Mathematical Modelling And Optimal Control Mechanisms Of Some Infectious Diseases

Infectious diseases are one of the most significant burdens the world has ever faced in its entire his-tory.Most of the infectious diseases have cure and vaccines,but some have neither cure nor vaccines to protect humanity.The cure and vaccines of some infectious diseases do not prevent reinfection or stop the disease transmission.Also,the onset of resistance of some of the infectious disease agents calls for a better understanding of the process of disease transmission and development of optimal control strategies for prevention and control of infectious diseases.In this dissertation,the transmission dynamics of some infectious diseases are modelled as a sys-tem of ordinary differential equations.The mathematical models used in this thesis aim to improve the understanding of infectious disease transmission by analysing and simulating the optimal control measures to control infectious diseases.This dissertation uses novel mathematical models that are specific to the transmission agents and the mode of transmission of the disease.An explicit expression for the basic reproduction number R₀is obtained via the next generation method for all the models considered.The basic reproduction number is interpreted in biological terms,as well as regarding the reproductive numbers for each type of interaction involved.The stability of the disease-free equi-librium and the existence and uniqueness of the endemic equilibriums of the models considered are computed and analysed.A sensitivity analysis reveals the critical parameters that are most sensitive in R₀which leads to the formulation of the optimal control model for each of the models considered.The first study formulates a mathematical model for the transmission dynamics of schistosomiasis using a multi-scaled 12-dimensional system of ordinary differential equations that includes vector-host and within-host dynamics of infection.A sensitivity analysis indicates that R₀is most sensitive to the natural death rate of the vector population.The numerical simulations of the optimal control strategies reveal that the most effective strategy for the control and possible elimination of schisto-somiasis should combine sanitary measures?access to safe water,improved sanitation and hygiene education?,large-scale treatment of infected population and vector control measures for a significant amount of time.The second study formulates and analyses a metapopulation model which explicitly integrates vector-borne and sexual transmission of an epidemic disease with passive and active movements between an urban city,and a satellite city.The basic reproduction number of the disease is explicitly determined as a combination of sexual and vector-borne transmission parameters.The sensitivity analysis reveals that the disease is primarily transmitted via the vector-borne mode,rather than via sexual transmis-sion and that sexual transmission by itself may not initiate or sustain an outbreak.Also,increasing the population movements from one city to the other leads to an increase in the basic reproduction num-ber of the later city but a decrease in the basic reproduction number of the former city.The influence of other significant parameters is also investigated via the analysis of suitable partial rank correlation coefficients.After gauging the effects of mobility,we explore the potential effects of optimal control strategies relying upon several distinct restrictions on population movement.The third study deals with cigarette smoking on college campuses which has become a significant public health issue,which led to an increasing focus on establishing programs to reduce its preva-lence.In this study,a compartmental model depicting the spread and cessation of the smoking habit on college campuses,obtained using theoretical principles often employed in mathematical epidemi-ology,is proposed and analysed.The existence and stability of the habitual smoking-free and habitu-al smoking-persistent equilibria,respectively,are explored in terms of a threshold parameter,hereby called the smokers generation number and denoted by R_c.A sensitivity analysis indicates that R_cis the most sensitive to the contact rate between habitual-smokers and occasional-smokers and to the rate of successfully quitting smoking.Numerical simulations of the proposed optimal control strategies reveal that the most effective approach to reduce the prevalence of cigarette smoking and possibly achieve a smoking-free campus should combine both control measures,namely allocating mandatory smoking rooms together with educating the public on the harmful effects of smoking and providing large scale guidance,counselling and support therapy to help students quit smoking.The admission and graduation rates are considered as impulsive functions and used to formulate an impulsive model to analyse the dynamics of smoking cigarettes.It was realised from the simulation results that both the continuous and the impulsive models fit perfectly and present the same results.From the sensitivity analysis of the models in the first and second studies,it is observed that the basic reproduction number of most vector-borne diseases are most sensitive to the vector death rate,vector birth rate and the vector contact rate with the human population,which indicates that the most effective ways to control vector-borne diseases are to control the vector population and to reduce the contact rate between the human and vector population.A similar analogy is observed for human behavioural addictions.In the second study,it is realised that restriction of movements could be use to control the spread of an infectious disease but significant control measures should be aimed at the larger population patch for effective control of the disease.In the third study,the model's basic repro-duction number is most sensitive to the recovery rate of the addicts and the contact rate between the addicts and the non-addicts indicating that the most effective way to control human addictions is by treating the addicts to recover and by preventing contact between the addicts and the non-addicts.