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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.

The present investigation centers around a numerical study of the impact of the presence of phototactic microorganisms on the thermo-bioconvection inside a wavy-walled horizontal porous annulus that is saturated with a Newtonian fluid. Consequently, an equation describing the movement of the microorganism cells is added to the mathematical model of natural convection. The thermal equilibrium, Oberbeck–Boussinesq approximations and Darcy law are taken into consideration. The transformed dimensionless partial differential equations with the temperature-concentration-stream function formulation are discretized applying the alternate direction implicit finite difference scheme. Results confirm that in the absence of microorganisms the Nusselt number decreases with the increase of the amplitude of wall waviness. It was also observed that the wall waviness influence the thermo-bioconvective instabilities and the flow regime structure.

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.

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.

In this paper, we present a novel SICVS COVID-19 pandemic model incorporating a vaccination strategy and crowding effect. First, we investigate the steady states and proved the local stability of the corresponding deterministic problem. Then we study the proposed stochastic epidemic model; we have considered a random incidence highlighting all the improbable fluctuations that appear in the infection process. Hence, the COVID 19 transmission parameter in the considered model will be stochastically perturbed by the white noise. We show the disease extinction of the pandemic according to the stochastic noise intensity. It was established, in the stochastic study, the existence of a possible extinction of the pandemic even if the base reproduction number is greater than unity. Furthermore we discuss the persistence in mean of the studied infection in terms of the stochastic model parameters. Our numerical simulations confirm the theoretical findings. The proposed model was validated and supported by a comparison, during the second wave of the pandemic in Morocco, between the trajectories of the clinical data of Covid-19 and those which represent our SICVS model. In addition, the impact of vaccination and social distancing strategies on reducing the infection severity were observed.

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0. Furthermore, using other appropriate Lyapunov functionals, the global stability results of the endemic equilibria are established depending on the strain 1 reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^{1}_0$$\end{document}R01 and the strain 2 reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^{2}_0$$\end{document}R02. 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.

The objective of this paper is to investigate a mathematical model describing the interactions between hepatitis B virus with DNA-containing capsids, liver cells (hepatocytes) and the adaptive immune response. This adaptive immunity will be represented by cytotoxic T-lymphocytes and antibody immune responses. The positivity and boundedness of solutions for non negative initial data are proved which is consistent with the biological studies. The local stability of the equilibria is established. In addition to this, the global stability of the disease-free equilibrium and the endemic equilibria is fulfilled by using appropriate Lyapunov functions. Finally, numerical simulations are performed to support our theoretical findings.

This paper deals with an optimal control problem for an human immunodeficiency virus (HIV) infection model with cytotoxic T-lymphocytes (CTL) immune response and latently infected cells. The model under consideration describes the interaction between the uninfected cells, the latently infected cells, the productively infected cells, the free viruses and the CTL cells. The two treatments represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. Existence of the optimal control pair is established and the Pontryagin's minimum principle is used to characterize these two optimal controls. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to show the role of optimal therapy in controlling the infection severity.

In this paper, we propose a bi-criteria optimization model for a flow shop scheduling problem with permutation, blocking and sequence dependent setup time. Indeed, these constraints are the most encountered in the industrial field, which demands high command flexibility. The objective is the minimization of two criteria, in our case the makespan and the total tardiness combined in a single objective function with a weighting coefficient for each criterion. To solve this problem, we propose a mixed integer linear programming method and a set of different metaheuristics. The suggested metaheuristics are; the genetic algorithm, the iterated greedy metaheuristic and the iterative local search algorithm. This last algorithm is proposed in two ways of exploration of the neighborhood. To verify the effectiveness of our resolution algorithms, a set of instances with n jobs and m machines is randomly generated from small instances to relatively large size ones. The analysis of the suggested simulation model allowed us to note that the iterative local search algorithm gives good results compared to the iterative greedy algorithm. Moreover, it was found that the weighting parameter plays an essential role in the problem decision making. However, it was established that it is difficult to find a good solution that minimizes both criteria at once, a suitable compromise will be necessary to be adopted using the weighting coefficient.

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 R0, the CTL immune response reproduction number R1z, the antibody immune response reproduction number R1w, the antibody immune competition reproduction number R2w, and the CTL immune response competition reproduction number R3z. 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 R0 is performed in order to check the impact of different input parameters.

In this paper, we study the influence of the presence of phototactic microorganisms on thermal bioconvection in horizontal porous annulus. To this end, an additional equation describing the concentration of the microorganism cells will be added to the classical ones of heat and hydrodynamics. The Boussinesq and Darcy approximations will be under consideration due to the incompressibility of the fluid in the porous medium. The derived problem with the stream function will be descritized numerically using the alternate direction implicit finite difference method and solved via Thomas algorithm. Numerical simulations have shown that the presence of the microorganisms enhances the heat transfer and has a destabilizing effect on the natural convection.

Natural convection and heat transfer inside a porous concentric horizontal cylindrical annulus saturated by a Cu–water nanofluid was investigated in this study. The governing equations were discretized using the finite difference technique and solved using the alternating direction implicit (ADI) method. It was found that the increase in nanoparticles had a favorable effect on the enhancement of the heat transfer for low Rayleigh numbers. However, exceeding a given threshold of Rayleigh number, the addition of nanoparticles had an adverse effect on the heat transfer. Also, the study revealed the effect of porosity on Nusselt number, where the increase in volume fraction of nanoparticles in the porous medium leads to a considerable increase in the Nusselt number ratio when the porosity equals 0.5 and 0.8, an opposite behavior was observed when this latter equals 0.1. The bifurcation between the unicellular and the bicellular regimes was investigated for several nanoparticles volume fractions and a wide range of porosity. The results revealed that regardless of the porosity, the critical Rayleigh number for the transition from the unicellular to the bicellular regime increases with volume fraction of nanoparticles for the whole range of the radius ratio. Several numerical correlations were developed for the critical Rayleigh number in terms of volume fraction of nanoparticles and porosity of the medium.

Ovarian cancer in women represents a mortality rate of >4% worldwide. In cancer, CpG islet hypermethylation of some gene promoters often leads to the inactivation of tumor suppressor genes. Ovarian cancer is characterized by a targeted hypermethylation, hence the need to study this hypermethylation. We have adopted the hypothesis that a very high mutation rate can cause the epimutation phenomenon, causing a 5-fold deamination methyl-cytosine in thymine, consequence of the mutation. To test this, we chose the P53, RASSF1A, WAF1, ARHI, CAV1, DOC-2, BRCA2 and BRCA1genes. The methylation status was first determined by the Chargaff coefficient, which represents a high percentage of about 50% for all the sequences chosen, secondly, the CpGo/e ratio was calculated. ANOVA test has been applied in order to be sure of the correlation between these two calculations andshowed thatthe Chargaff coefficientis significant. Lastly, we adopted Hidden Markov Models (HMM). To indicate exactly the CGI region in the promoters of the studied genes. This model maximizes the mathematical probability of having a hypermethylated region.

In this paper a numerical analysis of in situ-biorestoration is presented. Improved stability and error estimates are derived for a P 1 finite-element method applied to coupled system of non linear partial differential equations modeling flow in porous media. These results improve upon previously derived stability and error estimates in two respects: first, a stability of the approximate solution is demonstrated in the case of a non linear diffusive flux λ ( u ) ∇ u with weaker norm assumptions than before and depends only on the initial conditions and time interval, and second, error estimates are optimal as in linear case. Extensions include the finite element approximation of flow field, described by Darcy’s law under the stream function–vorticity formulation because of its multiple advantages. Finally, numerical simulations for a correlation exponential plume are presented.

We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy., This is a preprint of a paper whose final and definite form is with 'Discrete and Continuous Dynamical Systems - Series S' (DCDS-S), ISSN 1937-1632 (print), ISSN 1937-1179 (online), available at [http://aimsciences.org/journal/1937-1632]. Paper Submitted 01-June-2019; Revised 10-May-2021; Accepted 21-May-2021. arXiv admin note: text overlap with arXiv:1801.10048

Oncology is the science that deals with the prevention, diagnosis and study the different possible remedies in order to treat and fight the cancer disease. In this paper, immunotherapy and virotherapy are included to mathematical new model which expresses the interaction between tumor, immunity elements (Cytotoxic T cell and Interleukin-2 (IL-2)) and oncolytic virus. After the well-posedness of model, we show that the disease-free equilibrium point (DFE) and the endemic equilibrium point can exist under specific conditions depending on treatment parameters: adoptive cellular therapy (ACT) or IL-2 treatment or the case of oncolytic virus treatment (OVT) and combined treatments. We will also discuss the local stability of the equilibrium points. The oncolytic dynamic will be analyzed numerically by using MATLAB software in order to interpret the different cancer evolution situations. A set of numerical results are obtained in order to show the performance of each treatment to reduce the size and the speed of tumor proliferation, although it leaves treatment residues for a long time.

The disease caused by COVID-19 was declared a pandemic by the World Health Organization on March 11, 2020. Identified for the first time in the city of Wuhan, the Chinese authorities have mandated a quarantine strategy in the city of Wuhan in order to limit the spread of COVID-19 in a wider spectrum. Inspired by this prevention protocol against this new virus, the majority of the countries in the world have also adopted strict quarantine measures to fight against this new coronavirus pneumonia. In this article, we will establish a six-compartment SLIR model taking into account quarantine strategy, in which the dynamics of the COVID-19 epidemic is modeled by a system of six nonlinear differential equations, describing the interactions between susceptible, exposed, infected, and recovered. The basic reproduction number R0 depending on the quarantine strategy efficacy is calculated. We give the equilibrium points of the system, then we discuss, according to the value of R0, the global stability of the equilibrium solutions. Numerical simulations are presented in order to validate our theoretical results and we discuss the effectiveness of quarantine measures in the fight against the spread of the pandemic caused by COVID-19.

In this paper, we present some effective metaheuristics for solving hybrid flow shop scheduling with uniform parallel machines and sequence independent setup time. We implement three metaheuristics, the variable neighborhood search algorithm, the greedy randomized adaptive search procedure and the iterative greedy algorithm. The objective function is the minimization of the total flow time taking into account the availability constraints of the machines during scheduling. For each metaheuristic we choose the appropriate parameters to obtain the optimal solution by exploring the space of the neighborhood of the current solution. We conducted a simulation study on a set of randomly generated instances in order to test the effectiveness of different metaheuristics. We found that the iterative greedy algorithm gives good results compared to the other two metaheuristics.

In this paper, we study the dynamic of a multi-strain SEIR model with both saturated incidence and treatment functions. Two basic reproduction numbers are extracted from the epidemic model, noted $$R_{0,1}$$ and $$R_{0,2}$$ . Using the Lyapunov method, we investigate the global stability of the disease free equilibrium and prove that it is globally asymptotically stable when $$R_{0,1}$$ and $$R_{0,2}$$ are less than one. Moreover, we formulate the optimal control problem, solve it, and perform some numerical simulations, to support the analytical results and test how well the proposed model may be applied in practice.

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.

International audience; In this paper, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the Radial Basis Functions Method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion.