The model () is derived from a classical susceptible–infected–recovered (SIR) model with a few additions. First, we consider two strains of the pathogen which interact through cross-infection. In the following, strain 1 is initially endemic, whereas strain 2 is a new strain introduced into the population. Second, a vaccine is used to immunize a proportion *p* of all newborns, conferring full protection against strain 1, but partial protection only against strain 2. Third, cross-protection (conferred by the vaccine or the primary infection) is modelled as either reduced infection rate or shorter infectious period (); for each of the four cross-protection parameters, a value of zero represents no change, and cross-immunity increases as the value approaches 1, which represents total protection. Lastly, immunity wanes at a constant rate (*σ* for natural infection, *σ*_{V} for vaccination).

| **Table 1**The four types of cross-protection. |

In order to keep the model reasonably simple, we make a number of simplifying assumptions, which could be easily modified in line with relevant empirical data. Here, these parsimonious assumptions reflect the lack of information currently available on pertussis. In particular, both strains are assumed to have the same infection rate *β*, infectious period 1/*γ* and immune period 1/*σ*, and all infected individuals are equally infectious. In addition, births and deaths occur at the same rate *μ*, so that the population size remains constant in the deterministic model.

We successively used three techniques to explore the dynamics of this system. First, we analysed the initial invasion rate of the novel strain, using a deterministic version of the model, as described by the following system of ordinary differential equations (see for a description of symbols):

Second, we ran deterministic simulations by numerically integrating the above system, starting from endemic equilibrium of strain 1 with vaccine coverage *p* and introducing a fraction of 10^{−6} infected with the novel strain 2. We recorded the depth of the post-epidemic trough, i.e. the minimum fractions infected with strain 1 or 2 following introduction.

Third, we implemented a stochastic version of the model using a Gillespie-type algorithm (

Bartlett 1953;

Gillespie 1977;

Gibson & Bruck 2000), as described by

Restif & Grenfell (2006). Simulations were run with an initial population size of one million, starting from the (deterministic) endemic equilibrium for strain 1 and introducing one infected individual with strain 2 at time

*t*=0. For each set of parameter values, we ran a series of 1000 simulations for 10 years, during which we recorded all infections, as well as extinctions of either strain. Simulations were then classified based on the events of extinction within 10 years of introduction: extinction of both strains; initial extinction of strain 2 (i.e. without causing an outbreak); trough extinction of strain 2 only; replacement by strain 2 (i.e. only strain 1 went extinct after the outbreak); or coexistence (no extinction). Because all recorded extinctions occurred within 5 years of the introduction, we are confident that the outcome after 10 years is reliable as to the long-term persistence of strains in the population. However, that may not be the case in small populations, as endemic levels would be much lower and more prone to fade-outs.

Overall, we focused on the effect of vaccine parameters (coverage and cross-protection). Numerical values were inspired by pertussis, with a high basic reproductive ratio

*R*_{0}=17, an average infectious period of three weeks and average immune periods of several years (

Anderson & May 1991;

Esposito *et al*. 2001;

Broutin *et al*. 2004).