MODELING AND SIMULATION OF INFECTIOUS DISEASE USING FRACTIONAL CALCULUS

Abstract

In recent years, the world has faced a sharp rise in infectious diseases, which continue to be a serious threat to public health. Despite progress in medical science, surveil- lance systems, and control measures, outbreaks such as influenza, SARS, and most recently COVID-19 have shown that our societies remain highly vulnerable. These events have also revealed some of the limitations of the classical models used to study and predict the spread of infections. In particular, standard models often ignore mem- ory effects, individual behaviour, and environmental influences. To overcome these gaps, this thesis applies fractional calculus in the modeling and simulation of infec- tious diseases. Fractional-order models have the advantage of incorporating memory and history, which makes them more realistic for studying epidemics where past expo- sure, immunity, and behavioural changes play an important role. The work begins with a study of vaccination strategies followed in five countries that were badly affected during the first half of 2022: the USA, India, Brazil, France, and the UK. A detailed comparison shows that most countries gave priority first to frontline workers and health professionals, and then to elderly or immunocompromised people. The main difference was how countries divided the age groups for priority. By comparing these strategies with confirmed cases and deaths per population, as well as with population density and median age, the study highlights how vaccine distribution policies must be designed carefully to suit the demographics of each country. Motivated by these findings, different fractional-order models are developed in this thesis. The first is an SIS model with Beddington-De Angelis incidence, used to capture the effect of fear-driven behaviour. When people become afraid of infection, they may self-isolate or reduce contact with others. Such actions can strongly influence disease spread, and fractional calculus is especially suitable to model this because fear and behaviour are shaped by past experiences. A second contribution is an SVIR model that divide vaccinated people into two groups: partially vaccinated (those who did not complete the prescribed course of the doses) and fully vaccinated (those who completed the vaccination schedule and followed health guidelines). This distinction is important, as many people worldwide xiii xiv ACKNOWLEDGMENTS showed hesitancy in taking vaccines, often due to doubts about safety or mistrust of governments. The model allows us to study how partial vaccination affects recovery compared with full vaccination, giving a clearer picture of real vaccination outcomes. The thesis also extends the SEIQR model by including two realistic features: psy- chological effects during transmission (using Monod-Haldane incidence) and a limited quarantine capacity (Holling type-III function). These changes reflect how quarantine in practice cannot be increased indefinitely and is often constrained by resources. An associated fractional optimal control problem is studied using Pontryagin’s principle, showing how time-dependent controls can be used to reduce infections at minimum cost. Beyond vaccination and quarantine, the thesis considers environmental effects. A Susceptible-Pollution affected-Infected-Recovered (SPIR) model is proposed to study how exposure to pollutants weakens immunity and increases vulnerability to infec- tions. This model even accounts for prenatal exposure in newborns, reflecting the long-term consequences of pollution. A fractional optimal control problem with two controls is solved to examine how information campaigns and other interventions can help reduce infections in polluted environments. Another area studied is the role of bacteria. Due to rising household waste and urbanization, bacterial populations in the environment are growing, leading to more bacterial and vector-borne diseases. To address this, a fractional SIR model with bac- teria in the environment and in organisms is developed. An optimal control problem with three controls is analyzed to show how disease transmission can be reduced effi- ciently. Across all these models, the unifying theme is the use of fractional-order sys- tems. By including memory, they allow us to model more realistic epidemic be- haviours, whether due to human psychology, environmental stress, or bacterial growth. Numerical simulations are carried out using the Adams-Bashforth-Moulton predictor- corrector method, which validates the theoretical results and demonstrates how the models behave under different conditions. In summary, this thesis presents a set of new fractional-order models that bring together vaccination strategies, fear and behaviour, quarantine measures, environmen- tal pollution, and bacterial effects in infectious disease dynamics. The results show that fractional models are not only mathematically richer but also practically more meaningful, as they reflect the role of memory and history in epidemic processes. By combining theory, simulations, and control strategies, the thesis provides insights that can support better decision-making in managing infectious diseases and preparing for future outbreaks.