Mathematical Analysis And Optimization Strategies Of Several Types Of Infectious Disease Models

Infectious diseases threaten the lives and livelihood of people all over the world.The development of mathematical models helps to understand the var-ious transmission paths,and the means to control the spread of these diseases.The dissertation formulates new compartmental models to prevent/eradicate the spread of COVID-19,bacterial meningitis and query fever(Q fever)and gives optimal control strategies,which complements the existing models.Eco-nomic evaluation is pivotal in controlling infectious diseases,and policymakers need information about the effectiveness of a control intervention to evaluate the cost-benefits of intervention measures.The outbreak of COVID-19 in De-cember 2019 has pushed governments and health authorities to take drastic measures to curb the spread of COVID-19.Bacterial meningitis is one of the major causes of death in sub Saharan Africa and Q fever is a bacterial disease caused by Coxiella burnetii(C.burnetii),which is very infectious and very durable in the environment.Therefore,for these three infectious diseases,we formulate a series of dynamical models to study their dynamical behaviours and related optimal control measures.The main contents are as follows:(1)For CO VID-19 in Ghana,I collaborated with a student from the African Institute for Mathematical studies.Thus,we build the dynamical model con-sidering the transmission of human to human and between human and environ-ment.We calculate the basic reproduction number without control and apply Lyapunov's function to analyze the global stability of the proposed model.We fit the model to real data from Ghana with the aid of python programming language using the least-squares method.The average basic reproduction num-ber without controls is approximately 2.68.Based on the sensitivity analysis,we formulated optimal measures to verify the theoretical results numerically.We finally obtain the optimal cost-benefits strategy of six non-pharmaceutical control interventions.(2)We established a deterministic COVID-19 model incorporating im-migrant and local susceptibility,we found that when the basic reproduction number R0 is less than one,disease-free equilibrium is globally stable,and when R0>1,the endemic equilibrium is globally stable.We further found that COVID-19 spread is highly sensitive to personal protection and personal hygiene.(3)We established COVID-19 transmission model which includes sus-ceptible,asymptomatic,isolated,severe infected,recovered,dead and self-protected.We used the real data in Ghana and Egypt to fit the model,estimated the parameters and carried out sensitivity analysis.It was found that the imple-mentation of strict health protection measures over a long period can prevent multiple peaks in Ghana and Egypt.Sensitivity analysis showed that the en-hanced exposure tracking,isolation,wearing masks,etc.,were still the most effective measure in reducing the infection.(4)We propose a mathematical model for bacterial meningitis to include nonlinear recovery rate.The studies show instances for forward and backward bifurcation.We use Latin hypercube sampling to test for influential parameters in the basic reproduction number,R0.We use sensitivity heat map and param-eter sensitivity spectrum to test for the group sensitivity of all the state variables and parameters.The sensitivity heat map shows that the most sensitive state variable to all parameters in the model during none seasonal transmission is the recovery class followed by the susceptible class;and that the most sensitive state variable during seasonal transmission is the susceptible class followed by the carrier-class.Meanwhile,we use optimal control theory to study the impact of inoculation in an endemic setting with a limited number of antibiotics and hospital beds.The work provides a potential framework for the control of the disease spread in limited-resource settings.Potential extensions of the model are likewise examined.(5)We established a dynamical model describing the propagation of Q fever.Using the matrix theory and Lyapunov functions we prove the local and global asymptotic stability of the equilibrium point.We studied the optimal strategy of vaccination,environmental disinfection and infected cattle elimina-tion.The results provide theoretical guidance for animal epidemic prevention and the cost-benefits of the control measures.(6)We finally studied the transmission dynamics of Q fever with peri-odic transmission rate as well as different recovered rates and treatment rates.The control reproduction number under different conditions is calculated,and the Lyapunov function is used to study the global asymptotic stability of the non-seasonal model.For the seasonal model,the existence of positive periodic solutions is proved.We also studied the optimized control of time-dependent treatment,disinfection,and separate facilities for animal birthing.We calcu-lated the average and incremental cost-effectiveness,which provides theoreti-cal supports for other control interventions of Q fever in livestock.