2.1 Epidemic Dynamics in Anti-bioterrorism System
2.1.2 SIQRS Epidemic Diffusion Model
(1) Modeling assumptions and notations specification
To facilitate the model formulation in the following section, three assumptions are specified as follows:
(1) Once a bioterror attack is suffered, the epidemic area can be isolated from other areas to avoid the spread of the disease.
(2) Natural birth and death coefficient of the population in the epidemic area are not considered.
(3) Epidemic diffusion will not be disrupted by itself, which means the infection rate is a constant.
In this section, we consider the situation that epidemic diffusion without incuba- tion period. Notations used in the following model are specified as follows (Table2.1).
Table 2.1 Parameters specification Parameters Specification
N The whole nodes in the affected area
S(t) Susceptible nodes in the affected area which may become infected.
s(t)=S(t)/N represents its density
I(t) Infected nodes in the affected area which are infective with strong infectivity, but have not yet been quarantined.i(t)=I(t)/Nrepresents its density
Q(t) Quarantined nodes in the affected area which have been infected, and have been quarantined.q(t)=Q(t)/N represents its density
R(t) Recovered nodes in the affected area which have recovered from the disease.
r(t)=R(t)/Nrepresents its density
<k> Average degree distribution of the network β Infection rate of the biological epidemic
γ Rate of the recovered nodes transform to the susceptible nodes δ Rate of the infective nodes which will be found and quarantined μ Rate of quarantined nodes transform to the recovered nodes d1 Death rate of the infective nodes
d2 Death rate of the quarantined nodes
Furthermore,S(t)+I(t)+Q(t)+R(t)=N,s(t)+i(t)+q(t)+r(t)=1.
(2) Model formulation
Since quarantine is a common response measure when an epidemic outbreaks, here we divide people in the epidemic area into four groups: susceptible people (S), infected people (I), quarantined people (Q) and recovered people (R). The survey by Tham [24] shows that some of the recovered people who are discharged from the emergency department will be re-infected again. Thus, epidemic diffusion model in this section can be illustrated as Fig.2.1.
For epidemic diffusion, models based on a small-world network match the actual social network much better. A great deal of attention has been paid to studying these models. Therefore, based on the mean-filed theory [25], the time-based parameter s(t)meets the following equation from timettot+t:
s(t+t)−s(t)= −β<k>s(t)i(t)t+γr(t)t. (2.1)
Fig. 2.1 Framework of SIQRS model
δI d2Q d1I
β<k>SI
γR
S I Q μQ R
Thus, we get:
ds(t)
dt = −β<k>s(t)i(t)+γr(t). (2.2) Similarly, we have the other three ordinary differential equations as follows:
di(t)
dt =β<k>s(t)i(t)−d1i(t)−δi(t). (2.3) dq(t)
dt =δi(t)−d2q(t)−μq(t). (2.4)
dr(t)
dt =μq(t)−γr(t). (2.5)
Thus, the following SIQRS epidemic diffusion model can be formulated:
⎧⎪
⎪⎨
⎪⎪
⎩
ds(t)
dt = −β<k>s(t)i(t)+γr(t)
di(t)
dt =β<k>s(t)i(t)−d1i(t)−δi(t)
dq(t)
dt =δi(t)−d2q(t)−μq(t)
dr(t)
dt =μq(t)−γr(t)
. (2.6)
Here,β,<k>,γ,δ,μ,d1,d2 >0. Initial conditions for this epidemic diffusion model are demonstrated as follows:
i(0)=i01, s(0)=s0=1−i0, q(0)=r(0)=0.
(3) Analysis of the epidemic diffusion model
As is well known,i(0)=i01 ands(0)=s0 =1−i0, are initial percentage of infected people and susceptible people in the population, respectively. Obviously, when wide spread of the epidemic takes place, the following condition should be satisfied:
di(t) dt
t=0
>0. (2.7)
Considering Eq. (2.3), we have:
s0> d1+δ
β<k>. (2.8)
Equation (2.8) means that epidemic diffusion will take place whens0 meets the above condition. Generally, it is difficult to get the analytic solution for Eq. (2.6).
Thus, we consider the stable state of Eq. (2.6). Ass(t)+i(t)+q(t)+r(t)=1, and considering Eqs. (2.2), (2.3) and (2.4), then we have:
⎧⎨
⎩
ds(t)
dt = −β<k>s(t)i(t)+γ[1−s(t)−i(t)−q(t)]
di(t)
dt =β<k>s(t)i(t)−(d1+δ)i(t)
dq(t)
dt =δi(t)−(d2+μ)q(t)
. (2.9)
Let dsdt(t) =0, didt(t) =0 and dqdt(t) =0, we can get an obvious equilibrium point for the epidemic diffusion model I as follows:
P1 =(s, i, q)=(1, 0, 0). (2.10) Equation (2.10) shows that both the number of infected people and the number of quarantined people are equal to zero, which indicates that epidemic diffusion in the disaster area does not happen. All people in the area are susceptible at last. Thus, we refer to this as the disease-free equilibrium point.
Furthermore, according to Eq. (2.9), we can get another equilibrium point for the epidemic diffusion system as follows:
P2=(s,i,q)
= d1+δ
β<k>, γ[β<k>−(d1+δ)](d2+μ)
β<k>[(d1+δ+γ )(d2+μ)+γ δ], γ δ[β<k>−(d1+δ)]
β<k>[(d1+δ+γ )(d2+μ)+γ δ]
. (2.11) Equation (2.11) shows that when the epidemic diffusion system is stable, a certain amount of infected people and a certain amount of quarantined people exist in the epidemic area. Thus, we refer to this as the endemic equilibrium point.
Lemma 2.1 Disease-free equilibrium point P1in the epidemic diffusion network is stable whenβ < d<1k+δ>.
Proof Considering P1 = (s,i,q) =(1,0,0), we can obtain the Jacobi matrix of Eq. (2.9) as follows:
JP1=
⎡
⎢⎣
∂P11
∂s ∂P11
∂i ∂P11
∂q
∂P12
∂s ∂P12
∂i ∂P12
∂q
∂P13
∂s ∂P13
∂i ∂P13
∂q
⎤
⎥⎦=
⎡
⎣−γ −β<k>−γ −γ 0 β<k>−(d1+δ) 0
0 δ −(d2+μ)
⎤
⎦. (2.12)
Here P11, P12 and P13 represent the three differential equations in Eq. (2.9), respectively. Thus, it is easy to get the secular equation for the Jacobi matrix as follows:
(λ+γ )(λ−β<k>+d1+δ)(λ+d2+μ)=0. (2.13) Obviously, three characteristic roots for this secular equation are−γ,β<k>− d1−δ, and−d2−μ. Based on Routh-Hurwiz stability criterion, whenβ < d<1k+δ>, real parts of these three characteristic roots will be negative at the same time. Thus, the disease-free equilibrium point P1=(s,i,q)=(1,0,0)is stable whenβ < d<1k+δ>.
Lemma 2.2 Endemic equilibrium point P2in the epidemic diffusion network is sta- ble whenβ > d<1k+δ>.
Proof Similarly as Lemma2.1, coupling with Eq. (2.11), we can get the Jacobi matrix of Eq. (2.9) again as follows:
JP2=
⎡
⎢⎣
−γ[([β<d1+δ+γ )(k>−(d1d+δ)](2+μ)+γ δ]d2+μ) −γ −(d1+δ)−γ −γ
γ[β<k>−(d1+δ)](d2+μ)
[(d1+δ+γ )(d2+μ)+γ δ] 0 0
0 δ −(d2+μ)
⎤
⎥⎦. (2.14)
Then, the secular equation for Eq. (2.14) is
a0λ3+a1λ2+a2λ+a3=0. (2.15) Here,a0=1,a1=(d2+μ)+(A+γ ),a2 =(d2+μ)(A+γ )+(d1+δ+γ )A, a3=(d2+μ)(d1+δ+γ )A+γ δA, andA= γ[β<[(d1+δ+γ )(k>−(d1d+δ)](2+μ)+γ δ]d2+μ).
Obviously, whenβ > d<1k+δ>,we haveA>0.Then, we havea1>0,a2>0,a3>
0.Therefore,
a1a2−a0a3=(d2+μ)(A+γ )(d2+μ+A+γ ) +(d1+δ+γ )A2+γA(d1+γ ) >0
According to Routh-Hurwiz stability criterion, Eq. (2.11) contains three charac- teristic roots with negative real part. Thus, the endemic equilibrium pointP2is stable whenβ > d<1k+δ>.
Remark 2.1 From Lemmas2.1and2.2, we have the first conclusion: without con- sideration of incubation period, threshold of the epidemic diffusion not only depends on topological structure of the small-world network (<k>), but also relies on other two key parameters, the quarantined rate (δ) and the death rate of infected people (d1).