The state flow diagram of the model is given in Figure , which represents either the population counts in various compartments in the homogeneous model or the state of any given node (labelled by infection state, degree- and age-class) in the network model. The model describes the evolution of two concurrent strains of influenza infection, over a duration short compared to the natural lifespan of an individual in the population, and for this reason birth and natural death processes are ignored, and furthermore the number of individuals in each age class remains constant. Since we consider a static contact network, the degree class of each individual is also fixed. Therefore, each individual in the population belongs to a unique class (k,a), where k denotes the number of contacts, and a the age class, and the total number of individuals in each (k,a) class is constant. S denotes the susceptibles and V denotes individuals receiving vaccination either prior to the onset of the first infection or after this onset.
Figure 1 State flow diagram for the two-strain influenza model. S denotes the susceptible state, without prior vaccination. Other susceptible individuals may receive vaccination prior to the onset of infection, or after infection has appeared, and are denoted (more ...)
Vaccination prior to the onset of infection is specified by the fraction of susceptibles in each age class receiving vaccination. For vaccination occurring during an outbreak, the following model is used: for individuals in any given (k,a) class, the rate of vaccination at any given time is (i) proportional to the current number of susceptibles in the class; (ii) an increasing function of the total current (symptomatic) infection in the population as a whole, saturating at a prescribed rate. This was done to attempt to model the social response to an outbreak in the population, in which the greater the number of infected individuals the more likely that susceptible individuals would avail themselves of existing vaccination opportunities. The precise mathematical specification of this response is given in the Appendix.
The baseline transmission rate of infection between a susceptible-infected pair of individuals is denoted by τ. The actual rate will depend on the age-classes that these individuals belong to, and whether the susceptible individual of the pair is seeing infection (by either strain) for the first or second time. These various possibilities are accounted for by expressing the actual transmission rate as τ times a factor, which depends on age classes involved, whether this is the first or second infection, and whether the individual has received prior vaccination. Details are given in the Appendix and in Table .
Table 1 Model parameters and their values .
States labelled with I denote symptomatic infection, and those labelled with A denote asymptomatic infection. The P states describe immunity to one strain but not the other: Pj is the state with immunity to strain j (j = 1, 2), and R the state with immunity to both strains. In this model, we exclude co-infection: at any given time, an individual may be infected with at most one strain. State Ij denotes infection with strain j; and Ijk denotes previous infection with (and subsequent recovery from) strain j and current infection with strain k (where k ≠ j). A similar notation applies to the A-classes. The efficacy of the vaccine against strain j is denoted by σj.
’ denotes states of infection (or partial recovery) arising from failure of the vaccine; and as before, labels states with infection due to, or partial recovery from, one of the strains. Following vaccination, infection due to strain j
occurs with probability (1-σj
). In general, for seasonal influenza, the vaccine is targeted against the earlier-occurring strain 1 virus; its efficacy against the later-occurring strain 2 (mutated) virus is expected to be less, i.e., σ2
. As in [6
], the delay T*
in appearance of strain 2 in the population is a parameter of the model.
In Figure , the diverging pairs of directed edges are labelled with branching ratios for each strain of infection, with two pairs of such edges emanating from S and V classes. For example, if S is infected with one of the strains, it has a probability p of being symptomatically infected, and 1-p of being asymptomatically infected. (We assume that p is the same for both strains). Since S may be infected with either strain, there are two pairs of branches emanating from S in Figure . Similarly, there are two branch pairs for V, representing infection due to failure of the vaccine.
After recovery from one strain of infection, an individual is still, in general, susceptible to infection by the other strain: individuals in state Pj (i.e., recovered from infection with strain j), can become infected with strain k (≠ j) but with diminished probability δjk. The probability of such infection being symptomatic is denoted by pjk. Similarly, for individuals who have received prior vaccination but still become infected by strain j, the probability of strain k infection is denoted by pVjk. Finally, the model allows for the possibility of disease-induced death, denoted by the state D. The rates at which these occur are assumed to be d or dA for symptomatic and asymptomatic infections, respectively, regardless of which of the disease states precede death; furthermore, the death rates - as with other parameters of the model – may depend on the age group in which the death occurs.
The converging directed edges in this Figure are labelled with the recovery rates from infection: either µ (symptomatic infection) or µA (asymptomatic infection), where we assume that these rates are the same for both strains, regardless of whether this is the first or second infection for that individual. The parameter values used in the simulations are given in Table .
For the homogeneous model, we may apply the technique of the next-generation matrix [17
] to derive the basic reproductive number R0
. In general, the second strain appears after infection due to the first strain has begun, so that R0
can be calculated using a one-strain sub-model. With this assumption, we find
denote the initial numbers of susceptible and vaccinated individuals, respectively, in the population, and β is the transmission coefficient. To establish a relationship between β and the baseline transmission rate τ between individuals in contact, we construct a (single-age class) network in which the ‘edge probability’ of randomly choosing an edge, one of whose vertices has degree k
, is uniform. By relating this to the mean field model we derive (see Appendix)
= vertex degree of population sub-class into which the Strain-1 infection is introduced at time t
= 0, and kmax
= maximum vertex degree in the finite network (kmax
= 20 in the simulations). If we choose for V0
= 0.2, a conservative value R0
= 1.9 for influenza [10
], then using the above expressions for β and R0
we derive τ = 3.5 d-1
for the transmission rate to be used in the simulations. The value of R0
corresponding to this τ in the absence of vaccination is R0
In keeping with the definition of the two age class model (see Appendix), the estimates of death rates [18
] arising from symptomatic or asymptomatic infection (d, dA
, respectively) for the two age-class model correspond to the general population above and below the median of the age distribution Pa
which, for the city of Vancouver, is about 38 years [20
]. We assume that the death rates due to natural causes are negligible, and choose nominal values for the disease-induced rates: d
) = d
) = 0.002 d-1
]). These rates vary with the particular circulating influenza strains. Furthermore, we set d = dA
in this illustrative study.
In the model described above, the total number of individuals Nk,a in each (k,a) class is fixed, and hence the total population N (summed over all (k,a) classes) is constant. Therefore, by dividing the number of individuals in class (k,a) in state X at any given time by N, we may express the model in terms of the probability Xk,a(t) that a randomly chosen individual is in state X, and belongs to class (k,a), at time t. The resulting set of ordinary differential equations describing this deterministic model is given in the Appendix.