Fractional optimal control of childhood pneumonia disease

Pneumonia is a infectious lung inflammation caused by pathogens such as viruses or bacteria. It is one of the diseases that cause the most deaths in human history. Moreover, the impact of this on countries and households cannot be underestimated, as most individuals afflicted with the disease are children aged 0-5. Especially in recent years, mathematical epidemiology, which is complementary to clinical studies, has had a wide research area and interest. The first aim of the thesis is to explain the dynamics of the spread of pneumonia in a population of children with a mathematical model, considering the state of breastfeeding, which creates a strong immunity against the disease. The crossover behavior of many diseases may render integer degree derivatives and fractional derivatives with singular kernels inadequate for modelling. Therefore, the Atangana-Baleanu fractional derivative provides better insight into disease evolution at the beginning and end of disease spread, thanks to the nonsingular Mittag-Leffler kernel. Based on this, the model on which the optimal control problem is built is equipped with the Atangana-Baleanu derivative. The main purpose of this thesis is to discuss the effects of four control strategies such as preventive measures, vaccination, screening and treatment in preventing the epidemic of the disease. It is aimed to minimize both the proportion of carriers and infected children and the costs of control measures. Analysis, stability and numerical solutions of the model with control interventions are presented.