In this paper, a mathematical model of breast cancer governed by a system of ordinary differential equations in the presence of chemotherapy treatment and ketogenic diet is discussed. Several comprehensive mathematical analysis was carried out using varieties of analytical methods to study the stability of the breast cancer model. Also, sufficient conditions on parameter values to ensure cancer persistence in the absence of anti-cancer drugs ketogenic diet and cancer emission when anti-cancer drugs, immune-booster, ketogenic diet are included were established. Furthermore, optimal control theory is applied to find out the optimal drug adjustment as an input control of the system therapies to minimize the number of cancerous cells by considering different controlled combinations of administering the chemotherapy agent and ketogenic diet using the popular Pontryagin's Maximum Principle. Numerical simulations were presented to validate our theoretical results.

Novel Coronavirus is a highly infectious disease, with over one million confirmed cases and thousands of deaths recorded. The disease has become pandemic, affecting almost all nations of the world, and has caused enormous economic, social and psychological burden on countries. Hygiene and educational campaign programmes have been identified to be potent public health interventions that can curtail the spread of the highly infectious disease. In order to verify this claim quantitatively, we propose and analyze a non-linear mathematical model to investigate the effect of healthy sanitation and awareness on the transmission dynamics of Coronavirus disease (COVID-19) prevalence. Rigorous stability analysis of the model equilibrium points was performed to ascertain the basic reproduction number R 0 , a threshold that determines whether or not a disease dies out of the population. Our model assumes that education on the disease transmission and prevention induce behavioral changes in individuals to imbibe good hygiene, thereby reducing the basic reproduction number and disease burden. Numerical simulations are carried out using real life data to support the analytic results.

In this paper, a deterministic mathematical model of the Dengue virus with a nonlinear incidence function in a population is presented and rigorously analysed. The model incorporates control measures at the aquatic and adult stages of the vector (mosquito). The stability of the system is analysed for the disease-free equilibrium and the existence of endemic equilibria under certain conditions. The local stability of the Dengue-free equilibrium is investigated via the threshold parameter (reproduction number) that was obtained using the next-generation matrix techniques. The Routh-Hurwitz criterion, along with Descartes' rule of signs change, established the local asymptotically stability of the model whenever R 0 < 1 and was unstable otherwise. The comparison theorem was used to establish the global asymptomatically stability of the model.

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