Based on the spreading process of H1N1, we propose an SEIAR model by classifying the population as susceptible (

*S*), exposed (

*E*), asymptomatically infected (

*A*), symptomatically infected (

*I*) and removed/immune (

*R*)

*.* The asymptomatically infected compartment contains those who fail to show noticeable symptoms or with light flu-like symptoms; they are not identified as H1N1 cases, but are able to spread the infection. We assume that a susceptible individual becomes infected if they come into contact with an asymptomatically or symptomatically infective individual. Then, the susceptible enters the exposed class

*E* of those in the latent period. The period of incubation for H1N1 is 1-3 days [

3]. After the latent period, the individual enters the class

*I* or

*A* of infectives, who are infectious in the sense that they are capable of transmitting the infection. When the infectious period ends, the individual enters the recovered class

*R.* We assume that a removed individual will never become susceptible or infected again. In our model, new births, natural deaths and migrations are ignored. The flow diagram of the individuals is depicted in Figure .

In contrast to classical compartment models, we consider the whole population and their contacts in networks. Each individual in the community can be regarded as a vertex in the network, and each contact between two individuals is represented as an edge (line) connecting the vertices. The number of edges emanating from a vertex — that is, the number of contacts a person has — is called the degree of the vertex. Therefore, we assume that the population is divided into

*n* distinct groups of sizes

*N*_{k} (

*k* = 1, 2, …,

*n*) such that each individual in group

*k* has exactly

*k* contacts per day. If the whole population size is

*N* (

*N* =

*N*_{1}*+ N*_{2} +

+

*N*_{n}), then the probability that a uniformly chosen individual has

*k* contacts is

*P*(

*k*) =

*N*_{k}/

*N*, which is called the degree distributions of the network. Empirical studies have shown that many real networks have scale-free (SF) degree distributions

*P*(

*k*)

*≈ k*^{–γ} with 2

*≤* γ

*≤* 3 where the epidemic model does not show an epidemic threshold (see [

18]) and Poisson degree distributions

*P*(

*k*) =

*µ*^{k}/

*k*! exp(

*–µ*) (see [

19]). If

*S*_{k} ,

*E*_{k},

*A*_{k},

*I*_{k} and

*R*_{k} represent the number of susceptible, exposed, asymptomatically infected, symptomatically infected and recovered individuals within group

*k* (where

*S*_{k} +

*E*_{k} +

*A*_{k} +

*I*_{k} +

*R*_{k} =

*N*_{k}), then the following system of differential equations captures disease spread for arbitrarily large networks (

*N → ∞*), for both transmission through the network and the mean-field type transmission

where

represent the expectation that any given edge points to an infected and asymptomatically infected vertex respectively. Note that

; thus,

*S*_{k}(

*t*) +

*E*_{k}(

*t*) +

*A*_{k}(

*t*) +

*I*_{k}(

*t*) +

*R*_{k}(

*t*) =

*N*_{k} is constant.

The densities of susceptible, exposed, asymptomatically infected, symptomatically infected and recovered nodes of degree *k* at time *t*, are denoted by *s*_{k}, *e*_{k}, *a*_{k}, *i*_{k} and *r*_{k}, respectively. If *S*_{k}, *E*_{k}, *A*_{k}, *I*_{k} and *R*_{k} are used to represent *s*_{k}, *e*_{k}, *a*_{k}, *i*_{k}, and *r*_{k} respectively, we can still use system (1)-(5) to describe the spread of disease on the network. Clearly, these variables obey the normalization condition

*S*_{k} +

*E*_{k} +

*A*_{k} +

*I*_{k} +

*R*_{k} = 1, and also

All parameters are positive constants and we summarize them in Table .

The mathematical formulation of the epidemic modelling on the network is completed with the initial conditions given as *S*_{k}(0) = *S*_{k}_{0}, *I*_{k}(0) = *I*_{k}_{0}, *E*_{k}(0) = *A*_{k}(0) = *R*_{k}(0) = 0*.*