## Arts & Sciences Articles

Article

Mathematics

#### Journal Title

Applied Mathematical Modelling

2013

37

8

5962

#### Abstract

The viral lytic cycle is an important process in oncolytic virotherapy. Most mathematical models for oncolytic virotherapy do not incorporate this process. In this article, we propose a mathematical model with the viral lytic cycle based on the basic mathematical model for oncolytic virotherapy. The viral lytic cycle is characterized by two parameters, the time period of the viral lytic cycle and the viral burst size. The time period of the viral lytic cycle is modeled as a delay parameter. The model is a nonlinear system of delay differential equations. The model reveals a striking feature that the critical value of the period of the viral lytic cycle is determined by the viral burst size. There are two threshold values for the burst size. Below the first threshold, the system has an unstable trivial equilibrium and a globally stable virus free equilibrium for any nonnegative delay, while the system has a third positive equilibrium when the burst size is greater than the first threshold. When the burst size is above the second threshold, there is a functional relation between the bifurcation value of the delay parameter for the period of the viral lytic cycle and the burst size. If the burst size is greater than the second threshold, the positive equilibrium is stable when the period of the viral lytic cycle is smaller than the bifurcation value, while the system has orbitally stable periodic solutions when the period of the lytic cycle is longer than the bifurcation value. However, this bifurcation value becomes smaller when the burst size becomes bigger. The viral lytic cycle may explain the oscillation phenomena observed in many studies. An important clinic implication is that the burst size should be carefully modified according to its effect on the lytic cycle when a type of a virus is modified for virotherapy, so that the period of the viral lytic cycle is in a suitable range which can break away the stability of the positive equilibria or periodic solutions. (C) 2012 Elsevier Inc. All rights reserved.

#### DOI

10.1016/j.apm.2012.12.004

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