Modelling the effects of temporary immune protection and vaccination against infectious diseases q Silvia Martorano Raimundo a, * , Hyun Mo Yang b , Alejandro B. Engel c a Universidade de Sa ˜o Paulo – Faculdade de Medicina – DIM, Instituto Oscar Freire, Rua Teodoro Sampaio, 115, CEP: 05405-000, Sa ˜o Paulo, SP, Brazil b Universidade de Campinas, DMA – IMECC Campinas, SP, Brazil c Department of Mathematics and Statistic, Rochester Institute of Technology Rochester, NY, USA Abstract In this paper, we develop a mathematical model to describe the dynamics of reinfection under the assumption that immune protection may wane over time. As a disease control strategy a schedule of primary and secondary (booster) vac- cination is studied, with vaccine induced immunity declining over time. A distinction is made between infection in immu- nological naive individuals (primary infection) and infection in individuals whose immune system has been primed by vaccination or infection (reinfection). Using the model we analyze the association between prevalence of infection and immunity, induced either by infection or by vaccine. The model shows that eradication depends on vaccination coverage as well as on vaccine efficacy. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Epidemic model; Vaccination; Booster; Vaccination coverage; Immunological memory 1. Introduction The basic question of how we acquire immunity has been investigated for a century or more. Even so, sev- eral concepts in immunology still remain unclear. For instance, it is known that the first exposure of an indi- vidual to an antigen elicits a primary response, the initial antigen response. The primary immune response combats the infection with the production of antibodies, which appears after a time lag of a few days. After recovering from an infection, the concentration of antibodies against the infectious agent gradually declines over time, but the individual is often still protected against a second occurrence of the disease. That is, the person is immune. When an individual is exposed to the same antigen for a second time, a much larger number of antibodies is produced quickly to combat the infection. The response elicited when the individual encoun- ters the same antigen at a later time is the secondary immune response. This mechanism is possible because the primary immune response produces memory cells, which are specialized to mount an immune response against 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.12.051 q Grant Fapesp (Projeto Tema ´tico). * Corresponding author. E-mail address: silviamr@dim.fm.usp.br (S.M. Raimundo). Applied Mathematics and Computation 189 (2007) 1723–1736 www.elsevier.com/locate/amc