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In this article, a non-integer nonlinear mathematical model for toxoplasmosis disease in human and cat population is proposed and studied. The basic concepts of the model's dynamic are given. The study of qualitative dynamics is done by the basic threshold parameter $ R_{0} $ . Local and global stabilities are done and the system's disease free equilibrium point is an attractor when $ R_{0}1 $ . The sensitivity analysis of $ R_{0} $ shows which parameter has positive/negative impact on the model. Numerical simulation of the model for the parameters occurred in threshold parameter is also discussed. The techniques of Adams Bashforth Moulton will be considered to justify all the derived theoretical results which will help in understanding to study the effect of various parameters to both the transient and steady-state dynamics of the disease infection.

In this article, a stochastic SACR hepatitis B epidemic model is taken to be under consideration. We develop a stochastic epidemic model by considering the effect of environmental fluctuation on the hepatitis B dynamics and distribute the transmission rate by white noise. Using the stochastic Lyapunov function theory, we have shown the existence and uniqueness of the global positive solution. The extinction and persistence for our proposed model are derived with sufficient conditions. The numerical simulations are carried out using first-order Itô-Taylor stochastic scheme in the last section for the verification of our theoretical results.

In this paper, we formulate a stochastic model for hepatitis B virus transmission with the effect of fluctuation environment. We divide the total population into four different compartments, namely, the susceptible individuals in which the disease transmission rate is distributed by white noise, the acutely infected individuals in which the same perturbation occur, the chronically infected individuals and the recovered individuals. We use the stochastic Lyapunov function theory to construct a suitable stochastic Lyapunov function for the existence of positive solution. We also then establish the sufficient conditions for the hepatitis B extinction and the hepatitis B persistence. At the end numerical simulation is carried out by using the stochastic Runge–Kutta method to support our analytical findings.

Jordan quasigroups are commutative quasigroups satisfying the identity $x^{2}(yx)=(x^{2}y)x$. In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is idempotent, (iii) an idempotent entropic quasigroup satisfies Jordan's identity, (iv) a unipotent quasigroup Q satisfying Jordan's identity satisfies left nuclear square property, (vi) if a quasigroup satisfies LC identity, then alternativity ⇔ Jordan's identity, (vii) for a unipotent Jordan quasigroup Q, $x^{3}y=y^{3}x\ \forall \ x,y\in Q$ and (viii) every Steiner quasigroup is Jordan quasigroup. We also prove some properties of the autotopism of Jordan loops. Moreover, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6.

We propose an epidemic model for the transmission of hepatitis B virus along with the classification of different infection phases and hospitalized class. We formulate the model and discuss its basic mathematical properties, e.g. existence, positivity, and biological feasibility. Exploiting the next generation matrix approach, we find the basic reproductive number of the model. We perform sensitivity analysis to illustrate the effect of various parameters on the transmission of the disease. We investigate stability of the equilibria of the model in terms of the basic reproduction number. Conditions for the stability of the proposed model are obtained using various approaches. Finally, we perform the numerical simulations to discuss sensitivity analysis and to support our analytical work.

This study proposes a mathematical model of Anthroponotic visceral leishmaniasis epidemic with saturated infection rate and recommends different control strategies to manage the spread of this disease in the community. To do this, first, a model formulation is presented to support these strategies, with quantifications of transmission and intervention parameters. To understand the nature of the initial transmission of the disease, the reproduction number $ R_0 $ is obtained by using the next-generation method. On the basis of sensitivity analysis of the reproduction number $ R_0 $ , four different control strategies are proposed for managing disease transmission. For quantification of the prevalence period of the disease, a numerical simulation for each strategy is performed and a detailed summary is presented. Disease-free state is obtained with the help of control strategies. The threshold condition for globally asymptotic stability of the disease-free state is found, and it is ascertained that the state is globally stable. On the basis of sensitivity analysis of the reproduction number, it is shown that the disease can be eradicated by using the proposed strategies.

In this article, we present the transmission dynamic of the acute and chronic hepatitis B epidemic problem and develop an optimal control strategy to control the spread of hepatitis B in a community. In order to do this, first we present the model formulation and find the basic reproduction number $ R_0 $ . We show that if $ R_0\leq 1, $ then the disease-free equilibrium is both locally as well as globally asymptotically stable. Then, we prove that the model is locally and globally asymptotically stable, if $ R_0>1 $ . To control the spread of this infection, we develop a control strategy by applying three control variables such as isolation of infected and non-infected individuals, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis B individuals and maximize the number of susceptible and recovered individuals. Finally, we present numerical simulation to illustrate the feasibility of the control strategy.