The Influence of Different Forms of Cross-Protective Immunity on the Population Dynamics of Antigenically Diverse Pathogens

  • Neil Ferguson
  • Viggo Andreasen
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 126)


We develop simple epidemic models of co-circulating strains of an infectious disease in which the strains interact immunologically via cross-protective acquired immune responses. Two limiting forms of cross-protective immunity are explored: reduction of infectivity on infection with a strain that against which a degree of cross-protective immunity exists from prior excposure to a heterologous strain, and reduction of susceptibility to infection after exposure to the second strain. After developing a generic model framework capable of representing both forms of action, we show that model formulation can be simplified for some simple cross-immunity structures in the case of infectivity reduction. We then discuss equilbria and stability properties of the generic model, before investigating in detail the special case of allele-based cross-immunity, where antigenic relatedness depends on the number of alleles shared between two strains of a haploid pathogen. For this system, we present conditions for the stability of the symmetric and boundary equilibria in the case of purely infectivity-mediated cross-immunity, and illustrate numerically the wide range of complex limit cycle or chaotic dynamics that dominate a large region of parameter space. Finally, we describe the similarities between the dynamics exhibited by systems with each form of immunity action, and discuss biological applications of such models.

Key words

strains cross-protection epidemic model limit cycles stability analysis 


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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Neil Ferguson
    • 1
  • Viggo Andreasen
    • 2
  1. 1.Institute of GeneticsUniversity of Nottingham, Queen’s Medical CentreNottinghamUK
  2. 2.Department of Mathematics and PhysicsRoskilde UniversityRoskildeDenmark

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