XUEJUAN LU, SHAOKAI WANG, SHENGQIANG LIU AND JIA LI people's response to the threat of disease is dependent on their perception of risk, which is ...
MATHEMATICAL BIOSCIENCES AND ENGINEERING Volume 14, Number 5&6, October & December 2017
doi:10.3934/mbe.2017068 pp. 1317–1335
AN SEI INFECTION MODEL INCORPORATING MEDIA IMPACT
Xuejuan Lu Department of Mathematics, Harbin Institute of Technology Harbin 150001, China
Shaokai Wang Shenzhen Graduate School, Harbin Institute of Technology Shenzhen 518055, China
Shengqiang Liu1 Department of Mathematics, Harbin Institute of Technology Harbin 150001, China
Jia Li Department of Mathematical Sciences, University of Alabama in Huntsville Huntsville, AL 35899, USA
Abstract. To study the impact of media coverage on spread and control of infectious diseases, we use a susceptible-exposed-infective (SEI) model, including individuals’ behavior changes in their contacts due to the influences of media coverage, and fully investigate the model dynamics. We define the basic reproductive number 0, E(t) > 0, ∀t > t0 , which contradicts φ(t) ∈ ∂X0 , ∀t ≥ 0. We now let \ w(x). Ψ0 := x∈Z0
Here S Z0 is the global attractor of φ(t) restricted to ∂X0 . We claim that Ψ0 = {E0 } {E1 }. In fact, Ψ0 ⊆ Ω∂ = {(S(t), 0, 0)}. From system (2), we obtain S = 0 or S = K. Thus E0 , E1 ∈ Ψ0 . Since {E0 }, {E1 } are two isolated invariant sets of φ(t) in Ω∂ , using the similar arguments from Theorem 3.1 and 1, we can prove that E1 is asymptotically stable in Ω∂ , defined in (4). Hence Ψ0 has an acyclic covering. Next, we prove that W s ((0, 0, 0)) ∩ X0 = φ. Suppose that it is not true. Then for any 1 > 0, there exists T0 > 0 such that (S(t), E(t), I(t)) < ξ1 := (1 , 1 , 1 ), as t > T0 . By the first equation of (2), we have S˙ S µI 1 =b 1− − > b 1 − − µ1 > 0, t > T0 , S K 1 + aI 2 K where we let a = 0 and 1 small enough. Thus S(t) → ∞ as t → ∞, which leads to a contradiction. Next, we prove W s ((K, 0, 0)) ∩ X0 = φ. Suppose that it is not true. For any 2 > 0, there exists T1 > 0 such that (|S(t) − K|, E(t), I(t)) < ξ2 := (2 , 2 , 2 ), as t > T1 . By the second equation and the third equation in (2) and a = 0, we have µSI E˙ −(c + d) µ(K − 2 ) E 1+aI 2 − (c + d)E = ≥ · . c −γ I cE − γI I˙ Let
−(c + d) µ(K − 2 ) A= . c −γ The characteristic polynomial of A takes the form λ + (c + d) −µ(K − 2 ) = λ2 + (c + d + γ)λ + (c + d)γ − cµ(K − 2 ). −c λ+γ µcK > 1, we have (c + d)γ − cµ(K − 2 ) < 0. As 2 small enough and 0 such that limt→∞ inf(S(t), E(t), I(t)) ≥ ξ. This shows the uniform persistence of solutions of system (2).
4. Equilibria and media impact. Letting the right hand side of (2) equal zero, we find that the origin E0 = (0, 0, 0) is an equilibrium with eigenvalues b, −(c + d), −γ, and model (2) has one disease free equilibrium at E1 = (K, 0, 0). Clearly, E0 is a hyperbolic saddle point.
INFECTION MODEL WITH MEDIA IMPACT
From model (2), an endemic equilibrium satisfies the following equations: (c + d)γI S µSI = bS 1 − = , 2 K 1 + aI c
1323
(5)
which leads to µS − α(1 + aI 2 ) = 0,
(6)
bS(K − S) − αKI = 0,
(7)
where we write α := (c+d)γ . c Solving (6) for I yields s I = f1 (S) = which is a parabola for S ≥
α µ.
α S− , µ
(8)
b (K − S), Kα
(9)
µ aα
Define r
I = f2 (S) =
for 0 ≤ S ≤ K. α Clearly, the curves of functions f1 and f2 intersect once if K 2 < µ < K, that is α K 1 <
