On a two-strain epidemic mathematical model with vaccination.

In this paper, we study mathematically a two strains epidemic model taking into account non-monotonic incidence rates and vaccination strategy. The model contains seven ordinary differential equations that illustrate the interaction between the susceptible, the vaccinated, the exposed, the infected and the removed individuals. The model has four equilibrium points, namely, disease free equilibrium, endemic equilibrium with respect to the first strain, endemic equilibrium with respect to the second strain and the endemic equilibrium with respect to both strains. The global stability of the equilibria has been demonstrated using some suitable Lyapunov functions. The basic reproduction number is found depending on the first strain reproduction number R01 and the second reproduction number R02. We have shown that the disease dies out when the basic reproduction number is less than unity. It was remarked that the global stability of the endemic equilibria depends, on the strain basic reproduction number and on the strain inhibitory effect reproduction number. We have also observed that the strain with high basic reproduction number will dominate the other strain. Finally, the numerical simulations are presented in the last part of this work to support our theoretical results. We notice that our suggested model has some limitations and does not predicting the long-term dynamics for some reproduction numbers cases.

Local and global stability of an HCV viral dynamics model with two routes of infection and adaptive immunity.

The aim of this article is to formulate and study a mathematical model describing hepatitis C virus (HCV) infection dynamics. The model includes two essential modes of infection transmission, namely, virus-to-cell and cell-to-cell. The effect of therapy and adaptive immunity are incorporated in the suggested model. The adaptive immunity is represented by its two categories, namely, the humoral and cellular immune responses. Our article begins by establishing some mathematical results through proving the model's well-posedness in terms of existence, positivity and boundedness of solutions. We present all the steady states of the problem that depend on specific reproduction numbers. It moves then to the theoretical investigation of the local and global stability analysis of the free disease equilibrium and the four disease equilibria. The local and global stability analysis of the HCV mathematical model are established via the Routh-Hurwitz criteria and Lyapunov-LaSalle invariance principle, respectively. Finally, our article presents some numerical simulations to validate the analytical study of the global stability. Numerical simulations have shown the effect of the drug therapies on the system's dynamical behavior.

Modelling disease spread with spatio-temporal fractional derivative equations and saturated incidence rate.

The global analysis of a spatio-temporal fractional order SIR infection model with saturated incidence function is suggested and studied in this paper. The dynamics of the infection is described by three partial differential equations including a time-fractional derivative order for each one of them. The equations of our model describe the evolution of the susceptible, the infected and the recovered individuals with taking into account the spatial diffusion for each compartment. We will choose a saturated incidence rate in order to describe the nonlinear force of the infection. First, we will prove the well-posedness of our suggested model in terms of existence and uniqueness of the solution. Also in this context, the boundedness and the positivity of solutions are established. Afterward, we will give the forms of the disease-free equilibrium and the endemic one. It was demonstrated that the global stability of the each equilibrium depends mainly on the basic reproduction number. Finally, numerical simulations are performed to validate the theoretical results and to show the effect of vaccination in reducing the infection severity. It was shown that the fractional derivative order has no effect on the equilibria stability but only on the convergence speed towards the steady states. It was also observed that vaccination is amongst the good strategies in controlling the disease spread.

Hepatitis C virus fractional-order model: mathematical analysis.

Mathematical analysis of epidemics is crucial for the prediction of diseases over time and helps to guide decision makers in terms of public health policy. It is in this context that the purpose of this paper is to study a fractional-order differential mathematical model of HCV infection dynamics, incorporating two fundamental modes of transmission of the infection; virus-to-cell and cell-to-cell along with a cure rate of infected cells. The model includes four compartments, namely, the susceptible hepatocytes, the infected ones, the viral load and the humoral immune response, which is activated in the host to attack the virus. Each compartment involves a long memory effect that is modeled by a Caputo fractional derivative. Our paper starts with the investigation of some basic analytical results. First, we introduce some preliminaries about the needed fractional calculus tools. Next, we establish the well-posedness of our mathematical model in terms of proving the existence, positivity and boundedness of solutions. We present the different problem steady states depending on some reproduction numbers. After that, the paper moves to the stage of proving the global stability of the three steady states. To evaluate the theoretical study of the global stability, we apply a numerical method based on the fundamental theorem of fractional calculus as well as a three-step Lagrange polynomial interpolation method. The numerical simulations show that the free-endemic equilibrium is stable if the basic reproduction number is less than unity. In addition, the numerical tests demonstrate the stability of the other endemic equilibria under some optimal conditions. It is observed that the numerical simulations and the founding theoretical results are coherents.

Theoretical and numerical results of a stochastic model describing resistance and non-resistance strains of influenza.

In this world, there are several acute viral infections. One of them is influenza, a respiratory disease caused by the influenza virus. Stochastic modelling of infectious diseases is now a popular topic in the current century. Several stochastic epidemiological models have been constructed in the research papers. In the present article, we offer a stochastic two-strain influenza epidemic model that includes both resistant and non-resistance strains. We demonstrate both the existence and uniqueness of the global positive solution using the stochastic Lyapunov function theory. The extinction of our research sickness results from favourable circumstances. Additionally, the infection's persistence in the mean is demonstrated. Finally, to demonstrate how well our theoretical analysis performs, various noise disturbances are simulated numerically.

A fractional-order model of coronavirus disease 2019 (COVID-19) with governmental action and individual reaction.

The deadly coronavirus disease 2019 (COVID-19) has recently affected each corner of the world. Many governments of different countries have imposed strict measures in order to reduce the severity of the infection. In this present paper, we will study a mathematical model describing COVID-19 dynamics taking into account the government action and the individuals reaction. To this end, we will suggest a system of seven fractional deferential equations (FDEs) that describe the interaction between the classical susceptible, exposed, infectious, and removed (SEIR) individuals along with the government action and individual reaction involvement. Both human-to-human and zoonotic transmissions are considered in the model. The well-posedness of the FDEs model is established in terms of existence, positivity, and boundedness. The basic reproduction number (BRN) is found via the new generation matrix method. Different numerical simulations were carried out by taking into account real reported data from Wuhan, China. It was shown that the governmental action and the individuals' risk awareness reduce effectively the infection spread. Moreover, it was established that with the fractional derivative, the infection converges more quickly to its steady state.

Mathematical analysis and simulation of a stochastic COVID-19 Lévy jump model with isolation strategy.

This paper investigates the dynamics of a COVID-19 stochastic model with isolation strategy. The white noise as well as the Lévy jump perturbations are incorporated in all compartments of the suggested model. First, the existence and uniqueness of a global positive solution are proven. Next, the stochastic dynamic properties of the stochastic solution around the deterministic model equilibria are investigated. Finally, the theoretical results are reinforced by some numerical simulations.

Stability analysis and optimal control of HPV infection model with early-stage cervical cancer.

Cervical cancer cells may develop from any cell infected by human papillomavirus (HPV). The aim of this paper is to study whether an optimal control of HPV infection can reduce those resulting cancer cells. To this end, the problem will be modelled by five differential equations that describe the interactions between healthy cells, infected cells, free virus, precancerous cells and cancer cells. A saturated infection rate and two treatments are incorporated into the model. The first therapy stands for the efficacy of drug treatment in blocking new infections, whereas the second serves as the drug effectiveness in inhibiting viral production. First, The problem well-posedness is fulfilled in terms of existence, positivity and boundedness of solution. Next, the existence for the two optimal control pair is established, Pontryagin's maximum principle is used to characterize these two optimal controls. Moreover, the optimality system is derived and solved numerically using the forward and backward difference approximation scheme. Finally, numerical simulations are established in order to show the role of optimal therapy in controlling cancer cells proliferation. It was revealed that the antiviral drug therapies do not act only on the viral infection spread but also on reducing the amount of precancerous and cancerous cells. Consequently, the antiviral therapies can be considered amongst the most promising measures to reduce cervical cancer cells invasion.

Global dynamics of a multi-strain SEIR epidemic model with general incidence rates: application to COVID-19 pandemic.

This paper investigates the global stability analysis of two-strain epidemic model with two general incidence rates. The problem is modelled by a system of six nonlinear ordinary differential equations describing the evolution of susceptible, exposed, infected and removed individuals. The wellposedness of the suggested model is established in terms of existence, positivity and boundedness of solutions. Four equilibrium points are given, namely the disease-free equilibrium, the endemic equilibrium with respect to strain 1, the endemic equilibrium with respect to strain 2, and the last endemic equilibrium with respect to both strains. By constructing suitable Lyapunov functional, the global stability of the disease-free equilibrium is proved depending on the basic reproduction number R 0 . Furthermore, using other appropriate Lyapunov functionals, the global stability results of the endemic equilibria are established depending on the strain 1 reproduction number R 0 1 and the strain 2 reproduction number R 0 2 . Numerical simulations are performed in order to confirm the different theoretical results. It was observed that the model with a generalized incidence functions encompasses a large number of models with classical incidence functions and it gives a significant wide view about the equilibria stability. Numerical comparison between the model results and COVID-19 clinical data was conducted. Good fit of the model to the real clinical data was remarked. The impact of the quarantine strategy on controlling the infection spread is discussed. The generalization of the problem to a more complex compartmental model is illustrated at the end of this paper.

Mathematical Analysis and Clinical Implications of an HIV Model with Adaptive Immunity.

In this paper, a mathematical model describing the human immunodeficiency virus (HIV) pathogenesis with adaptive immune response is presented and studied. The mathematical model includes six nonlinear differential equations describing the interaction between the uninfected cells, the exposed cells, the actively infected cells, the free viruses, and the adaptive immune response. The considered adaptive immunity will be represented by cytotoxic T-lymphocytes cells (CTLs) and antibodies. First, the global stability of the disease-free steady state and the endemic steady states is established depending on the basic reproduction number R 0, the CTL immune response reproduction number R 1 z , the antibody immune response reproduction number R 1 w , the antibody immune competition reproduction number R 2 w , and the CTL immune response competition reproduction number R 3 z . On the other hand, different numerical simulations are performed in order to confirm numerically the stability for each steady state. Moreover, a comparison with some clinical data is conducted and analyzed. Finally, a sensitivity analysis for R 0 is performed in order to check the impact of different input parameters.

Mathematical Analysis and Treatment for a Delayed Hepatitis B Viral Infection Model with the Adaptive Immune Response and DNA-Containing Capsids.

We model the transmission of the hepatitis B virus (HBV) by six differential equations that represent the reactions between HBV with DNA-containing capsids, the hepatocytes, the antibodies and the cytotoxic T-lymphocyte (CTL) cells. The intracellular delay and treatment are integrated into the model. The existence of the optimal control pair is supported and the characterization of this pair is given by the Pontryagin's minimum principle. Note that one of them describes the effectiveness of medical treatment in restraining viral production, while the second stands for the success of drug treatment in blocking new infections. Using the finite difference approximation, the optimality system is derived and solved numerically. Finally, the numerical simulations are illustrated in order to determine the role of optimal treatment in preventing viral replication.

Prediction of DNA methylation in the promoter of gene suppressor tumor.

The epigenetics methylation of cytosine is the most common epigenetic form in DNA sequences. It is highly concentrated in the promoter regions of the genes, leading to an inactivation of tumor suppressors regardless of their initial function. In this work, we aim to identify the highly methylated regions; the cytosine-phosphate-guanine (CpG) island located on the promoters and/or the first exon gene known for their key roles in the cell cycle, hence the need to study gene-gene interactions. The Frommer and hidden Markov model algorithms are used as computational methods to identify CpG islands with specificity and sensitivity up to 76% and 80%, respectively. The results obtained show, on the one hand, that the genes studied are suspected of developing hypermethylation in the promoter region of the gene involved in the case of a cancer. We then showed that the relative richness in CG results from a high level of methylation. On the other hand, we observe that the gene-gene interaction exhibits co-expression between the chosen genes. This let us to conclude that the hidden Markov model algorithm predicts more specific and valuable information about the hypermethylation in gene as a preventive and diagnostics tools for the personalized medicine; as that the tumor-suppresser-genes have relative co-expression and complementary relations which the hypermethylation affect in the samples studied in our work.