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In this paper, the mathematical modeling of the novel corona virus (COVID-19) is considered. A brief relationship between the unknown hosts and bats is described. Then the interaction among the seafood market and peoples is studied. After that, the proposed model is reduced by assuming that the seafood market has an adequate source of infection that is capable of spreading infection among the people. The reproductive number is calculated and it is proved that the proposed model is locally asymptotically stable when the reproductive number is less than unity. Then, the stability results of the endemic equilibria are also discussed. To understand the complex dynamical behavior, fractal-fractional derivative is used. Therefore, the proposed model is then converted to fractal-fractional order model in Atangana-Baleanu (AB) derivative and solved numerically by using two different techniques. For numerical simulation Adam-Bash Forth method based on piece-wise Lagrangian interpolation is used. The infection cases for Jan-21, 2020, till Jan-28, 2020 are considered. Then graphical consequences are compared with real reported data of Wuhan city to demonstrate the efficiency of the method proposed by us.

This study presents a novel numerical method to solve PDEs with the fractional Caputo operator. In this method, we apply the Newton interpolation numerical scheme in Laplace space, and then, the solution is returned to real space through the inverse Laplace transform. The Newton polynomial provides good results as compared to the Lagrangian polynomial, which is used to construct the Adams–Bashforth method. This procedure is used to solve fractional Buckmaster and diffusion equations. Finally, a few numerical simulations are presented, ensuring that this strategy is highly stable and quickly converges to an exact solution.

In this paper, a stochastic HIV/AIDS epidemic model has been studied numerically. A discussion among the solutions related to deterministic HIV/AIDS model and stochastic HIV/ AIDS epidemic model has shown that the stochastic solution is more realistic than the deterministic solution. To control the diseases, the threshold parameter R 0 plays a key role in the stochastic HIV/AIDS epidemic model. If R 0 1 then disease is under control while the disease is out of control if R 0 > 1 . The explicit approaches such as the Milstein scheme, stochastic Euler scheme, and stochastic Runge-Kutta 4 are dependent on temporal step size, whereas non-standard finite difference approaches are independent of step size. The results for numerical approaches like the Milstein scheme, stochastic Euler scheme, and stochastic Runge-Kutta 4 scheme fail for outsized step size. The stochastic non-standard finite difference scheme conserves dynamic features like confinedness, consistency and positivity.

In this paper, we consider a fractional-order model of a brushless DC motor. To develop a mathematical model, we use the concept of the Liouville–Caputo noninteger derivative with the Mittag-Lefler kernel. We find that the fractional-order brushless DC motor system exhibits the character of chaos. For the proposed system, we show the largest exponent to be 0.711625. We calculate the equilibrium points of the model and discuss their local stability. We apply an iterative scheme by using the Laplace transform to find a special solution in this case. By taking into account the rule of trapezoidal product integration we develop two iterative methods to find an approximate solution of the system. We also study the existence and uniqueness of solutions. We take into account the numerical solutions for Caputo Liouville product integration and Atangana–Baleanu Caputo product integration. This scheme has an implicit structure. The numerical simulations indicate that the obtained approximate solutions are in excellent agreement with the expected theoretical results.

In this article, a mathematical model of the COVID-19 pandemic with control parameters is introduced. The main objective of this study is to determine the most effective model for predicting the transmission dynamic of COVID-19 using a deterministic model with control variables. For this purpose, we introduce three control variables to reduce the number of infected and asymptomatic or undiagnosed populations in the considered model. Existence and necessary optimal conditions are also established. The Grünwald-Letnikov non-standard weighted average finite difference method (GL-NWAFDM) is developed for solving the proposed optimal control system. Further, we prove the stability of the considered numerical method. Graphical representations and analysis are presented to verify the theoretical results.

It is very important to note that a mathematical model plays a key role in different infectious diseases. Here, we study the dynamical behaviors of both hepatitis B virus (HBV) and hepatitis C virus (HCV) with their co-infection. Actually, the purpose of this work is to show how the bi-therapy is effective and include an inhibitor for HCV infection with some treatments, which are frequently used against HBV. Local stability, global stability and its prevention from the community are studied. Mathematical models and optimality system of nonlinear DE are solved numerically by RK4. We use linearization, Lyapunov function and Pontryagin’s maximum principle for local stability, global stability and optimal control, respectively. Stability curves and basic reproductive number are plotted with and without control versus different values of parameters. This study shows that the infection will spread without control and can cover with treatment. The intensity of HBV/HCV co-infection is studied before and after optimal treatment. This represents a short drop after treatment. First, we formulate the model then find its equilibrium points for both. The models possess four distinct equilibria: HBV and HCV free, and endemic. For the proposed problem dynamics, we show the local as well as the global stability of the HBV and HCV. With the help of optimal control theory, we increase uninfected individuals and decrease the infected individuals. Three time-dependent variables are also used, namely, vaccination, treatment and isolation. Finally, optimal control is classified into optimality system, which we can solve with Runge–Kutta-order four method for different values of parameters. Finally, we will conclude the results for implementation to minimize the infected individuals.

Infection of hepatitis B virus (HBV) is a global health problem. We provide the study about hepatitis B virus dynamics that can be controlled by education campaign (awareness), vaccination, and treatment. Initially we bring constant controls in considerations for treatment, vaccination, and education campaign (awareness). In the case of constant controls, we study the stability and existence of the disease-free and endemic equilibria model’s solutions. Afterwards, we take time as a control and formulate the suitable optimal control problem, acquire optimal control strategy in order to reduce the number of humans that are infected and the costs associated. At the end, results of numerical simulations show that the optimal combination of education campaign (awareness), treatment, and vaccination is the most efficient way to control the infection of hepatitis B virus (HBV) infection.

In this paper, we develop the theory of fractional order hybrid differential equations involving Riemann–Liouville differential operators of order [Formula: see text]. We study the existence theory to a class of boundary value problems for fractional order hybrid differential equations. The sum of three operators is used to prove the key results for a couple of hybrid fixed point theorems. We obtain sufficient conditions for the existence and uniqueness of positive solutions. Moreover, examples are also presented to show the significance of the results.

We describe the formulation of hepatitis B epidemic model with saturated incidence rate. Once we formulate the epidemic problem then we discuss the basic properties (existence of positive solutions and positively invariant set etc.) to show the mathematical as well as the biological feasibility. We find the basic reproduction number R0. The disease free and endemic equilibria of the model will be obtained to investigate the local stability and backward bifurcation. We also perform the local sensitivity analysis and find the sensitive parameter. On the basis of this, we develop the optimal control strategy for the elimination of hepatitis B virus (HBV) spreading with the help of optimization theory. Finally, a large scale numerical simulation will be performed to verify our analytical findings.

In this paper, an optimal control problem of HIV infection model of delay differential equations is taken into account. Then we set a control function which represents the efficiency of reverse transcriptase inhibitors. Objective functional is constructed to minimize the virus concentration as well as treatment costs.Adjoint system is derived using Pontryagins Maximum Principle. Optimality system is calculated and numerical simulation is carried out to illustrate the theoretical results. Finally, conclusion is drawn

In this article, the Adomian decomposition method (ADM) is applied to find the solution of HIV infection model of latently infected CD4+T cells. This method investigates the solution of ordinary differential equation which is calculated in the form of the components of an infinite series. These components can be easily calculated. The efficiency and the reliability of proposed method is demonstrated in different time intervals by numerical example. The derived results indicate that the approximate solution by using the ADM can be obtained in a more efficient way. All computations have been carried out by computer code written in Mathematica.

In this paper, the analysis of an HIV-1 epidemic model is presented by incorporating a distributed intracellular delay. The delay term represents the latent period between the time that the target cells are contacted by the virus and the time the virions penetrated into the cells. To understand the analysis of our proposed model, the Rouths–Hurwiz criterion and general theory of delay differential equations are used. It is shown that the infection free equilibrium and the chronic-infection equilibrium are locally as well as globally asymptotically stable, under some conditions on the basic reproductive number R 0 {R_{0}} . Furthermore, the obtained results show that the value of R 0 {R_{0}} can be decreased by increasing the delay. Therefore, any drugs that can prolong the latent period will help to control the HIV-1 infection.

In this paper an arbitrary order model for Toxoplasmosis ailment in the humanoid and feline is verbalized and explored. The dynamics of this ailment is discovered using an epidemiology type paradigm. We have proposed the fractional order multistage differential transform method for the Toxoplasmosis model. It is employed to analyze and find the elucidation for the model, and the numerical simulations have been conducted in order to study the effectiveness of the technique. The suggested algorithm can be considered as a fractional extension of the well know method known as Multistage Differential Transform Method. The sensitivity analysis of the strictures of the specimen is discussed. The numeric imitations of the projected non-integer specimens are conceded out to illustrate different dynamics of the model, which depend on R 0 .

In this research article, a new mathematical model of delayed differential equations is developed which discusses the interaction among CD4T cells, human immunodeficiency virus (HIV), and recombinant virus with cure rate. The model has two distributed intracellular delays. These delays denote the time needed for the infection of a cell. The dynamics of the model are completely described by the basic reproduction numbers represented byR0,R1, andR2. It is shown that ifR0<1, then the infection-free equilibrium is locally as well as globally stable. Similarly, it is proved that the recombinant absent equilibrium is locally as well as globally asymptotically stable if1

JUNG, IL HYO/0000-0002-6086-5124 WOS: 000482145300001 This article studies the dynamical behavior of the analytical solutions of the system of fraction order model of HIV-1 infection. For this purpose, first, the proposed integer order model is converted into fractional order model. Then, Laplace-Adomian decomposition method (L-ADM) is applied to solve this fractional order HIV model. Moreover, the convergence of this method is also discussed. It can be observed from the numerical solution that (L-ADM) is very simple and accurate to solve fraction order HIV model. Department of Mathematics, Pusan National University, South Korea; Higher Education Commission (HEC) of PakistanHigher Education Commission of Pakistan [21-1657/SRGP/RD/HEC/2017] The authors ensure the originality of this research. Upon acceptance, the authors will provide all source files to the journal. This research work has been financially supported by the Department of Mathematics, Pusan National University, South Korea, and the Higher Education Commission (HEC) of Pakistan under grant number 21-1657/SRGP/R&D/HEC/2017.

Background and Objective Dengue viral infections are a standout amongst the supreme critical mosquito-borne illnesses nowadays. They create problems like dengue fever (DF), dengue stun disorder (DSS) and dengue hemorrhagic fever (DHF). Lately, the frequency of DHF has expanded considerably. Dengue may be caused by one of serotypes DEN-1 to DEN-4. For the most part, septicity with one serotype presents upcoming defensive resistance against that specific serotype yet not against different serotypes. When anyone is infected for a second time with different serotypes, a serious ailment will occur. The proposed model focused on the dynamic interaction between susceptible cells and free virus cells. The ailment free steady states of the specimen are determined. The steadiness of the steady states has been examined by using Laplace transform. Methods We introduce an appropriate numerical technique based on an Adams Bash-forth Moulton method for non-integer order delay differential equations. The numerical simulations validate the accuracy and efficacy of the numerical method. Results In this paper, we study a non-integer order model with temporal delay to elaborate the dynamics of Dengue internal transmission dynamics. The temporal delay is presented in the susceptible cell and free virus cell. Centered on non-integer Laplace transform, some environs on firmness and Hopf bifurcation are derived for the model. Beside these global stability analysis is also done. Lastly, the imitative theoretical results are justified by few numerical simulations. Conclusion The study spectacles that the non-integer order with temporal-delay can successfully enhance the dynamics and rejuvenate the steadiness terms of non-integer order septicity prototypes. Both the ailment free equilibrium (AFE) node and ailment persistent equilibrium (APE) node are steady for the given system. We deduce a recipe that regulates the critical value at threshold.

Background and Objective: As we all know, mathematical models provide very important information for the study of the human immunodeficiency virus type. Mathematical model of human immunodeficiency virus type-1 (HIV-1) infection with contact rate represented by Crowley-Martin function response is taken into account. The aims of this novel study is to checkthe local and global stability of the disease and also prevent the outbreak from the community. Methods: The mathematical model as well as optimal system of nonlinear differential equations are tackled numerically by Runge-Kutta fourth-order method. For global stability we use Lyapunov-LaSalle invariance principle and for the description of optimal control, Pontryagin’s maximum principle is used. Results: Graphical results are depicted and examined with different parameters values versus the basic reproductive number R0 and also the plots with and without control. The density of infected cells continued to increase without treatment, but the concentration of these cells decreased after treatment. The intensity of the pathogenic virus before and after the optimal treatment. This indicates a sharp drop in the rate of pathogenic viruses after treatment. It prevents the production of viruses by preventing cell infection and minimizing side effects. Conclusions: We analysed the model by defining the basic reproductive number, showing the boundedness, positivity and permanence of the solution, and proving the local and global stability of the infection-free state. We show that the threshold quantity R0 1, HIV-1 infection remains in the host. When the threshold quantity R0 > 1, then it shows that the steady-state of chronic disease is globally stable. Optimal control strategies are developed with the optimal control pair for the description of optimal control. To reduce the density of infected cells and viruses as well as maximize the density of healthy cells is determined by the objective functional.

This manuscript is concerned with the investigation of three-point boundary value problem (BVP) of nonlinear fractional differential equations (FDEs) as provided by $$\begin{aligned} \begin{aligned} -&\mathbf {D}^\omega w(t)=\varphi (t,w(t));\ t\in (0, 1),\ \omega \in (1, 2],\\ {}&\mathbf {D}^\theta w(0)=0, \ w(1)= \gamma \mathbf {D}^{\theta }w(\eta ),\ \theta \in (0, 1], \end{aligned} \end{aligned}$$ where usual Riemann–Liouville derivative is denoted by $$\mathbf {D}^q$$ with order $$\omega $$ and $$\gamma , \eta \in (0, 1)$$ . Also $$\varphi :[0,1]\times [0,\infty )\rightarrow [0,\infty )$$ is continuous nonlinear function. Thank to classical Leggett–Williams fixed point theorem, appropriate conditions are established for the existence and multiplicity results. Further some useful results about Hyers–Ulam stability for the obtained solutions are also provided. The establish theory is properly justified by providing two examples.

In this manuscript, we investigate some appropriate conditions which ensure the existence of at least one solution to a class of fractional order differential equations (FDEs) provided by $$\textstyle\begin{cases} -{}^{C}\mathbf{D}^{q} z(t)=\theta (t,z(t)); \quad t\in \mathfrak{J}=[0,1], q\in (1, 2], \\ z(t)\vert _{t=0}=\phi (z),\qquad z(1)=\delta{}^{C}\mathbf{D}^{p} z(\eta ),\quad p, \eta \in (0,1). \end{cases} $$ The nonlinear function $\theta :\mathfrak{J}\times \mathbf{R} \rightarrow \mathbf{R}$ is continuous. Further, $\delta \in (0, 1)$ and $\phi \in C(\mathfrak{J},\mathbf{R})$ is a non-local function. We establish some adequate conditions for the existence of at least one solution to the considered problem by using Gronwall’s inequality and a priori estimate tools called the topological degree method. We provide two examples to verify the obtained results.

In this work, an optimal control strategy is developed to eliminate the spread of HIV-1. To do this, two control variables are used such as the efficaciousness of drug therapy in reducing the infection of new cells and decreasing the production of new viruses. Existence for the optimal control pair is accomplished and the Pontryagins Maximum Principle is used to characterize these optimal controls. Objective functional is constituted to minimize the densities of infected cells and free virus and to maximize the density of healthy cells. The optimality system is derived and solved numerically.

The aim of this thesis is to study complex nature of the Azerbaijani Foreign policy and to analyze relations of Azerbaijan with other countries in regional and global context in order to determine which countries are more prioritized. The author concluded that the balanced Foreign policy is no longer pursued and its orientation is now changed towards West. The study had also determined the reasons behind the Western orientation. Thesis concluded that Azerbaijan pursues economic integration with the West as well as sustains political dominance in the South Caucasus region.

In this study, asymptotic analysis of an HIV-1 epidemic model with distributed intracellular delays is proposed. One delay term represents the latent period which is the time when the target cells are contacted by the virus particles and the time the contacted cells become actively infected and the second delay term represents the virus production period which is the time when the new virions are created within the cell and are released from the cell. The infection free equilibrium and the chronic-infection equilibrium have been shown to be locally asymptotically stable by using Rouths Hurwiths criterion and general theory of delay differential equations. Similarly, by using Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction ratio is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable. Finally, numerical results with conclusion are discussed.

A mathematical model for HIV-1 infection with multiple delays is proposed. These delays account for (i) the delay in contact process between the uninfected cells virus, (ii) a latent period between the time target cells which are contacted by the virus particles and the time the virions enter the cells, and (iii) a virus production period for new virions to be produced within and released from the infected cells. For this model, the basic reproductive number is identified and its threshold property is discussed. The uninfected and infected steady states are shown to be locally as well as globally asymptotically stable. The value of the basic reproductive number shows that increasing any one of these delays will decrease this number. This may suggest a new direction for new drugs that can prolong the infection process and spreading of virus. The proved results have potential applications in HIV-1 therapy.