## 1 Introduction  
In a large variety of ecosystems and in humans, which are the most affected species, epidemics are known to pose a significant threat to living organisms. Every year, millions of humans die from epidemics, which has motivated researchers in mathematical and biological sciences to develop control and mitigation strategies in the fight against epidemic diseases. Using various mathematical modeling techniques, epidemiologists have gained deeper insights into the dynamics by using this mathematical and cost-effective analytical setup, whereas traditional experimental techniques are time-consuming and more expensive. This has encouraged many mathematicians and biologists to develop competent epidemic models that can present a vivid picture of the reality (see, for details, [1, 2, 3, 4, 5, 6]). The first SIR model describing the dynamics of the three principal populations: the susceptible \((S)\), the infected \((I)\) and the recovered \((R)\), was studied by [PERSON] and [PERSON] in 1927 (see [7]). Their proposed model played a crucial role in initiating different research works in the field of disease dynamics. Based upon the natural history of hepatitis C, a recent study proposed a Susceptible-Infectious-Chronic-Recovered (SICR) type model (see [8]). A natural phenomenon is always affected by environmental factors that can aggravate or mitigate the spread of the epidemic. The stochastic quantification of several real-lifephenomena has been immensely helpful in understanding the random nature of their incidence or occurrence. It has also helped to find solutions to those problems that arise from it, either in the form of minimizing their undesirability or maximizing their rewards. In addition, infectious diseases are subject to chance and uncertainty in terms of the normal progression of the infection. Therefore, stochastic models are more suitable when compared to deterministic models, keeping in view the fact that stochastic systems consider not only the moving mean, but also the standard deviation behavior around it. On the other hand, deterministic systems generate similar results for fixed initial values, but stochastic systems may give different predicted results. Several stochastic infection models describing the effect of the Brownian motion on viral dynamics have been developed in the literature (see, for example, [9; 10; 11; 12; 13]; see also [14]). Motivated essentially by the above-mentioned and other recent developments (see [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]), we propose here a mathematical model for the transmission of an SICR model. Our goal is to show the effect of the Levy jump in population dynamics, in which the Levy noise is used to describe contingency and outburst. It will, therefore, be interesting to consider the following stochastic model jointly driven by white noise and Levy noise:  
\[\left\{ \begin{aligned} dS(t)=&[\lambda-\beta I(t)S(t )-\mu S(t)]dt-\sigma I(t)S(t)dW_{t}\\ &-\int_{U}Q(u)I(t-)S(t-)\dot{N}(dt,du),\\ dI(t)=&[\beta I(t)S(t)-(\gamma_{1}+\gamma_{2}+\mu)I(t )+\varepsilon R(t)+\delta C(t)]dt+\sigma I(t)S(t)dW_{t}\\ &+\int_{U}Q(u)I(t-)S(t-)\dot{N}(dt,du),\\ dC(t)=&[\gamma_{2}I(t)-(\delta+\mu+\vartheta)C(t)] dt,\\ dR(t)=&[\gamma_{1}I(t)-(\varepsilon+\mu)R(t)]dt, \end{aligned} \right. \tag{1}\]  
where \(S(t)\) denotes the number of susceptible individuals at time \(t\). The infected individuals with no clinical symptoms (that is, the virus is living or developing in the individuals, but without producing symptoms or only mild ones) are able to transmit the infection to other individuals \(I(t)\). The infected individuals under treatment (that is, those in the so-called chronic stage) with a viral load remaining low \(C(t)\), and \(R(t)\) denotes the number of the recovered individuals at time \(t\).