The present paper describes the mathematical modeling and dynamics of a novel corona virus (2019-nCoV). We describe the brief details of interaction among the bats and unknown hosts, then among the peoples and the infections reservoir (seafood market). The seafood marked are considered the main source of infection when the bats and the unknown hosts (may be wild animals) leaves the infection there. The purchasing of items from the seafood market by peoples have the ability to infect either asymptomatically or symptomatically. We reduced the model with the assumptions that the seafood market has enough source of infection that can be effective to infect people. We present the mathematical results of the model and then formulate a fractional model. We consider the available infection cases for January 21, 2020, till January 28, 2020 and parameterized the model. We compute the basic reproduction number for the data is R 0 ≈ 2.4829 . The fractional model is then solved numerically by presenting many graphical results, which can be helpful for the infection minimization.

Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative

In this paper, we study on the conformable (2+1)-dimensional Ablowitz-KaupNewell-Segur equation in order to show the existence of complex combined dark-bright soliton solutions. To this purpose an effective method which is the sine-Gordon expansion method is used. The 2D and 3D surfaces under some suitable values of parameters are also plotted.

Complex solitons in the conformable (2+1)-dimensional Ablowitz-Kaup-Newell-Segur equation

In the present paper, we formulate a new mathematical model for the dynamics of COVID-19 with quarantine and isolation. Initially, we provide a brief discussion on the model formulation and provide relevant mathematical results. Then, we consider the fractal-fractional derivative in Atangana–Baleanu sense, and we also generalize the model. The generalized model is used to obtain its stability results. We show that the model is locally asymptotically stable if 

$\mathcal{R}_{0}<1$
. Further, we consider the real cases reported in China since January 11 till April 9, 2020. The reported cases have been used for obtaining the real parameters and the basic reproduction number for the given period, 

$\mathcal{R}_{0}\approx 6.6361$
. The data of reported cases versus model for classical and fractal-factional order are presented. We show that the fractal-fractional order model provides the best fitting to the reported cases. The fractional mathematical model is solved by a novel numerical technique based on Newton approach, which is useful and reliable. A brief discussion on the graphical results using the novel numerical procedures are shown. Some key parameters that show significance in the disease elimination from the society are explored.

/pdf/the-dynamics-of-covid-19-with-quarantined-and-isolation-37uyp99ee5.pdf

The dynamics of COVID-19 with quarantined and isolation

In this paper, a nonlinear fractional order epidemic model for HIV transmission is proposed and analyzed by including extra compartment namely exposed class to the basic SIR epidemic model Also, the infected class of female sex workers is divided into unaware infectives and the aware infectives The focus is on the spread of HIV by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the HIV transmission The exposed class contains those susceptible males in the population who have sexual contact with the female sex workers and are exposed to the infection directly or indirectly The Caputo type fractional derivative is involved and generalized Adams-Bashforth-Moulton method is employed to numerically solve the proposed model Model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle Analysis of the model demonstrates that the population is free from the disease if R 0 1 and disease spreads in the population if R 0 > 1  Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed Furthermore, for the fractional optimal control problem associated with the control strategies such as condom use for exposed class, treatment for aware infectives, awareness about disease among unaware infectives and behavioral change for susceptibles, we formulated a fractional optimality condition for the proposed model The existence of fractional optimal control is analyzed and the Euler-Lagrange necessary conditions for the optimality of fractional optimal control are obtained The effectiveness of control strategies is shown through numerical simulations and it can be seen through simulation, that the control measures effectively increase the quality of life and age limit of the HIV patients It significantly reduces the number of HIV/AIDS patients during the whole epidemic

/pdf/global-dynamics-of-a-fractional-order-model-for-the-1ucmhdwvi4.pdf

Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control

Number of well-known contagious diseases exist around the world that mainly include HIV, Hepatitis B, influenzas etc., among these, a recently contested coronavirus (COVID-19) is a serious class of such transmissible syndromes. Abundant scientific evidence the wild animals are believed to be the primary hosts of the virus. Majority of such cases are considered to be human-to-human transmission, while a few are due to wild animals-to-human transmission and substantial burdens on healthcare system following this spread. To understand the dynamical behavior such diseases, we fitted a susceptible-infectious-quarantined model for human cases with constant proportions. We proposed a model that provide better constraints on understanding the climaxes of such unseen disastrous spread, relevant consequences, and suggesting future imperative strategies need to be adopted. The main features of the work include the positivity, boundedness, existence and uniqueness of solution of the model. The conditions were derived under which the COVID-19 may extinct or persist in the population. Sensitivity and estimation of those important parameters have been carried out that plays key role in the transmission mechanism. To optimize the spread of such disease, we present a control problem for further analysis using two control measures. The necessary conditions have been derived using the Pontryagin's maximum principle. Parameter values have been estimated from the real data and experimental numerical simulations are presented for comparison as well as verification of theoretical results. The obtained numerical results also present the verification, accuracy, validation, and robustness of the proposed scheme.

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China.

It is prominent that memory has a prodigious influence on the development of every process associated with human societies. More specifically, the growth of an epidemic process is directly associated with the individuals’ experiences. In fact, the real epidemic process is obviously sustained by non-Markovian dynamics: heredity properties and memory effects perform a critical role in the subsequent spread of infection. These additional properties increase the accuracy and reliability of fractional order systems than the other ordinary systems. In this current study, a dengue infection model with asymptomatic carriers through Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) fractional derivatives is introduced. Analytic skills are used to obtain the basic reproduction number for the proposed dengue model, denoted by R0. We use partial rank correlation coefficient (PRCC) method to detect the effect of input parameters on the outcomes of R0. In addition, we have proven sufficient condition for the existence and uniqueness of solution for the suggested fractional dynamics of dengue infection. To explore the intricate dynamics of dengue infection with the effect of asymptomatic carriers, we perform numerical simulations of the suggested dengue model by varying the fractional order l. Fractional order model offers realistic information about the dynamics of the suggested dengue model and sharply decrease infected individuals by decreasing the fractional order parameter l for the case of Caputo–Fabrizio model while a rapid decrease in the case of Atangana–Baleanu model. We show that the Atangana–Baleanu model gives good decrease for the infected compartments in case of decreasing the fractional order parameter than that of the Caputo–Fabrizio model. It is shown that the asymptomatic fraction can be greatly decreased by decreasing the parameter l. Furthermore, the influence of the biting rate of mosquitoes on infected humans is investigated numerically, and it is suggested to the control policymakers that controlling the biting rate can significantly reduce the level of dengue infection.

Modeling the transmission of dengue infection through fractional derivatives

Asymptomatic carriers in transmission dynamics of dengue with control interventions

Effect of partial immunity on transmission dynamics of dengue disease with optimal control

Human immunodeficiency virus (HIV) is a life life-threatening and serious infection caused by a virus that attacks CD4+ T-cells, which fight against infections and make a person susceptible to other diseases. It is a global public health problem with no cure; therefore, it is highly important to study and understand the intricate phenomena of HIV. In this article, we focus on the numerical study of the path-tracking damped oscillatory behavior of a model for the HIV infection of CD4+ T-cells. We formulate fractional dynamics of HIV with a source term for the supply of new CD4+ T-cells depending on the viral load via the Caputo–Fabrizio derivative. In the formulation of fractional HIV dynamics, we replaced the constant source term for the supply of new CD4+ T-cells from the thymus with a variable source term depending on the concentration of the viral load, and introduced a term that describes the incidence of the HIV infection of CD4+ T-cells. We present a novel numerical scheme for fractional view analysis of the proposed model to highlight the solution pathway of HIV. We inspect the periodic and chaotic behavior of HIV for the given values of input factors using numerical simulations.

https://www.mdpi.com/1420-3049/26/6/1806/pdf

Fractional Dynamics of HIV with Source Term for the Supply of New CD4+ T-Cells Depending on the Viral Load via Caputo–Fabrizio Derivative

In this research paper, a novel approach in dengue modeling with the asymptomatic carrier and reinfection via the fractional derivative is suggested to deeply interrogate the comprehensive transmission phenomena of dengue infection. The proposed system of dengue infection is represented in the Liouville–Caputo fractional framework and investigated for basic properties, that is, uniqueness, positivity, and boundedness of the solution. We used the next-generation technique in order to determine the basic reproduction number R 0 for the suggested model of dengue infection; moreover, we conduct a sensitivity test of R 0 through a partial rank correlation coefficient technique to know the contribution of input factors on the output of R 0. We have shown that the infection-free equilibrium of dengue dynamics is globally asymptomatically stable for R 0 < 1 and unstable in other circumstances. The system of dengue infection is then structured in the Atangana–Baleanu framework to represent the dynamics of dengue with the non-singular and non-local kernel. The existence and uniqueness of the solution of the Atangana–Baleanu fractional system are interrogated through fixed-point theory. Finally, we present a novel numerical technique for the solution of our fractional-order system in the Atangana–Baleanu framework. We obtain numerical results for different values of fractional-order ϑ and input factors to highlight the consequences of fractional-order ϑ and input parameters on the system. On the basis of our analysis, we predict the most critical parameters in the system for the elimination of dengue infection.

Rashid Jan

Papers

Modeling the transmission of dengue infection through fractional derivatives

Asymptomatic carriers in transmission dynamics of dengue with control interventions

Effect of partial immunity on transmission dynamics of dengue disease with optimal control

Fractional Dynamics of HIV with Source Term for the Supply of New CD4+ T-Cells Depending on the Viral Load via Caputo–Fabrizio Derivative

Fractional-calculus analysis of the transmission dynamics of the dengue infection.