In this paper, we have modified the White and Comiskey heroin epidemic model (White and Comiskey in Math. Biosci. 208:312-324, 2007) into a nonautonomous heroin epidemic model with distributed time delay. We have introduced some new threshold values R * and R * and further obtained that the heroin-using career will be permanent when R * > 1 and the heroin-using career will be going to extinct when R * < 1. Using the method of Lyapunov functional, some sufficient conditions are derived for the global asymptotic stability of the system. The aim of this modification is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

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We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune components-helper (resting) T-cells which stimulate CTLs and convert them into active (hunting) CTL cells and active (hunting) CTL cells which attack, destroy, or ingest the tumor cells. In our model there are four compartments, namely, tumor cells, active CTL cells, helper T-cells, and chemotherapeutic drug. We have discussed the behaviour of the solutions of our system. The dynamical behaviour of our system by analyzing the existence and stability of the system at various equilibrium points is discussed elaborately. We have set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic drug administration. Here we used a quadratic control to quantify this goal and have considered the administration of chemotherapy drug as control to reduce the spread of the disease. The important mathematical findings for the dynamical behaviour of the tumor-immune model with control are also numerically verified using MATLAB. Finally, epidemiological implications of our analytical findings are addressed critically.

This paper aims to study an SIR epidemic model with an asymptotically homogeneous transmission function. The stability of the disease-free and the endemic equilibrium is addressed. Numerical simulations are carried out. Implications of our analytical and numerical findings are discussed critically.

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COVID-19 has spread around the world since December 2019, creating one of the greatest pandemics ever witnessed. According to the current reports, this is a situation when people need to be more careful and take the precaution measures more seriously, unless the condition may become even worse. Maintaining social distances and proper hygiene, staying at isolation or adopting the self-quarantine method are some of the common practices that people should use to avoid the infection. And the growing information regarding COVID-19 and its symptoms help the people to take proper precautions. In this present study, we consider an SEIRS epidemiological model on COVID-19 transmission which accounts for the effect of an individual's behavioural response due to the information regarding proper precautions. Our results indicate that if people respond to the growing information regarding awareness at a higher rate and start to take the protective measures, then the infected population decreases significantly. The disease fatality can be controlled only

In the present work we have studied a prey–predator model with logistic growth of the prey in the absence of the predator. We have also considered the fear effect and have investigated the impact of fear of the predator on prey when the predator is provided additional food. Functional responses of the predator towards prey and additional food are derived in this text. Death rates of both prey and predator have been considered as stochastic parameters due to the effect of the fluctuating environment. Existence and uniqueness, boundedness and uniform continuity of the global positive solution of the proposed model have been established. The conditions for extinction and persistence of the system have been derived. In the investigation, it is found that environmental noise plays a vital role in extinction as well as in persistence. Our analytical derivations are justified through numerical simulations which show the reliability of the model from the ecological point of view. We have also investigated the impact of intense fear as well as the absence of fear on this model by numerical simulation. Several interesting numerical results have been obtained based on different fear functions.

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