Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Xingfu Zou

Abstract

In this thesis, we use mathematical models to study the problems about the evolution of hosts and parasites. Firstly, we study a within-host age-structured model with mutation and back mutation which is in the form of partial differential equations with double-infections by two strains of viruses. For the case when the production rates of viruses are gamma distributions, the PDE model can be transferred into an ODE one. Then, we analyze our model in two cases: one is without mutation, and the other is with mutation. In the first case, we prove that the two strains of viruses without mutation would die out if both of the individual reproductive numbers are less than one; otherwise, their evolution will comply with competitive exclusion principle meaning that the stronger one will survive finally. In the second case, we verify that they can coexist under some specific conditions in the sense that there exists a coexistence equilibrium which is globally asymptotically stable.

Secondly, we explore the viral evolutionary strategies by using a within-host model under body immune response. We consider two types of trade-offs involving the viral production rate, the host death rate caused by infection (i.e., virulence), and the transmission rate. By choosing appropriate fitness, we show that the evolutionary and convergent stability of an evolutionary singular strategy can ne affected by the shapes of the trade-off functions. We also find that the evolutionary branching may occur at the singular strategy for some special trade-off functions. The results imply that the immune response has an important effect on viral evolution. Finally, two classes of trade-off functions are specified which yield some more detailed information on the virus evolutionary strategies.

Thirdly, we investigate the cost of immunological up-regulation caused by infection in a between-host transmission dynamical model with superinfection, which describes disease transmission between a single host and two parasites. After introducing mutant hosts to original model, we explore this problem in two cases: (A) monomorphic case; (B) dimorphic case. For (A), mutant hosts have two possible infections: one is by parasite $1$; the other is by parasite $2$. In each of these two cases, we identify an appropriate fitness for the invasion of the mutant hosts by analyzing the local stability of the mutant free equilibrium. Then, We consider the trade-off between the production rate of infected hosts and their recovery rate. By employing the adaptive dynamical approach, we analyze the evolutionary stability and convergence stability of this singular point, leading to some the conditions for continuously stable strategy, evolutionary branching point and repeller. For (B), we define a new fitness to measure the invasion of mutant hosts with parasite $1$ and $2$ by the same method. When the trade-off function is chosen to be linear, we are able to obtain conditions for isoclinic stability and absolute convergence stability through simulations. We find that although immune response is benign to hosts, the host evolution would not favor high degree of immunological up-regulation, implying that an intermediate degree of immunological response will be helpful to the host evolution. Moreover, superinfection would help weaker virulent parasite exist in hosts.

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