be possible to attain infection-free status, where diseases die out exponentially with a probability of one. Some numerical simulations are conducted with the Euler–Maruyama scheme in order to verify the results. White noise intensity is a key factor in treating infectious diseases