Jatropha curcas is a plant with a variety of potential and ecological applications. The seeds of this plant contain a high amount of oil that can be used to obtain a better quality of alternative fuel biodiesel. But the Jatropha plants are seriously affected by the mosaic virus (Begomovirus) that is carried by infected vector whiteflies. It severely affects the Jatropha plants by causing leaf damage, yellowing leaves and sap drainage. In particular, it attacks its fruits considerably reducing the production of seeds. In this paper, we formulate a model for the dynamics of this disease and its possible control via insecticide spraying. We identify the parameters that are most important for vector-borne disease control. Pontryagin minimum principle is employed to minimize the cost of spraying. The findings indicate that the optimal spraying policy does not require insecticide application during the first ten days of the epidemic outbreak and that instead the spraying must be continued for the following three months to eradicate the disease.

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In this research article, we establish a fractional-order mathematical model to explore the infections of the coronavirus disease (COVID-19) caused by the novel SARS-CoV-2 virus. We introduce a set of fractional differential equations taking uninfected epithelial cells, infected epithelial cells, SARS-CoV-2 virus, and CTL response cell accounting for the lytic and non-lytic effects of immune responses. We also include the effect of a commonly used antiviral drug in COVID-19 treatment in an optimal control-theoretic approach. The stability of the equilibria of the fractional ordered system using qualitative theory. Numerical simulations are presented using an iterative scheme in Matlab in support of the analytical results.

Research reveals that most HIV infections come from undiagnosed people and untreated people, but principally from unaware people. Human awareness results in the reduction of susceptibility to infection, naturally, in the epidemiological study this factor should be included. Researchers have produced a wealth of information about the disease, including a number of critical tools and interventions to diagnose, prevent and treat HIV. Broadcast media have tremendous reach and influence, particularly with young people, so in this study, we incorporate the influence of media and investigate the effect of awareness program in disease outbreaks. We divided the whole susceptible and infected populations into two sub-classes: 'unaware class' and 'aware class' . A detailed mathematical analysis has been carried out and numerical simulations have been performed to show the role of some of the important model parameters.