Capturing the dynamics of pathogens with many strains

Abstract

Pathogens that consist of multiple antigenic variants are a serious public health concern. These infections, which include dengue virus, influenza and malaria, generate substantial morbidity and mortality. However, there are considerable theoretical challenges involved in modelling such infections. As well as describing the interaction between strains that occurs as a result cross-immunity and evolution, models must balance biological realism with mathematical and computational tractability. Here we review different modelling approaches, and suggest a number of biological problems that are potential candidates for study with these methods. We provide a comprehensive outline of the benefits and disadvantages of available frameworks, and describe what biological information is preserved and lost under different modelling assumptions. We also consider the emergence of new disease strains, and discuss how models of pathogens with multiple strains could be developed further in future. This includes extending the flexibility and biological realism of current approaches, as well as interface with data.

Keywords: Cross-immunity; Evolution; Influenza; Multi-strain pathogens; Transmission model.

Fig. 1

Possible routes of infection in a two strain history-based model. a Two strain model in which a host’s new infection history is obtained upon recovery from infection (Castillo-Chavez et al. 1989). $S_i$ denotes the proportion of hosts who have previously been infected—and recovered from—the set of strains $\{i\}$; $I_i$ denotes hosts who are experiencing a primary infection with strain $i$; $J_i$ denotes hosts who are experiencing a secondary infection with strain $i$; ${\mathrm{\Lambda }}_i$ is the force of infection for strain $i$; $\mathit{\gamma }$ is the rate of recovery; and $\mathit{\tau }$ is the relative susceptibility of hosts who have previously been infected with a heterologous strain. Births and deaths are not shown. b Two strain model in which infection history obtained immediately upon infection (Andreasen et al. ; Gupta et al. 1996). Here $S_i$ denotes hosts who have been infected with the set of strains $\{i\}$, and $\mathit{\sigma }$ denotes the relative infectiousness of hosts who have previously been infected with a heterologous strain

Fig. 2

Possible infection histories in a three strain model. Sets are disjoint, with subscripts indicating which collection of strains have previously been seen

Fig. 3

Example of cross-immunity in nested model. Circles show antigenic neighbourhoods for strain $i$, indicated by black dot. Blue crosses show strains in infection history. As the nearest strains are in the set $N_2$ but not in $N_1$, cross-immunity would be equal to ${\mathit{\sigma }}_2$ (colour figure online)

Fig. 4

Potential sources of data. a Phylogenetic tree for influenza subtype H3N2 [adapted from Holmes and Grenfell (2009)]; b percent of sampled individuals in Britain with immunity to 2003 H3N2 strain in 2003 and 2004 [adapted from Johnson et al. (2009)]; c results of contact survey in Great Britain [adapted from Mossong et al. (2008)], with lighter colours representing a larger number of reported contacts between those age groups; d age specific incidence of ILI, as factor of all-age incidence, for 2003/4 influenza season in Britain [adapted from Johnson et al. (2009)]