The model is based on a standard SIR (Susceptible, Infective, Recovered) compartmental model [

20]. We allow for two different strains, seasonal (denoted by 1) and pandemic (denoted by 2), vaccination coverage for the seasonal strain, relatively long-lived immunity to the infecting strain and temporary immunity to any strain after infection. Because we consider only a short time span, the population size is considered constant and we can refer to the proportion of people in each class rather than the number in the class in the standard way. In the interests of model simplicity, the population is assumed to be homogeneous and well mixed and we have made no attempt to distinguish between different age classes or different transmission properties in sub-populations. We have assumed that seasonal vaccination provides protection against seasonal influenza infection to a proportion of vaccinated people, depending on vaccine effectiveness and vaccine coverage, but that it provides no protection against infection with pH1N1.

As vaccination alters the susceptibility to the seasonal strains we partition the population into 3 broad groups: the proportion of hosts vaccinated with the seasonal influenza vaccine (susceptible to the pandemic strain only); unvaccinated hosts (susceptible to both pandemic and seasonal influenza); and hosts susceptible to seasonal influenza only. Initially this latter group only comprises individuals who have some cross-immunity to the pandemic strain such as was seen with older people during the pH1N1 2009 outbreak [

21]. Because vaccination is not completely effective, we model effective vaccination coverage which we define as vaccination coverage multiplied by the vaccine effectiveness for the seasonal strain. Throughout we let sub/super script 1 refer to seasonal and 2 to pandemic strains. We further divide each of the broad groups into the following classes: susceptible (

*S*), infective (

*I*), temporarily immune to all strains (

*T*) and recovered (

*R*). Superscripts are used to identify strain susceptibility type and subscripts the infecting strain. Hence the classes considered are given by

*S*^{12}**Unvaccinated**, susceptible to 1 (seasonal) and 2 (pandemic)

*S*^{1} Susceptible to 1 (seasonal) only

*S*^{2}**Vaccinated**, susceptible to 2 (pandemic) only

Was susceptible to 1 and 2 and now infected with 1

Was susceptible to 1 and 2 and now infected with 2

Was susceptible to 1 only and now infected with 1

Was susceptible to 2 only and now infected with 2

Temporary immune to all strains, was susceptible to 1 and 2 and recently infected with 1

Temporary immune to all strains, was susceptible to 1 and 2 and recently infected with 2

Temporary immune to all strains, was susceptible to 1 only and recently infected with 1

Temporary immune to all strains, was susceptible to 2 only and recently infected with 2

*R* Recovered and immune to 1 and 2

For example, depending on the effective vaccine coverage, a proportion of hosts vaccinated against seasonal influenza will be susceptible only to the new pandemic strain, and will be a member of the

*S*^{2} class. Once this host is infected with pandemic influenza he/she will move to the

class and will then recover to be in the temporary immunity

class. Over time the host loses temporary immunity to all strains and moves to the

*R* class. Of particular importance in this model are those hosts who are unvaccinated and initially a member of the

*S*^{12} class, being susceptible to both seasonal and pandemic influenza. A member of this class who becomes infected with seasonal influenza will then move to the

class, and recover to the

class before moving to the

*S*^{2} class. Membership of the

class is important since these hosts have developed temporary immunity to pH1N1 infection. All possible disease progression paths are shown schematically in Figure .

For a fixed population size

*S, I, T* and

*R* can also represent the proportion of the population in each group. If

*β* is the transmission rate,

*γ* is the disease recovery rate (hence 1/

*γ* is the average infectious period) and

*δ* the recovery rate from the temporary immunity (hence 1/

*δ* is the average length of the immunity period) then the disease dynamics can be modelled by

For example, equation (1) states that unvaccinated individuals (

*S*^{12}) can be either infected by seasonal influenza infectious individuals

or pandemic strain infectious individuals

equation (3) states that the proportion of individuals only susceptible to the pandemic strain (

*S*^{2}), which is originally a proportion of vaccinated hosts, can decrease by being infected by pandemic strain infectious individuals and can also increase due to individuals who were originally susceptible to both strains leaving the temporary immunity class after being infected with the seasonal strain

(and hence are susceptible to only the pandemic strain). As the population size is fixed there is no need to keep track of the

*R* class.

The system of ordinary differential equations (1)-(11) is solved numerically using MATLAB. Different initial conditions are used depending on the effective vaccination coverage and any prior immunity. Residual cross-immunity from prior infection with related strains is incorporated in the initial conditions. The Canadian studies reported a vaccine coverage of approximately 30% and a vaccine effectiveness of 56% resulting in an effective vaccination coverage of 17%. Other studies report vaccine effectiveness for healthy young adults, the age group most often infected with pH1N1, of approximately 70% [

22]. The different timings of the seasonal and pandemic strains are simulated by seeding the population with a small proportion (0.01%) of one strain and at some later time introducing the other strain. Values of

*β* and

*γ* are chosen to give a basic reproduction number (

*R*_{0}) [

23] typical of influenza of between 1.3 and 1.8. For simplicity it is assumed that both the seasonal and pandemic strains have the same basic reproduction number, consistent with estimates of R for both seasonal [

24] and pandemic influenza [

25]. The average duration of strain transcending temporary immunity is modelled to be 120 days. The results presented here are relatively insensitive to this parameter, except for the case where there is a long interval between the introductions of seasonal and pandemic influenza.

To compare our results with the observational studies [

1,

3-

6] we calculate the ratio of the odds of pH1N1 infection for vaccinated versus unvaccinated individuals. If

*p*_{v} is the probability of a vaccinated individual being infected with pH1N1 and

*p*_{u} is the probability of an unvaccinated individual being infected with pH1N1 then the odds ratio (OR) is defined to be