Abstract
Understanding the dynamics of an infectious disease, such as malaria, helps us to reduce the number of deaths and to achieve better control of its spread. Mathematical models can help to predict the outcomes of new ideas for containing disease spread. This thesis constitutes an extension of previous attempts to understand, via mathematical modelling, the dynamics of mosquito populations and malaria. We propose 5 distinct non-linear age dependent mathematical models. The first model uses, as a starting point, an approach to modelling nonlinear effects in agestructured models due to Gurtin and MacCamy [31]. However, the model we derive (which takes the form of a system of delay differential equations) is much more complex. We demonstrate analytical results on linear stability of both zero and positive equilibria in various cases. We then examine a more complex equation which incorporates competition among larval mosquitoes. Furthermore, results on boundedness of solutions and on the existence of positive equilibria are proved. Numerical simulations show that for specific values of several parameters three equilibrium points can be achieved as well as that the equilibria decrease as we increase the larval competition coefficient. In the second mathematical model, we examine a neutral delay differential equation. This specific type of equation is consequent upon the assumption that an adult mosquito lays a batch of eggs immediately upon maturation, followed possibly by further batches (not necessarily containing the same number of eggs) on reaching the particular ages τi + nτ, n = 1,2,..., with no egg laying in between these ages. This models the idea that egg laying follows blood meals. A particular case is the case when adult mosquito lays all of its eggs immediately on maturation, and none at all later in life. In that case the non-trivial equilibrium is locally stable but the roots of the characteristic equation are not bounded away from the imaginary axis. More generally, with adults laying eggs at ages τi +nτ, we may show under some conditions that the unique positive equilibria is linearly stable. Results on the existence of positive equilibria and boundedness of solutions are proved in this general case. Moving to the third mathematical model of the thesis, in Chapter 4, we examine two strains for the mosquito population, the vulnerable and the resistant strain. This model is based on the assumption that mosquitoes may become resistant to insecticides. One particular idea that we examine is the possibility that the parameter values such as the per-capita death rate, maturation time and the kernel g(a) which describes the adult mosquito egg laying activity are different in the two strains. We present analytical results on the global stability of the zero equilibrium and the linear stability of the boundary equilibrium. Numerical simulations show that for several parameter values either of the two strains can win the competition and drive the other one to extinction. In Chapter 5, the fourth mathematical model that we propose has similarities to the model of Chapter 4 but also allows the possibility of an adult vulnerable mosquito to die due to the insecticide. We propose a model for the case of an insecticide that attacks a mosquito with increasing potency as it ages, eventually giving us a system of four-integral equations. We compare two kinds of insecticides, late-life acting (LLA) insecticides and conventional insecticides, and try to find under what circumstances the LLA insecticide will slow down the evolution of insecticide resistance. The final part of the thesis examines the interaction between the host (human) and the vector (mosquito). Our fifth model provides analytical and numerical results for an eight-dimensional system of equations, consisting of two differential equations and six integral equations. For this model we find a set of conditions sufficient for the eradication of malaria.