Some diseases have a long latent period. Once a person is infected, he or she does not become infectious immediately until the incubation period has passed. This is the idea of an SEIR model.The standard form of SEIR epidemic model is given by
dS
dt = −βSI, dE
dt = βSI − δE, dI
dt = δE − γI, dR dt = γI,
where δ is the incubation rate and E represents the exposed class which means that the population in this class is infected but not infectious. Again a closed population is as-sumed with S + E + I + R = N , where N denotes the total population size.
Figure 1.4: Phase plane of the susceptibles and the infectives in the SIR epidemic model. All classes are normalized to the total population. An epidemic occurs if I(t) > I(0) for some time t > 0. In (b), epidemic occurs when the initial value of u > R1
0; in (a) where R0 < 1, R0u − 1 is always smaller than 1 because 0 ≤ u ≤ 1 and thus the infectives decay for all time so that the disease dies out.
0 0.2 0.4 0.6 0.8 1 two functions indicates the size of the epidemic.
The equation for the exposed indicates that an exposed individual will eventually become an infective before recovering from the disease. Thus to prevent an epidemic, we require E + I to decrease with time. Non-dimensionalising the system as before, we require that R0 = N βγ < 1 so that d(E+I)dt < 0 for all time. Similar analysis to that of the SIR model can be carried out and hence will not be repeated here.
To study the avian influenza disease dynamics, we extend the SIR model above. A more complete model will be presented in appendix which we will simplify with further assump-tions and present in the next section.
2 The mathematical model
For the model we investigate here, the incubation period is assumed negligible thus the susceptibles immediately become infectious when infected. Therefore the population is divided into three main classes, namely susceptibles(S), infectives(I) and removed class(R) which we will further divide into subclasses with the following assumptions. Only a small proportion of the susceptibles are considered to have close contact with the domesticated birds and will catch bird flu which we call the chicken-farmers(denoted by Sc). The rest of the susceptibles are considered as non-chicken-farmers(denoted by Sn) who will not usually have contact with domesticated birds. Although a chicken-farmer might catch bird flu, the disease is not infectious to other humans until the virus mutates to a transmissible form among humans. Thus two subclasses of the infectives are assumed: Ib denotes those infected of non-transmissible bird-flu and Im denotes those of the mutated form. For
simplicity of the model, we will assume that once a human catches bird flu, he can not be infected by the mutated flu again, and vice versa; furthermore if a person recovers from either form of the diseases, he is immune from both. Here we consider the disease dynamics only among the human population, the changes in the disease status among the birds are ignored and thus the force of infection from birds is assumed fixed at a constant rate. The duration of the epidemic is generally short compared with the total population turnover and thus the total population is considered closed. The governing equations are as follows:
dSc
dt = −µSc− βImSc, (2.1)
dSn
dt = −βImSn, (2.2)
dIb
dt = µSc− αIb− γIb, (2.3)
dIm
dt = βIm(Sc+ Sn) + αIb− γIm, (2.4) dR
dt = γ(Ib+ Im), (2.5)
where S, I and R commonly denote the classes of population: susceptibles, infectives and removed, respectively. The subscripts {c,n} of S denote chicken-farmers and non-chicken-farmers; subscripts {b,m} of I denote the infective with bird flu and mutated flu respectively. The parameter µ is the infection rate constant of bird flu from birds to hu-mans; β is infectious contact rate constant, i.e. the rate of infection per contact between a susceptible and an infective. Furthermore, homogeneous mixing of the human population is assumed, i.e. the probability of each individual getting contact with each other is equal and therefore β is applied in both Scand Snhere. α represents the mutation rate from Ib to Im while γ is the recovery rate for both Ib and Im, assuming the two forms of diseases are similar enough. A schematic illustrating the interaction of equations (2.1)-(2.5) is shown in Figure 2.1.
Equations (2.1)-(2.5) satisfy the condition Sn(t) + Sc(t) + Ib(t) + Im(t) + R(t) = N , where N denotes the total population which is assumed constant. The equations also satisfy the initial conditions Sn(0) = S0, Sc(0) = N −S0, Ib(0) = Im(0) = R(0) = 0, where we assume that there are no infective or recovered at t = 0 and S0 is the number of the non-chicken farmers.
The equations are non-linear and will not be easy to solve. We thus use the asymptotic approach by assuming series solutions to the equations while exploiting the small
param-eters to simplify the equations. Some key characteristic of an epidemic as introduced in the SIR model before namely the maximum level of the infective and the epidemic size will be predicted. Some schemes for controlling the disease will also be shown in the later sections.
α
S c
S n
I b
I m
R
µ γ
γ β
β
Figure 2.1: A schematic of the model.
2.1 Non-dimensionalisation
To make analysis simpler, first we rescale the variables by putting Sn=N ˘Sn, Sc=N ˘Sc, Ib=N ˘Ib, Im=N ˘Im, R=N ˘R, t = γ1˘t. Subsequently, the parameters are rescaled by µ = γ ˘µ, βN = γ ˘β, α = γ ˘α. The mutation rate α is assumed slow compared with the recovery rate γ, i.e. α << γ, so we take ˘α = ² where ² << 1. Substituting these into equations (2.1)-(2.5), dropping the ” ˘ ” symbol for brevity, the dimensionless system is as follows :
dSc
dt = −µSc− βImSc, (2.6)
dSn
dt = −βImSn, (2.7)
dIb
dt = µSc− ²Ib− Ib, (2.8)
dIm
dt = βIm(Sc+ Sn) + ²Ib− Im, (2.9) dR
dt = Ib+ Im. (2.10)
The initial condition of the above system now becomes
Sn(0) = S0, Sc(0) = 1 − S0, Ib(0) = 0, Im(0) = 0, R(0) = 0, (2.11) where S0 is now the fraction of the non-chicken farmers.