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Abstract and Applied Analysis
Volume 2014, Article ID 891401, 10 pages
http://dx.doi.org/10.1155/2014/891401
Research Article

Adaptive Evolution of Virulence-Related Traits in a Susceptible-Infected Model with Treatment

1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Received 14 May 2014; Revised 9 July 2014; Accepted 14 July 2014; Published 7 August 2014

Academic Editor: Sher Afzal Khan

Copyright © 2014 Xinzhu Meng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Evolution problem is now a hot topic in the mathematical biology field. This paper investigates the adaptive evolution of pathogen virulence in a susceptible-infected (SI) model under drug treatment. We explore the evolution of a continuous trait, virulence of a pathogen, and consider virulence-dependent cure rate (recovery rate) that dramatically affects the outcome of evolution. With the methods of critical function analysis and adaptive dynamics, we identify the evolutionary conditions for continuously stable strategies, evolutionary repellers, and evolutionary branching points. First, the results show that a high-intensity strength drug treatment can result in evolutionary branching and the evolution of pathogen strains will tend towards a higher virulence with the increase of the strength of the treatment. Second, we use the critical function analysis to investigate the evolution of virulence-related traits and show that evolutionary outcomes strongly depend on the shape of the trade-off between virulence and transmission. Third, after evolutionary branching, the two infective species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist, and no further branching is possible, which is independent of the cure rate function.