Volume 11 Supplement 1
Mathematical Modelling of Influenza
Effects of vaccination and population structure on influenza epidemic spread in the presence of two circulating strains
- Murray E Alexander^{1, 2}Email author and
- Randy Kobes^^{2}
https://doi.org/10.1186/1471-2458-11-S1-S8
© Alexander and Kobes; licensee BioMed Central Ltd. 2011
Published: 25 February 2011
Abstract
Background
Human influenza is characterized by seasonal epidemics, caused by rapid viral adaptation to population immunity. Vaccination against influenza must be updated annually, following surveillance of newly appearing viral strains. During an influenza season, several strains may be co-circulating, which will influence their individual evolution; furthermore, selective forces acting on the strains will be mediated by the transmission dynamics in the population. Clearly, viral evolution and public health policy are strongly interconnected. Understanding population-level dynamics of coexisting viral influenza infections, would be of great benefit in designing vaccination strategies.
Methods
We use a Markov network to extend a previous homogeneous model of two co-circulating influenza viral strains by including vaccination (either prior to or during an outbreak), age structure, and heterogeneity of the contact network. We explore the effects of changes in vaccination rate, cross-immunity, and delay in appearance of the second strain, on the size and timing of infection peaks, attack rates, and disease-induced mortality rate; and compare the outcomes of the network and corresponding homogeneous models.
Results
Pre-vaccination is more effective than vaccination during an outbreak, resulting in lower attack rates for the first strain but higher attack rates for the second strain, until a “threshold” vaccination level of ~30-40% is reached, after which attack rates due to both strains sharply dropped. A small increase in mortality was found for increasing pre-vaccination coverage below about 40%, due to increasing numbers of strain 2 infections. The amount of cross-immunity present determines whether a second wave of infection will occur. Some significant differences were found between the homogeneous and network models, including timing and height of peak infection(s).
Conclusions
Contact and age structure significantly influence the propagation of disease in the population. The present model explores only qualitative behaviour, based on parameters derived for homogeneous influenza models, but may be used for realistic populations through statistical estimates of inter-age contact patterns. This could have significant implications for vaccination strategies in realistic models of populations in which more than one strain is circulating.
Background
Human influenza infection is characterized by seasonal epidemics. This occurs because influenza A is able to maintain its presence in human populations by evolutionary adaptations to population-wide immunity, resulting in mutations that gradually change viral antigens allowing the virus to evade immune detection, a process known as “antigenic drift”. On account of these rapid mutations, vaccination for influenza must be updated annually on a global basis, following surveillance to monitor the appearance of new strains [1]. Antigenic drift also diminishes vaccine efficacy for mutant strains, but may still confer partial immunity to these strains. Therefore, understanding the short-term evolution of influenza virus is crucial to developing seasonal vaccines. Conversely, vaccination of a population may influence the short-term evolution of the virus, for example by decreasing the number of hosts in which the virus may replicate.
In general, during a single influenza season, more than one viral strain is circulating. It is known [2, 3] that when suitable invasion conditions are satisfied, stable coexistence of two different strains is possible. The coexistence of two or more strains in a population will influence their individual evolution; and furthermore, the selective forces acting on the strains will be mediated by the transmission dynamics in the population. For example, the infection of hosts by one strain will reduce susceptibility to other strains, thereby limiting their spread in the population [4]. In addition, the time lag in emergence of a second strain following onset of an epidemic by a first strain will be influenced by the strategy and timing of vaccination [5]. It is clear that viral evolution and public health policy are strongly interconnected, and understanding the population-level dynamics of coexisting viral influenza infections, when vaccination of the population is to be undertaken, would be of great benefit in designing such vaccination strategies [6].
In [6], a homogeneous model of two viral strains was developed, incorporating cross-immunity and delay in emergence of the second strain. It was found that for small delay and large cross-immunity, infections with both strains appeared as a single epidemic wave; on the other hand, with sufficient delay, a second epidemic wave is possible. Further, for sufficient delay and high cross-immunity, the population of susceptible hosts may become so depleted as to prevent a second wave. These findings, together with possible impact of vaccination on antigenic drift, suggest that vaccination would be an important factor to include [6].
In large populations, contacts between individuals are not uniform, as assumed in the homogeneous model [6]. Typically, the number of contacts per day per individual is much smaller than the population size, and the structure of the corresponding ‘contact matrix’ plays an important role in the development of the pattern of the disease [7]. The effects of spatial correlations [8], such as occur when community structures are present [9], were illustrated in the spread of drug resistance in a network with mild clustering [10]: the spread of the resistant strain occurred more rapidly, and at significantly lower treatment levels, than was predicted by the homogeneous model.
The present paper extends the model in [6] in a number of ways. The model includes either pre-vaccination or vaccination during the epidemic, of a predetermined part of the population. The contact structure is modelled as a Markov network [11], in which the distribution of degrees of the nodes (i.e., number of contacts for individuals in the population) is specified. In addition, the model allows a distribution of ages in the population by incorporating a prescribed number of age classes. The Markov assumption for the contact network allows the specification of structural parameters such as assortativity [12] and clustering [13–15] that are important characteristics of social groupings. These generalizations enable vaccination to be targeted according to age group and ‘contact number’ (degree of node), which in general respond to the vaccine in different ways. The model inevitably contains many parameters and allows a wide range of network structures to be specified; in addition, initial conditions can be specified in many different ways. Therefore, in this paper, only a simplified network model will be investigated. The structure of the network is comprised of uncorrelated nodes, with degree distribution specified as a truncated scale-free form [7]. Furthermore, for simplicity only one or two age-classes are considered, where, for the latter, the median age is chosen to separate the two classes. While a detailed age distribution, characteristic of a real population, could be specified, the present results are intended to be illustrative only and to allow comparison with the corresponding homogeneous model. The network model can potentially be useful in describing specific populations, such as a small or large city, in which case the network structure and age distribution would need to be determined from statistical analysis of demographic and census data [16].
Section 2 describes the model in broad terms, and lists some of the parameter values used; technical details are given in the Appendix. Section 3 presents the results of simulations, in which the cross-immunity and delay in appearance of the second strain infection are varied. These results are also compared with those produced by the corresponding homogeneous model, to ascertain the importance of structure in the network for determining the time-course and final extent (“total attack rate”) of the disease. Finally, Section 4 discusses these results, some possible extensions of the model, and implications for vaccination strategies in more realistic models based on specific demographic data.
Methods
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.
Model parameters and their values [6].
Parameter | Value | Parameter | Value |
---|---|---|---|
τ | 3.5 d^{-1} | p _{ V } _{12} | 0.3 |
δ_{ A } | 0.142 | p _{ V } _{21} | 0.06 |
δ_{12}= δ_{21} (δ) | 0≤δ≤1 | µ | 0.244 d^{-1} |
δ_{V1} | 0.8 | µ_{ A } | 0.244 d^{-1} |
δ_{V2} | 0.9 | σ_{1} | 0.8 |
p | 0.6 | σ_{2} | 0.4 |
p_{12}, p_{21} | 0.3 | d | 0.002 d^{-1} |
p _{V1} | 0.12 | d _{ A } | 0.002 d^{-1} |
p _{V2} | 0.36 | T* | 10d, 60d |
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: P_{ j } 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 I_{ j } denotes infection with strain j; and I_{ jk } 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 σ_{ j }.
Subscript ‘V’ 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-σ_{ j }). 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., σ_{2} < σ_{1}. As in [6], the delay T* in appearance of strain 2 in the population is a parameter of the model.
In Figure 1, 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 1. 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 P_{ j } (i.e., recovered from infection with strain j), can become infected with strain k (≠ j) but with diminished probability δ_{ jk }. The probability of such infection being symptomatic is denoted by p_{ jk }. Similarly, for individuals who have received prior vaccination but still become infected by strain j, the probability of strain k infection is denoted by p_{ Vjk }. 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 d_{ A } 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 µ_{ A } (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 1.
where k_{1}= vertex degree of population sub-class into which the Strain-1 infection is introduced at time t = 0, and k_{ max } = maximum vertex degree in the finite network (k_{ max } = 20 in the simulations). If we choose for V_{0}= 0.2, a conservative value R_{0} = 1.9 for influenza [10], then using the above expressions for β and R_{0} we derive τ = 3.5 d^{-1} for the transmission rate to be used in the simulations. The value of R_{0} corresponding to this τ in the absence of vaccination is R_{0} = 2.34.
In keeping with the definition of the two age class model (see Appendix), the estimates of death rates [18, 19] arising from symptomatic or asymptomatic infection (d, d_{ A }, respectively) for the two age-class model correspond to the general population above and below the median of the age distribution P_{ a } 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(a_{1}) = d(a_{2}) = 0.002 d^{-1} (Ref.[10]). These rates vary with the particular circulating influenza strains. Furthermore, we set d = d_{ A } in this illustrative study.
In the model described above, the total number of individuals N_{ k }_{,}_{ 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 X_{ k }_{,}_{ 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.
Results
The initial state was specified as follows. For pre-vaccination, a prescribed fraction V_{0}(a) of individuals in each age class a receive vaccination. Infection by strain 1 is introduced into fraction ε_{1} of the remaining susceptibles residing in a single class (k_{1},a_{1}). After the strain 1 infection has spread through the population for a time T*, a strain 2 infection is introduced into a fraction ε_{2} of class (k_{2},a_{2}) individuals. In the simulations, we use ε_{1} = ε_{2} = 0.5; k_{1} = 5, k_{2} = 10, and for the two age class model, a_{1} = 1, a_{2} = 2. As previously mentioned, it is assumed that no individual may be infected with both strains simultaneously. The simulations were performed using three models: (1) network model with two age classes; (2) network model with one age class; and (3) the homogeneously-mixing 'mean field' model. For (1) and (2), the structure of the network was chosen to have a scale-free form [7], with the number of individuals (nodes of the contact network) with k contacts being proportional to k^{-2.5}[21], and 1 ≤ k ≤ k_{ max } = 20. Furthermore, the degrees of the nodes of the network were assumed to be uncorrelated: although real networks show significant correlation structure – e.g., clustering and associativity [12–15] - the purpose of the present simulations is to illustrate the general effects of departure from the homogeneous mixing assumption. A value R_{0} = 1.9 was fixed for the mean field model with V_{0}= 0.2.
As expected, if the second strain is introduced after the strain 1 infection has been largely cleared from the population (as is the case when T* = 60 days: right column), then the first and second waves behave as distinct, non-interacting one-strain epidemics. However, when T* is only 10 days (left column), there is still a significant presence of strain 1 infection in the population: the infections in the two age-class model merge into a single broad peak, whereas the other two models show two distinct peaks, with the second peak occurring in both models ~50 days after initial infection.
It is therefore apparent that the two age class network model exhibits a larger delay in peak infection – for both first and second waves – compared to the one age class and mean-field models. This can be accounted for by the reduced transmissibility between classes compared to within one class, as well as reduced transmissibility within the second age class (see the M matrix in the Appendix). (Recall that strain 1 and 2 infections are introduced into different age classes in the two age class network model). Such differences in delays between mean-field and structured models have been observed elsewhere [10], and underline the importance of spatial structure in determining the course of an epidemic.
Total attack rates for 2-age class network model
Attack rates: Prior vaccination V_{0}= 0.2, ω_{0} = 0.0 | Attack rates: Vaccination during epidemic: V_{0}= 0, ω_{0} = 1.0 | ||||||
---|---|---|---|---|---|---|---|
δ | T* | Strain 1+2 | Strain 1 | Strain 2 | Strain 1+2 | Strain 1 | Strain 2 |
0.9 | 10 | 0.149929 | 0.229126 | 0.178316 | 0.141735 | 0.393720 | 0.083434 |
0.9 | 60 | 0.115638 | 0.279763 | 0.117244 | 0.000858 | 0.541612 | 0.000404 |
0.4 | 10 | 0.037663 | 0.338132 | 0.082410 | 0.008601 | 0.528141 | 0.010630 |
0.4 | 60 | 0.000463 | 0.398912 | 0.000749 | 0.000001 | 0.542568 | 0.000003 |
Total attack rates for 1-age class network model
Attack rates: Prior vaccination V_{0} = 0.2, ω_{0} = 0.0 | Attack rates: Vacc. during epidemic: V_{0} = 0, ω_{0} = 1.0 | ||||||
---|---|---|---|---|---|---|---|
δ | T* | Strain 1+2 | Strain-1 | Strain-2 | Strain 1+2 | Strain-1 | Strain-2 |
0.9 | 10 | 0.706272 | 0.141649 | 0.098164 | 0.640962 | 0.139139 | 0.134720 |
0.9 | 60 | 0.706589 | 0.141371 | 0.098143 | 0.640968 | 0.139135 | 0.134721 |
0.4 | 10 | 0.136833 | 0.713552 | 0.027143 | 0.008607 | 0.775840 | 0.002266 |
0.4 | 60 | 0.000225 | 0.853501 | 0.000043 | 0.000058 | 0.785292 | 0.000015 |
Total attack rates for mean field model
δ | T* | Attack rates: Prior vaccination V_{0} = 0.2, ω_{0} = 0.0 | Attack rates: Vacc. during epidemic: V_{0} = 0, ω_{0} = 1.0 | ||||
---|---|---|---|---|---|---|---|
Strain 1+2 | Strain-1 | Strain-2 | Strain 1+2 | Strain-1 | Strain-2 | ||
0.9 | 10 | 0.334151 | 0.319792 | 0.163241 | 0.289341 | 0.370007 | 0.125953 |
0.9 | 60 | 0.335045 | 0.328586 | 0.152525 | 0.290336 | 0.372194 | 0.123627 |
0.4 | 10 | 0.031496 | 0.623353 | 0.038615 | 0.007640 | 0.653452 | 0.009348 |
0.4 | 60 | 0.001286 | 0.664957 | 0.001463 | 0.000537 | 0.664308 | 0.000645 |
Effects of varying pre-vaccination fraction on total attack rates for 2-age class network model
V _{0} | R _{0} | δ=0.4 | δ=0.9 | ||||
---|---|---|---|---|---|---|---|
Strain 1+2 | Strain 1 | Strain 2 | Strain 1+2 | Strain 1 | Strain 2 | ||
0 | 2.34 | 0 | 0.661707 | 0 | 0.001156 | 0.660372 | 0.000374 |
0.1 | 2.12 | 0.000016 | 0.539788 | 0.000013 | 0.035076 | 0.500271 | 0.018405 |
0.2 | 1.90 | 0.000463 | 0.398912 | 0.000749 | 0.115638 | 0.279763 | 0.117244 |
0.3 | 1.68 | 0.022769 | 0.205177 | 0.088641 | 0.091647 | 0.135334 | 0.209307 |
0.4 | 1.46 | 0.007655 | 0.019366 | 0.274655 | 0.014051 | 0.014563 | 0.286775 |
0.5 | 1.24 | 0.000339 | 0.004577 | 0.051839 | 0.000754 | 0.004131 | 0.054892 |
Effect on death rate of pre-vaccination
V _{0} | Fraction of deaths Pre-vaccination | EffectiveR_{0} | |
---|---|---|---|
δ = 0.4 | δ = 0.9 | ||
0 | 0.00542 | 0.00544 | 2.34 |
0.1 | 0.00442 | 0.00486 | 2.12 |
0.2 | 0.00328 | 0.00518 | 1.90 |
0.3 | 0.00279 | 0.00434 | 1.68 |
0.4 | 0.00254 | 0.00270 | 1.46 |
0.5 | 0.00047 | 0.00050 | 1.24 |
Effect on death rate of vaccination during epidemic
ω_{0} | Fraction of deaths Vaccination during epidemic | |
---|---|---|
(δ = 0.4) | (δ = 0.9) | |
0 | 0.00542 | 0.00544 |
0.1 | 0.00529 | 0.00530 |
0.2 | 0.00516 | 0.00517 |
0.4 | 0.00495 | 0.00496 |
0.8 | 0.00459 | 0.00460 |
1.0 | 0.00445 | 0.00446 |
Conclusions
We have considered extensions of the two viral strain mean field (homogenous) model introduced in [6], to explore the effects of both local network structure and the division of the population into different age classes. The present study used model parameter values (in particular, R_{0}) originally estimated for mean field models; and in order to translate these to the network models derived in this paper, a correspondence was established between the mean field model and a limiting case of the network model (see Appendix). The two age class model assumed the age boundary was located at the median age (about 38 years for Vancouver), with vaccine efficacy of 80% in the lower age group and 40% in the upper age group.
Several notable features were observed when comparing the network models to the corresponding mean field case. Firstly, the amount of cross-immunity present is significant in determining whether a second wave of infection occurs. Due to a lower transmission rate between age classes and within the second age class, compared to within the first age class, infection levels were found to be significantly less for the two age class model than for either the one age class or mean field models. The infections occurred as either a single wave or as two successive waves. A second wave is more likely to occur the longer the delay in introduction of the second strain, since when this delay is short (~10 days) infections due to both strains merge into a single, broad peak. When a second wave does occur, the shapes of the two waves depend on when the second strain infection is introduced. If it occurs well after the first infection has run its course, then the two waves behave as distinct, non-interacting infections. The second infection peak is delayed, and its amplitude reduced, in the network model, compared to the mean field case. This behaviour reflects a longer propagation time in the network model, and has been qualitatively observed in other models, reinforcing the importance of including local network structure in realistic models.
As expected, the amount of cross-immunity between the two strains is important in determining the size of the second-strain outbreak. It was found that its size decreased sharply with increasing cross-immunity. As the level of vaccination increases, strain 1 attack rates decrease, with a sharp drop occurring around 30-40% pre-vaccination coverage; at the same time, strain 2 infections increase with increasing vaccination coverage, reaching their maximum somewhere in this range, and drop off sharply for higher coverage levels. This phenomenon is reminiscent of the development of drug resistance, where there is an optimal level of drug treatment (compare: vaccination coverage) that minimizes the overall infection [10]. This could have significant implications for vaccination strategies in realistic models of populations in which more than one strain is circulating.
It was found that increasing either pre-vaccination or vaccination during an outbreak, reduces the disease-induced mortality. Furthermore, pre-vaccination appears to be more effective than vaccination during an outbreak in reducing overall mortality, though this needs further investigation as it may depend critically on how the latter is implemented. This study considered only a simple model in which at any given time vaccination rates during an outbreak were governed by the total infection in the population at that time, and considers only vaccination of the susceptible class S, neglecting vaccination of other classes (e.g., P_{1} and P_{2} and asymptomatic cases).
As mentioned earlier, the particular form of the terms included in the model to incorporate local network structure and the effects of age classes was chosen for illustrative purposes. This approach, though, can be used on a specific population if sufficient data are available to determine realistic estimates of the age classes and network structure present and of the parameters of the model. The main difficulty is in determining the form of the two-point correlations between vertices of the contact network for a realistic particular population, and this must be derived indirectly from estimates of network structure extracted from the data [16]. An intermediate approach is to explore the effects of a few network structure parameters – e.g., clustering, associativity, betweenness, and centrality [7, 16], obtaining expressions for the two-point probabilities defining the Markov network directly in terms of these parameters. This is currently under investigation.
Appendix: Effects of vaccination and population structure on influenza epidemic spread in the presence of two circulating strains
The various parameters in the model (Figure 1 of main text) are defined below:
τ = baseline transmission rate between a susceptible-infected pair
p = probability of developing symptomatic infection with no prior exposure
p_{ V }_{1}, p_{ V }_{2} = probabilities of pre-vaccinated individuals developing symptomatic infection from strains 1 and 2, respectively, with no prior exposure
σ_{1}, σ_{2} = effectiveness of vaccine to strains 1 and 2, respectively
δ_{ V }_{1}, δ_{ V }_{2} = reduction in transmissibility of strains 1 and 2, respectively, for vaccinated individuals
p_{12} = probability of developing symptomatic infection with prior exposure to strain 1
p_{ V }_{12} = probability of pre-vaccinated individuals developing symptomatic infection with prior exposure to strain 1
p_{21} = probability of developing symptomatic infection with prior exposure to strain 2
p_{ V }_{21} = probability of pre-vaccinated individuals developing symptomatic infection with prior exposure to strain 2
δ_{ A } = reduction in infectiousness due to asymptomatic infection
µ, µ_{ A } = recovery rates from symptomatic and asymptomatic infections
δ_{12}, δ_{21} = level of cross-immunity induced by previous exposure to strain 1 and strain 2, respectively
d, d_{ A } = disease-induced death rates, assumed to be age-dependent but the same for each type of infection.
for U ∈ {A_{1}, A_{ V }_{1},I_{1}, I_{ V }_{1},A_{21}, A_{ V }_{21},I_{21}, I_{ V }_{21}, A_{2}, A_{ V }_{2}, I_{2}, I_{ V }_{2}, A_{12}, A_{ V }_{12}, I_{12}, I_{ V }_{12}}, denote the force of infection for age-class a and degree-class k. Here, M(a,a′) denotes the relative transmission coefficient between age-groups, so that τM(a,a′) = transmission coefficient between a susceptible individual of age-class a in contact with an infected individual of age-class a′. Also, P(k′,a′|k,a) is the probability that an individual (node) of age-class a and degree-class k has a neighbour (adjacent node) of age-class a′ and degree-class k′.
- (i)
φ is a function of the total (symptomatic) infection in the population, I_{ tot } (summed over all k and a), and φ = 0 when I_{ tot } = 0;
- (ii)
φ is proportional to the population in class S(k,a,t);
- (iii)
φ eventually saturates at a maximum value as I_{ tot } increases.
where ω_{0} is the (age-class dependent) saturation rate of vaccination, α is the value of I_{ tot } at half-saturation, and n > 0 governs the steepness of the response curve. In the simulations, α = 0.4 (i.e., half-saturation occurs when 40% of population is infected), and n = 2.
In order to incorporate death due to infection, we add a set of classes {D(k,a,t)} to the model, and (similar to the recovery rates) assume that death rates are either d (for all symptomatic infections) or d_{ A } (for all asymptomatic infections).
This shows that the various Θ(k,a,t)’s describe the connectivity of a vertex of degree k and age-class a to all the infected adjacent vertices.
Comparison with mean field model
where is the mean degree of nodes in the network.
Using this approximate relationship enables us to compare the simulation of the behaviours of the network and mean-field models, by relating numerical values of the parameters β and τ through the simplified limiting case of a network in which the probability of drawing an edge at random from the network is uniform.
Initial conditions for the mean field model
In what follows, it is assumed that the total population (including deaths) is normalized to unity, which is permissible since for this model it is constant. The initial conditions for the mean field model must be consistent with those of the network model. The analysis that follows applies to an arbitrary number of age classes and degree distributions.
where are the changes in the susceptible and vaccinated sub-populations, respectively, and P_{ka2} ≡P(k_{2})P_{ a }(a_{2}) is the fraction of the total population in class (k_{2},a_{2}).
In order to allow comparison between Mean Field and network models, all age-dependent parameters δ_{ A }, δ_{ V }_{1}, δ_{V2}, σ_{1}, σ_{2}, µ_{ A }, µ, d_{ A }, d, etc., in the network model are replaced by their age-distributed averages: , etc., where without risk of ambiguity we may drop the ‘MF’ superscript.
For the network model, for all age classes we set ε_{1} = ε_{2} = 0.5, p = 0.6, V_{0}= 0.2, σ_{1} = 0.8, σ_{2} = 0.4. For the two age-class model, we chose (k_{1},a_{1}) = (5,1), (k_{2},a_{2}) = (10,2); and for the one-age class model k_{1} = 5, k_{2}= 10. The (truncated) scale-free distribution P(k) ~ k^{ - }^{3.5} with k_{ max } = 20 yields P_{ ka }_{1} = 0.0067, P_{ ka }_{2} = 0.0012 (so that, in a population of N = 10,000, the number of infections is 67 and 12, respectively), where we are assuming P_{ a } to be uniformly distributed in the 2-age-class model: P_{a}(a_{1}) = P_{ a }(a_{2}) = 0.5.
Substituting these values into the expression for R_{0} (Section 2 in main paper), and using R_{0} = 1.9, V_{0} = 0.2, k_{ max } = 20, and k_{1} = 5, yields the values β = 0.8765, and τ = 3.5 d^{-1}. For V_{0} = 0.4, using the same value τ = 3.5 d^{-1}, the corresponding value of R_{0} is 2.34.
Declarations
Acknowledgements
It is with great sadness that I report the sudden passing of my dear friend and co-author, Dr Randy Kobes, whilst this paper was in revision. He will be sorely missed by his many friends and colleagues. I would like to thank Dr Seyed Moghadas for his helpful comments and assistance in designing the mean field model, and the referee for incisive and useful comments that have greatly improved the paper. This work was supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.
This article has been published as part of BMC Public Health Volume 11 Supplement 1, 2011: Mathematical Modelling of Influenza. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2458/11?issue=S1.
Authors’ Affiliations
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