# Study of Virus Dynamics by Mathematical Models

## Degree

Doctor of Philosophy

## Program

Applied Mathematics

Xingfu Zou

## Abstract

This thesis studies virus dynamics within host by mathematical models, and topics discussed include viral release strategies, viral spreading mechanism, and interaction of virus with the immune system.

Firstly, we propose a delay differential equation model with distributed delay to investigate the evolutionary competition between budding and lytic viral release strategies. We find that when antibody is not established, the dynamics of competition depends on the respective basic reproduction numbers of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for both strains but the neutralization capacities are the same for both strains, consequence of the competition also depends only on the basic reproduction numbers of the budding and lytic viruses. Using two concrete forms of the viral production functions, we are also able to conclude that budding virus will outcompete if the rates of viral production, death rates of infected cells and neutralizing capacities of the antibodies are the same for budding and lytic viruses. In this case, budding strategy would have evolutionary advantage. However, if the antibody neutralization capacity for the budding virus is larger than that for the lytic virus, lytic virus can outcompete provided that its reproductive ratio is very high. An explicit threshold is derived.

Secondly, we consider model containing two modes for viral infection and spread, one is the diffusion-limited free virus transmission and the other is the direct cell-to-cell transfer of viral particles. By incorporating infection age, a rigorous analysis of the model shows that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number, which is identified explicitly. The formula for the basic reproduction number of our model reveals the effects of various model parameters including the transmission rates of the two modes, and the impact of the infection age. We show that basic reproduction number is underestimated in the existing models that only consider the cell-free virus transmission, or the cell-to-cell infection, ignoring the other. Assuming logistic growth for target cells, we find that if the basic reproduction number is greater than one, the infection can persist and Hopf bifurcation can occur from the positive equilibrium within certain parameter ranges.

Thirdly, the repulsion effect of superinfecting virion by infected cells is studied by a reaction diffusion equation model for virus infection dynamics. In this model, the diffusion of virus depends not only on its concentration gradient but also on the concentration of infected cells. The basic reproduction number, linear stability of steady states, spreading speed, and existence of traveling wave solutions for the model are discussed. It is shown that viruses spread more rapidly with the repulsion effect of infected cells on superinfecting virions, than with random diffusion only. For our model, the spreading speed of free virus is not consistent with the minimal traveling wave speed. With our general model, numerical computations of the spreading speed shows that the repulsion of superinfecting vision promotes the spread of virus, which confirms, not only qualitatively but also quantitatively, some recent experimental results.

Finally, the effect of chemotactic movement of CD8+ cytotoxic T lymphocytes (CTLs) on HIV-1 infection dynamics is studied by a reaction diffusion model with chemotaxis. Choosing a typical chemosensitive function, we find that chemoattractive movement of CTLs due to HIV infection does not change stability of the positive steady state of the model. However, chemorepulsion movement of CTLs destabilizes the positive steady state as the strength of the chemotactic sensitivity increases. In this case, Turing instability occurs, which can be Hopf bifurcation or steady state bifurcation, and spatial heterogeneous patterns may form.

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