Global stability of multigroup epidemic model with group mixing and ...
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Author's personal copy Applied Mathematics and Computation 218 (2011) 280–286
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Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates Ruoyan Sun, Junping Shi * Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA
a r t i c l e
i n f o
Keywords: Multigroup epidemic model Nonlinear incidence Group mixing Global stability Lyapunov function
a b s t r a c t In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R0. It shows that, the basic reproduction number R0 is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result. Ó 2011 Published by Elsevier Inc.
1. Introduction Multigroup models have been proposed in the literature to describe the transmission dynamics of infectious diseases in heterogeneous host populations, such as measles, mumps, gonorrhea, HIV/AIDS, West-Nile virus and vector borne diseases such as Malaria. Heterogeneity in host population can be the result of many factors. Groups can be geographical such as communities, cities, and countries, or epidemiological, to incorporate differential infectivity or co-infection of multiple strains of the disease agent. Much research has been done on multigroup models, for example, see [1–6] and references therein. It is well known that global dynamics of multigroup models with higher dimensions, especially the global stability of the endemic equilibrium, is a very challenging problem. The question of uniqueness and global stability of the endemic equilibrium, when the basic reproduction number R0 is greater than 1, has largely been open. Recently, the paper [7] proposed a graphtheoretic approach to the method of global Lyapunov functions and used it to establish the global stability of a unique endemic equilibrium of a multi-group SIR model with varying subpopulation sizes. Their result completely solved the open problem of the uniqueness and global stability of endemic equilibrium for this class of multi-group models. By using the results or ideas of [7], the papers [8–13] investigated uniqueness and global stability of the endemic equilibrium for several class of multigroup models, with the basic reproduction number R0 is greater than 1, and some open problems were resolved. In this paper, we consider a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. It covers many models in the literature, for example, the ones in [3,4,8,10,12,14,15]. The population is divided into n distinct groups (n P 1). For 1 6 k 6 n, the kth group is further partitioned into four compartments: the susceptible, exposed, infectious, and recovered, whose numbers of individuals at time t are denoted by Sk(t), Ek(t), Ik(t) and Rk(t), respectively. The new multigroup epidemic model with group mixing and nonlinear incidence rates as follows:
⇑ Corresponding author. E-mail address: [email protected] (J. Shi). 0096-3003/$ - see front matter Ó 2011 Published by Elsevier Inc. doi:10.1016/j.amc.2011.05.056
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8 n P > > S0k ¼ uk ðSk Þ bkj fkj ðSk ; Ij Þ; > > > j¼1 > > > < n P E0k ¼ bkj fkj ðSk ; Ij Þ lk g k ðEk Þ; > j¼1 > > > 0 > > I ¼ c > k g k ðEk Þ ak wk ðI k Þ; > : k0 Rk ¼ pk wk ðIk Þ qk ðRk Þ;
281
ð1Þ
where uk(Sk) denotes the net growth of the susceptible class in the kth group, the nonlinear term bkjf(Sk, Ij) represents the cross infection from group j to group k; The matrix B = (bij)nn is the irreducible contact matrix, where bij P 0; ckgk(Ek) accounts for the progression of individuals in group k from the exposed class into the infectious class; lkgk(Ek), akwk(Ik) and qk(Rk) denote the removal of the exposed, infectious and recovered classes in the kth group, respectively, which include the mortality of individuals in the above-mentioned classes; pkwk(Ik) denotes the production of the recovered individuals from infectious ones in the kth group. All constants lk, ck, ak and pk are assumed to be positive. In Section 2, we first obtain that the basic reproduction number R0 is a global threshold parameter in the sense that if it is less than or equal to one, then the disease free equilibrium is globally asymptotically stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally asymptotically stable and thus the disease persists in the population. And in Section 3, some numerical simulations are showed to illustrate the effectiveness of the proposed result. 2. Main results Since the variables Rk do not appear in the first three equations of (1), we can work on the reduced system as follows:
8 n P > > S0k ¼ uk ðSk Þ bkj fkj ðSk ; Ij Þ; > > > j¼1 < n P 0 Ek ¼ bkj fkj ðSk ; Ij Þ lk g k ðEk Þ; > > > j¼1 > > : 0 Ik ¼ ck g k ðEk Þ ak wk ðIk Þ:
ð2Þ
For the functions gk, wk and uk in (2), we assume that (G1) gk, wk are local Lipschitz on [0, 1) with gk(0) = wk(0) = 0, gk, wk are continuous, positive, on (0, 1), the function w uðuÞ is k non-increasing on (0,1), and limu!0þ w uðuÞ ¼ dk for positive constant dk > 0; k 0 (G2) uk are local Lipschitz on [0, 1) with uk(0) > 0, and the equation uk(u) = 0 admits a unique positive solution u ¼ Sk and uk ðSÞðS S0k Þ < 0 for S – S0k ; (G3) there exist constants Dk > 0 and Mk > 0 such that
1
max fu ðnÞg; for u P Dk ; lk n2½0;S0k k c wk ðuÞ P k max fuk ðnÞg; for u P Mk : ak lk n2½0;S0k
g k ðuÞ P
The basic assumptions on functions fkj(Sk, Ij) are as follows: (H1) 0 < limIj !0þ
fkj ðSk ;Ij Þ Ij
¼ C kj ðSk Þ 6 þ1, for 0 < Sk 6 S0k ;
(H2) fkj(Sk, Ij) 6 Ckj(Sk)Ij for all Ij > 0; (H3) C kj ðSk Þ 6 C kj ðS0k Þ, for 0 < Sk < S0k ; k; j ¼ 1; 2; . . . ; n. In addition, we assume that (G4)
R1
1 dx 0þ hðxÞ
¼ þ1, where h 2 {Ckk, gk, wk}, k = 1, 2, . . . , n.
Typical examples of gk and wk are gk(E) = bkE and wk(I) = ckI, and typical uk is uk(S) = dk ekS. Examples of fkj(Sk, Ij) satisfying (H1)–(H3) include common incidence functions such as
fkj ðSk ; Ij Þ ¼ Sk Ij ½7; 14—16; f kj ðSk ; Ij Þ ¼ fj ðIj ÞSk ½17; f kj ðSk ; Ij Þ ¼ Sqk Ij ½18; fkj ðSk ; Ij Þ ¼
gSk Ij pSk Ij bSk Ij ½2; f kj ðSk ; Ij Þ ¼ ½6; f kj ðSk ; Ij Þ ¼ ½19: 1 þ hSk uðIj Þ 1 þ aI2j
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For each k, adding the equations in (2), one obtains that
ðSk þ Ek Þ0 ¼ uk ðSk Þ lk g k ðEk Þ 6 max fuk ðuÞg lk g k ðEk Þ 6 0; u2½0;S0k
if Ek P Dk
and
0 l ak lk a k lk Sk þ Ek þ k Ik ¼ uk ðSk Þ wk ðIk Þ 6 max fuk ðuÞg wk ðIk Þ 6 0;
ck
ck
ck
u2½0;S0k
if Ik P Mk :
This indicates that the region
C ¼ ðS1 ; E1 ; I1 ; S2 ; E2 ; I2 . . . ; Sn ; En ; In Þ 2
R3n þ
: Sk 6
S0k ; Sk
þ Ek 6
S0k
l l þ Dk ; Sk þ Ek þ k Ik 6 S0k þ Dk þ k Mk ; k ¼ 1; 2; . . . ; n ck ck
ð3Þ denote the interior of C. is positively invariant with respect to (2). Let C It is clear that P 0 ¼ S01 ; 0; 0; S02 ; 0; 0; . . . ; S0n ; 0; 0 is a disease-free equilibrium of system (2). An equilibrium of C is called an endemic equilibrium, where S ; E ; I > 0 satisfy the folP ¼ S1 ; E1 ; I1 ; S2 ; E2 ; I2 ; . . . ; Sn ; En ; In in the interior C k k k lowing equilibrium equations
uk ðSk Þ ¼
n X
bkj fkj ðSk ; Ij Þ ¼ lk g k ðEk Þ;
ð4Þ
j¼1
ck g k ðEk Þ ¼ ak wk ðIk Þ:
ð5Þ
Set R0 = q(M0) denote the spectral radius of the following matrix
M0 ¼
MðS01 ; S02 ; . . . ; S0n Þ
bij dj ci C ij ðS0i Þ
!
:
ai li
nn
If C ij ðS0i Þ ¼ þ1 for some i and j, we set R0 = +1. The parameter R0 is referred to as the basic reproduction number. We have the following basic result: Theorem 2.1. Assume that the functions gk, wk, uk and fij satisfy (G1)–(G4) and (H1)–(H3), and the matrix B = (bij) is irreducible. (1) If R0 6 1, then P0 is the unique equilibrium of system (2) and it is globally asymptotically stable in C. . (2) If R0 > 1, then P0 is unstable and system (2) is uniformly persistent in C Proof. Let Q ðS; IÞ ¼
ci bij C ij ðSi ÞIj , ai li wj ðIj Þ nn
where S = (S1, S2, . . . , Sn) and I = (I1, I2, . . . , In). Since B = (bij)nn is irreducible, hence M0 is
also irreducible. One knows that there exist xk > 0, k = 1, 2, . . . , n, such that
ðx1 ; x2 ; . . . ; xn ÞqðM0 Þ ¼ ðx1 ; x2 ; . . . ; xn ÞM0 : Let V ¼
Pn
k¼1
0
V ¼
xk ðck Ek þlk Ik Þ , ak lk
n X k¼1
we have
" Pn
xk
ck
j¼1 bkj fkj ðSk ; Ij Þ
a k lk
#
n X
"
#
n c X wk ðIk Þ 6 xk k b C ðS ÞI wk ðIk Þ ¼ ðx1 ; x2 ; . . . ; xn Þ½Q ðS; IÞwðIÞ wðIÞ ak lk j¼1 kj kj k j k¼1
6 ðx1 ; x2 ; . . . ; xn Þ½M 0 wðIÞ wðIÞ ¼ ½qðM 0 Þ 1ðx1 ; x2 ; . . . ; xn ÞwðIÞ 6 0;
if qðM0 Þ 6 1:
Here w(I) = (w1(I1), w2(I2), . . . , wn(In)). If q(M0) < 1, then V0 = 0 if and only if I = 0. If q(M0) = 1, then V0 = 0 implies
"
#
n n X bkj C kj ðSk ÞIj c X xk k xk wk ðIk Þ: wj ðIj Þ ¼ ak lk j¼1 wj ðIj Þ k¼1 k¼1
n X
If S – S0 ðS01 ; S02 ; . . . ; S0n Þ, then
"
#
"
ð6Þ
#
n n n X bkj C kj ðSk ÞIj bkj C kj ðS0k ÞIj c X c X xk k xk k wj ðIj Þ < wj ðIj Þ 6 ðx1 ; x2 ; . . . ; xn ÞM0 wðIÞ ak lk j¼1 wj ðIj Þ ak lk j¼1 wj ðIj Þ k¼1 k¼1
n X
¼ ðx1 ; x2 ; . . . ; xn ÞqðM 0 ÞwðIÞ ¼ ðx1 ; x2 ; . . . ; xn ÞwðIÞ; which implies that (6) has only the trivial solution I = 0. Therefore, V0 = 0 if and only if I = 0 or S = S0 provided q(M0) 6 1. It can be verified that the only compact invariant subset of the set where V0 = 0 is the singleton {P0}. By LaSalle’s Invariance Principle, P0 is globally asymptotically stable in C if q(M0) 6 1.
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283
If R0 = q(M0) > 1 and I – 0, then
ðx1 ; x2 ; . . . ; xn ÞM 0 ðx1 ; x2 ; . . . ; xn Þ ¼ ½qðM0 Þ 1ðx1 ; x2 ; . . . ; xn Þ > 0 and thus, by continuity, n X
" Pn
ck
xk
j¼1 bkj fkj ðSk ; Ij Þ
a k lk
k¼1
# wk ðIk Þ > 0;
. This implies that P0 is unstable. Using a uniform persistence result from [20] and a similar arguin a neighborhood of P0 in C ment as in the Proof of Proposition 3.3 of [4], we can show that the instability of P0 implies the uniform persistence of system (2) when R0 > 1. This completes the Proof of Theorem 2.1. h Next we show that the endemic equilibrium P⁄ of system (2) is unique and globally asymptotically stable when R0 > 1. Note that the system (2) is uniformly persistent if R0 > 1 from the Theorem 2.1, together with the uniform boundedness , then system (2) admits at least one endemic equilibrium of solution of (2) in C
P ¼ ðS1 ; I1 ; E1 ; S2 ; I2 ; E2 ; . . . ; Sn ; In ; En Þ;
Si ; Ii
and Ei > 0 for 1 6 i 6 n:
To obtain our main global stability result, we further make the following assumptions: (G5) ðg k ðEk Þ g k ðEk ÞÞðEk Ek Þ > 0 for Ek – Ek , Ek P 0; (G6) ðwk ðIk Þ wk ðIk ÞÞðIk Ik Þ > 0 for Ik – Ik , Ik P 0; (G7) ðuk ðSk Þ uk ðSk ÞÞðSk Sk Þ 6 0 for Sk P 0. We notice that if gk (or wk) are increasing, then (G5) (or (G6)) holds; and if uk is decreasing, then (G7) holds. For examples given earlier, all these conditions are easily satisfied. However the monotonicity of functions gk, wk and uk are not necessary for (G5)–(G7) to hold. In Section 3, we give an example of non-monotone uk but (G7) is satisfied. Our main global stability result is: Theorem 2.2. Assume that the functions gk, wk, uk and fij satisfy (G1)–(G7) and (H1)–(H3), and the matrix B = (bij) is irreducible. If R0 > 1 and fkj(Sk, Ij) also satisfy the following conditions (H4) For Sk – Sk ,
½uk ðSk Þ uk ðSk Þ ½fkk ðSk ; Ik Þ fkk ðSk ; Ik Þ < 0; (H5) For Sk, Ij > 0,
ðfkk ðSk ; Ik Þfkj ðSk ; Ij Þ
fkj ðSk ; Ij Þfkk ðSk ; Ik ÞÞ
! fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkj ðSk ; Ij Þfkk ðSk ; Ik Þ 6 0; wj ðIj Þ wj ðIj Þ
. then there exists a unique endemic equilibrium P⁄ for system (2), and P⁄ is globally asymptotically stable in C , which implies that the endemic equilibrium is unique. Let Proof. We prove that P⁄ is globally asymptotically stable in C
Vk ¼
Z
Sk
Sk
fkk ðn; Ik Þ fkk ðSk ; Ik Þ dn þ fkk ðn; Ik Þ
Z
Ek
Ek
g k ðsÞ g k ðEk Þ l ds þ k g k ðsÞ ck
Z Ik
Ik
wk ðsÞ wk ðEk Þ ds: wk ðsÞ
Using equilibrium Eqs. (4) and (5), one obtains that
V 0k
# " # " n n X fkk ðSk ; Ik Þ g k ðEk Þ g k ðEk Þ X ¼ 1 uk ðSk Þ bkj fkj ðSk ; Ij Þ þ bkj fkj ðSk ; Ij Þ lk g k ðEk Þ fkk ðSk ; Ik Þ g k ðEk Þ j¼1 j¼1 þ
n X lk wk ðIk Þ wk ðIk Þ fkk ðSk ; Ik Þ fkk ðSk ; Ik Þ bkj fkj ðSk ; Ij Þ ½ck g k ðEk Þ ak wk ðIk Þ ¼ uk ðSk Þð1 Þþ þ lk g k ðEk Þ wk ðIk Þ fkk ðSk ; Ik Þ f ck ðS ; I Þ kk k k j¼1 n X
bkj fkj ðSk ; Ij Þ
j¼1
¼
n X j¼1
" bkj fkj ðSk ; Ij Þ
g k ðEk Þ ak lk w ðI Þ ak lk wk ðIk Þ lk g k ðEk Þ k k þ w ðI Þ g k ðEk Þ wk ðIk Þ ck ck k k
# fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þ a k lk w ðI Þ þ uk ðSk Þð1 2þ wk ðIk Þ lk g k ðEk Þ k k Þ fkk ðSk ; Ik Þ wk ðIk Þ ck fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þfkj ðSk ; Ij Þ
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# fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þ þ ½uk ðSk Þ uk ðSk Þ 1 ¼ 2þ fkk ðSk ; Ik Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þfkj ðSk ; Ij Þ j¼1 fkk ðSk ; Ik Þ ak lk w ðI Þ wk ðIk Þ lk g k ðEk Þ k k þ uk ðSk Þ 1 f"kk ðSk ; Ik Þ wk ðIk Þ ck # n X fkk ðSk ; Ik Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ wk ðIk Þ wk ðIk Þ g k ðEk Þ ¼ bkj fkj ðSk ; Ij Þ 3 þ þ ½uk ðSk Þ fkk ðSk ; Ik Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ wk ðIk Þ wk ðIk Þ g k ðEk Þ j¼1 fkk ðSk ; Ik Þ : uk ðSk Þ 1 fkk ðSk ; Ik Þ n X
"
bkj fkj ðSk ; Ij Þ
Let akj ¼ bkj fkj ðSk ; Ij Þ, and
F kj ðSk ; Ek ; Ik ; Ij Þ ¼ 3
fkk ðSk ; Ik Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ wk ðIk Þ wk ðIk Þ g k ðEk Þ þ : fkk ðSk ; Ik Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ wk ðIk Þ wk ðIk Þ g k ðEk Þ
Then, by (H4),
V 0k 6
n X
akj F kj ðSk ; Ik ; Ij Þ:
ð7Þ
j¼1
Let U(a) = 1 a + lna, then U(a) 6 0 for a > 0 and equality hold only at a = 1. Furthermore, under (H5),
! wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þ wk ðIk Þ g k ðEk Þ þU þU F kj ðSk ; Ek ; Ik ; Ij Þ ¼ Gk ðIk Þ Gj ðIj Þ þ U fkk ðSk ; Ik Þ wk ðIk Þ g k ðEk Þ wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ ! " # # " wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þ fkj ðSk ; Ij Þ þ þU 1 1 g k ðEk Þ fkj ðSk ; Ij Þ wj ðIj Þ fkk ðSk ; Ij Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ 6 Gk ðIk Þ Gj ðIj Þ;
ð8Þ
where Gk ðIk Þ ¼ wwk ðIðIk ÞÞ þ ln wwk ðIðIk ÞÞ. k
k
k
k
Obviously, the equalities in (7) and (8) hold if and only if
fkk ðSk ; Ik Þ ¼ 1; fkk ðSk ; Ik Þ
fkk ðSk ; Ik Þ ½uk ðSk Þ uk ðSk Þ 1 ¼0 fkk ðSk ; Ik Þ
and
! ! wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ 1 1 ¼ 0; wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ i.e., Sk ¼ Sk ; Ik ¼ Ik ; k ¼ 1; 2; . . . ; n. We can show that Vk, Fkj, Gk and akj satisfy the assumptions of Theorem 3.1 and Corollary 3.3 P in [10]. Therefore, the function V ¼ nk¼1 ck V k as defined in the Theorem 3.1 of [10] is a Lyapunov function for system (2), . One can only show that the largest invariant subset where V0 = 0 is namely, V0 6 0 for all (S1, I1, E1, S2, I2, E2, . . . , Sn, In, En) 2 C ⁄ the singleton {P } using the same argument as in [8,10]. By LaSalle’s Invariance Principle, P⁄ is globally asymptotically stable . This completes the Proof of Theorem 2.2. h in C Remark 1. We present a complete proof for global asymptotic stability of unique equilibrium P⁄ of system (2). The paper [12] gives part of the proof for that problem when fkj(Sk, Ij) = Ck(Sk)gj(Ij) in system (2). 3. Numerical example Consider the system (2) when k = 2, one has a two-group model as follows:
8 h i 0 S1 I 1 S1 I2 > > > S1 ¼ u1 ðS1 Þ b11 1þI21 þ b12 1þI22 ; > > h i > > > S1 I 1 S1 I2 > E01 ¼ b11 1þI > 2 þ b12 2 l1 g 1 ðE1 Þ; > 1þI > 1 2 > > < I0 ¼ c g ðE Þ a w ðI Þ; 1 1 1 1 1 1 1 h i > > S02 ¼ u2 ðS2 Þ b21 S2 I12 þ b22 S2 I22 ; > > 1þI1 1þI2 > > h i > > 0 S2 I 1 S2 I2 > > > E2 ¼ b21 1þI21 þ b22 1þI22 l2 g 2 ðE2 Þ; > > > : 0 I2 ¼ c2 g 2 ðE2 Þ a2 w2 ðI2 Þ;
ð9Þ
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where
fkj ðSk ; Ij Þ ¼
Sk I j 1 þ I2j
k; j ¼ 1; 2;
;
ð10Þ
u1 ðuÞ ¼ 2 u; u2 ðuÞ ¼ 3 u; g 1 ðuÞ ¼ g 2 ðuÞ ¼ w1 ðuÞ ¼ w2 ðuÞ ¼ u and
1 2
1 4
1 3
ð11Þ
5 ; 36
ð12Þ
c1 ¼ ; a1 ¼ 2; l1 ¼ ; c2 ¼ 1; a2 ¼ ; l2 ¼ 3: If bij are chosen as
b11 ¼
5 ; 24
b12 ¼
1 ; 24
b21 ¼
1 ; 36
b22 ¼
then we have R0 = 0.5 < 1, hence P0 = (2, 0, 0, 3, 0, 0) is the unique equilibrium of system (9) and it is globally stable in C from Theorem 2.1 (see Fig. 1 left panel). On the other hand, if bij are chosen as
b11 ¼
1 ; 2
b12 ¼ 1;
b21 ¼
2 ; 3
b22 ¼
1 ; 3
ð13Þ
then we have R0 = 3 > 1, hence P⁄ = (1.145, 3.421, 0.855, 2.005, 0.332, 0.995) is a unique endemic equilibrium for system (9) from Theorem 2.2 (see Fig. 1 right panel). and it is globally asymptotically stable in C As a second example, we still consider (9) but with
f1j ðS1 ; Ij Þ ¼ S21 Ij ; f 2j ðS2 ; Ij Þ ¼ S32 Ij ( 2 500 þ u þ 13 u 73 ; 27 u1 ðuÞ ¼ ð4 uÞð1 þ uÞ2 ; ( 2 256 þ u þ 16 u 23 ; 27 u2 ðuÞ ¼ ð2 uÞð2 þ uÞ2 ; g 1 ðuÞ ¼ g 2 ðuÞ ¼ u2 ;
j ¼ 1; 2; 0 6 u 6 73 ; u > 73 ;
ð14Þ
0 6 u 6 23 ; u > 23 ;
w1 ðuÞ ¼ w2 ðuÞ ¼ u
and
1 8
1 4
1 2
c1 ¼ ; a1 ¼ 1; l1 ¼ 2; c2 ¼ ; a2 ¼ 4; l2 ¼ :
ð15Þ
Notice here the growth rates of susceptible class uk are not monotone decreasing, but one can check that the condition (G7) is satisfied. If bij are chosen as
b11 ¼
3 ; 4
b12 ¼ 2;
b21 ¼ 0;
b22 ¼
2 ; 3
ð16Þ
8
8
7
7
6
6
5
5
4
4
3
1
I
1
0
10
S
1
I
1
E
0
E2 5
2
I2
1
I
1
S
2
E1
2
0 −1
3
S
2
E
1
S2
15
20
time t
25
30
35
40
−1
2
0
5
10
15
20
25
30
35
40
time t
Fig. 1. Numerical simulation of (9) with functions and parameters in (10) and (11). (Left) bij as in (12), and P0 is globally asymptotically stable; (Right) bij as in (13), and P⁄ is globally asymptotically stable. Initial condition in both graphs: S1(0) = 7, E1(0) = 3, I1(0) = 1.5, S2(0) = 5, E2(0) = 2, I2(0) = 1.
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14
12
12
10
10
8
8
6
6 S1
4
E
2
4
E
S
1
2
2 I1
0
I
2
E
E2
1
2
0
10
20
30
I1
40
50
time t
60
70
80
0
S
1
S
2
I
2
0
1
2
3
4
5
6
time t
Fig. 2. Numerical simulation of (9) with functions and parameters in (14) and (15). (Left) bij as in (16), and P0 is globally asymptotically stable; (Right) bij as in (17), and P⁄ is globally asymptotically stable. Initial condition in both graphs: S1(0) = 8, E1(0) = 4, I1(0) = 2.3, S2(0) = 10, E2(0) = 5.5, I2(0) = 3.
then we have R0 = 0.75 < 1, hence P0 = (4, 0, 0, 2, 0, 0) is the unique equilibrium of system (9) and it is globally stable in C from Theorem 2.1 (see Fig. 2 left panel). On the other hand, if bij are chosen as
b11 ¼ 1;
b12 ¼ 2;
b21 ¼ 1;
b22 ¼ 3;
ð17Þ
then we have R0 = 3.732 > 1, hence P⁄ = (2.422, 3.040, 1.159, 1.243, 3.990, 0.995) is a unique endemic equilibrium for system from Theorem 2.2 (see Fig. 2 right panel). (9) and it is globally asymptotically stable in C Acknowledgement This project is partially supported by a William and Mary Charles Center Biomathematics Summer Scholarship, NSF Grants DMS-1022648, DMS-0703532, EF-0436318 and William and Mary HHMI travel award. References [1] A. Lajmanovich, J.A. York, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976) 221–236. [2] R.M. Anderson, R.M. May, Population biology of infectious diseases I, Nature 280 (1979) 361–367. [3] W. Huang, K.L. Cooke, C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math. 52 (1992) 835–854. [4] M.Y. Li, J.R. Graef, L. Wang, J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci. 160 (1999) 191–213. [5] H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [6] D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci. 208 (2007) 419–429. [7] H. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q 14 (2006) 259–284. [8] H. Guo, M.Y. Li, Z. Shuai, A graphtheoretic approach to the method of global Lyapunov functions, Proc. Am. Math. Soc. 136 (2008) 2793–2802. [9] M.Y. Li, Z.S. Shuai, C.C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl. 361 (2010) 38–47. [10] M.Y. Li, Z.S. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Eqn. 248 (2010) 1–20. [11] C.C. McCluskey, Complete global stability for an SIR epidemic model with delay – distributed or discrete, Nonlinear Anal. Real World Appl. 11 (2010) 55–59. [12] Z.H. Yuan, L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. Real World Appl. 11 (2010) 995–1004. [13] Z.H. Yuan, X.F. Zou, Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population, Nonlinear Anal. Real World Appl. 11 (2010) 3479–3490. [14] A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol. 71 (2009) 75–83. [15] P. Georgescu, Y.H. Hsieh, H. Zhang, A Lyapunov functional for a stage structured predator–prey model with nonlinear predation rate, Nonlinear Anal. Real World Appl. 11 (2010) 3653–3665. [16] S.J. Gao, Z.D. Teng, D.H. Xie, The effects of pulse vaccination on SEIR model with two time delays, Appl. Math. Comput. 201 (2008) 282–292. [17] Y.N. Kyrychko, K.B. Blyuss, Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal. Real World Appl. 6 (2005) 495–507. [18] X. Wang, Y.D. Tao, X.Y. Song, Pulse vaccination on SEIR epidemic model with nonlinear incidence rate, Appl. Math. Comput. 210 (2009) 398–404. [19] L.M. Cai, X.Z. Lin, Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl. Math. Model. 33 (2009) 2919–2926. [20] H.I. Freedman, M.X. Tang, S.G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dyn. Differ. Eqn. 6 (1994) 583–600.
Author's personal copy Applied Mathematics and Computation 218 (2011) 280–286
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates Ruoyan Sun, Junping Shi * Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA
a r t i c l e
i n f o
Keywords: Multigroup epidemic model Nonlinear incidence Group mixing Global stability Lyapunov function
a b s t r a c t In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R0. It shows that, the basic reproduction number R0 is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result. Ó 2011 Published by Elsevier Inc.
1. Introduction Multigroup models have been proposed in the literature to describe the transmission dynamics of infectious diseases in heterogeneous host populations, such as measles, mumps, gonorrhea, HIV/AIDS, West-Nile virus and vector borne diseases such as Malaria. Heterogeneity in host population can be the result of many factors. Groups can be geographical such as communities, cities, and countries, or epidemiological, to incorporate differential infectivity or co-infection of multiple strains of the disease agent. Much research has been done on multigroup models, for example, see [1–6] and references therein. It is well known that global dynamics of multigroup models with higher dimensions, especially the global stability of the endemic equilibrium, is a very challenging problem. The question of uniqueness and global stability of the endemic equilibrium, when the basic reproduction number R0 is greater than 1, has largely been open. Recently, the paper [7] proposed a graphtheoretic approach to the method of global Lyapunov functions and used it to establish the global stability of a unique endemic equilibrium of a multi-group SIR model with varying subpopulation sizes. Their result completely solved the open problem of the uniqueness and global stability of endemic equilibrium for this class of multi-group models. By using the results or ideas of [7], the papers [8–13] investigated uniqueness and global stability of the endemic equilibrium for several class of multigroup models, with the basic reproduction number R0 is greater than 1, and some open problems were resolved. In this paper, we consider a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. It covers many models in the literature, for example, the ones in [3,4,8,10,12,14,15]. The population is divided into n distinct groups (n P 1). For 1 6 k 6 n, the kth group is further partitioned into four compartments: the susceptible, exposed, infectious, and recovered, whose numbers of individuals at time t are denoted by Sk(t), Ek(t), Ik(t) and Rk(t), respectively. The new multigroup epidemic model with group mixing and nonlinear incidence rates as follows:
⇑ Corresponding author. E-mail address: [email protected] (J. Shi). 0096-3003/$ - see front matter Ó 2011 Published by Elsevier Inc. doi:10.1016/j.amc.2011.05.056
Author's personal copy R. Sun, J. Shi / Applied Mathematics and Computation 218 (2011) 280–286
8 n P > > S0k ¼ uk ðSk Þ bkj fkj ðSk ; Ij Þ; > > > j¼1 > > > < n P E0k ¼ bkj fkj ðSk ; Ij Þ lk g k ðEk Þ; > j¼1 > > > 0 > > I ¼ c > k g k ðEk Þ ak wk ðI k Þ; > : k0 Rk ¼ pk wk ðIk Þ qk ðRk Þ;
281
ð1Þ
where uk(Sk) denotes the net growth of the susceptible class in the kth group, the nonlinear term bkjf(Sk, Ij) represents the cross infection from group j to group k; The matrix B = (bij)nn is the irreducible contact matrix, where bij P 0; ckgk(Ek) accounts for the progression of individuals in group k from the exposed class into the infectious class; lkgk(Ek), akwk(Ik) and qk(Rk) denote the removal of the exposed, infectious and recovered classes in the kth group, respectively, which include the mortality of individuals in the above-mentioned classes; pkwk(Ik) denotes the production of the recovered individuals from infectious ones in the kth group. All constants lk, ck, ak and pk are assumed to be positive. In Section 2, we first obtain that the basic reproduction number R0 is a global threshold parameter in the sense that if it is less than or equal to one, then the disease free equilibrium is globally asymptotically stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally asymptotically stable and thus the disease persists in the population. And in Section 3, some numerical simulations are showed to illustrate the effectiveness of the proposed result. 2. Main results Since the variables Rk do not appear in the first three equations of (1), we can work on the reduced system as follows:
8 n P > > S0k ¼ uk ðSk Þ bkj fkj ðSk ; Ij Þ; > > > j¼1 < n P 0 Ek ¼ bkj fkj ðSk ; Ij Þ lk g k ðEk Þ; > > > j¼1 > > : 0 Ik ¼ ck g k ðEk Þ ak wk ðIk Þ:
ð2Þ
For the functions gk, wk and uk in (2), we assume that (G1) gk, wk are local Lipschitz on [0, 1) with gk(0) = wk(0) = 0, gk, wk are continuous, positive, on (0, 1), the function w uðuÞ is k non-increasing on (0,1), and limu!0þ w uðuÞ ¼ dk for positive constant dk > 0; k 0 (G2) uk are local Lipschitz on [0, 1) with uk(0) > 0, and the equation uk(u) = 0 admits a unique positive solution u ¼ Sk and uk ðSÞðS S0k Þ < 0 for S – S0k ; (G3) there exist constants Dk > 0 and Mk > 0 such that
1
max fu ðnÞg; for u P Dk ; lk n2½0;S0k k c wk ðuÞ P k max fuk ðnÞg; for u P Mk : ak lk n2½0;S0k
g k ðuÞ P
The basic assumptions on functions fkj(Sk, Ij) are as follows: (H1) 0 < limIj !0þ
fkj ðSk ;Ij Þ Ij
¼ C kj ðSk Þ 6 þ1, for 0 < Sk 6 S0k ;
(H2) fkj(Sk, Ij) 6 Ckj(Sk)Ij for all Ij > 0; (H3) C kj ðSk Þ 6 C kj ðS0k Þ, for 0 < Sk < S0k ; k; j ¼ 1; 2; . . . ; n. In addition, we assume that (G4)
R1
1 dx 0þ hðxÞ
¼ þ1, where h 2 {Ckk, gk, wk}, k = 1, 2, . . . , n.
Typical examples of gk and wk are gk(E) = bkE and wk(I) = ckI, and typical uk is uk(S) = dk ekS. Examples of fkj(Sk, Ij) satisfying (H1)–(H3) include common incidence functions such as
fkj ðSk ; Ij Þ ¼ Sk Ij ½7; 14—16; f kj ðSk ; Ij Þ ¼ fj ðIj ÞSk ½17; f kj ðSk ; Ij Þ ¼ Sqk Ij ½18; fkj ðSk ; Ij Þ ¼
gSk Ij pSk Ij bSk Ij ½2; f kj ðSk ; Ij Þ ¼ ½6; f kj ðSk ; Ij Þ ¼ ½19: 1 þ hSk uðIj Þ 1 þ aI2j
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For each k, adding the equations in (2), one obtains that
ðSk þ Ek Þ0 ¼ uk ðSk Þ lk g k ðEk Þ 6 max fuk ðuÞg lk g k ðEk Þ 6 0; u2½0;S0k
if Ek P Dk
and
0 l ak lk a k lk Sk þ Ek þ k Ik ¼ uk ðSk Þ wk ðIk Þ 6 max fuk ðuÞg wk ðIk Þ 6 0;
ck
ck
ck
u2½0;S0k
if Ik P Mk :
This indicates that the region
C ¼ ðS1 ; E1 ; I1 ; S2 ; E2 ; I2 . . . ; Sn ; En ; In Þ 2
R3n þ
: Sk 6
S0k ; Sk
þ Ek 6
S0k
l l þ Dk ; Sk þ Ek þ k Ik 6 S0k þ Dk þ k Mk ; k ¼ 1; 2; . . . ; n ck ck
ð3Þ denote the interior of C. is positively invariant with respect to (2). Let C It is clear that P 0 ¼ S01 ; 0; 0; S02 ; 0; 0; . . . ; S0n ; 0; 0 is a disease-free equilibrium of system (2). An equilibrium of C is called an endemic equilibrium, where S ; E ; I > 0 satisfy the folP ¼ S1 ; E1 ; I1 ; S2 ; E2 ; I2 ; . . . ; Sn ; En ; In in the interior C k k k lowing equilibrium equations
uk ðSk Þ ¼
n X
bkj fkj ðSk ; Ij Þ ¼ lk g k ðEk Þ;
ð4Þ
j¼1
ck g k ðEk Þ ¼ ak wk ðIk Þ:
ð5Þ
Set R0 = q(M0) denote the spectral radius of the following matrix
M0 ¼
MðS01 ; S02 ; . . . ; S0n Þ
bij dj ci C ij ðS0i Þ
!
:
ai li
nn
If C ij ðS0i Þ ¼ þ1 for some i and j, we set R0 = +1. The parameter R0 is referred to as the basic reproduction number. We have the following basic result: Theorem 2.1. Assume that the functions gk, wk, uk and fij satisfy (G1)–(G4) and (H1)–(H3), and the matrix B = (bij) is irreducible. (1) If R0 6 1, then P0 is the unique equilibrium of system (2) and it is globally asymptotically stable in C. . (2) If R0 > 1, then P0 is unstable and system (2) is uniformly persistent in C Proof. Let Q ðS; IÞ ¼
ci bij C ij ðSi ÞIj , ai li wj ðIj Þ nn
where S = (S1, S2, . . . , Sn) and I = (I1, I2, . . . , In). Since B = (bij)nn is irreducible, hence M0 is
also irreducible. One knows that there exist xk > 0, k = 1, 2, . . . , n, such that
ðx1 ; x2 ; . . . ; xn ÞqðM0 Þ ¼ ðx1 ; x2 ; . . . ; xn ÞM0 : Let V ¼
Pn
k¼1
0
V ¼
xk ðck Ek þlk Ik Þ , ak lk
n X k¼1
we have
" Pn
xk
ck
j¼1 bkj fkj ðSk ; Ij Þ
a k lk
#
n X
"
#
n c X wk ðIk Þ 6 xk k b C ðS ÞI wk ðIk Þ ¼ ðx1 ; x2 ; . . . ; xn Þ½Q ðS; IÞwðIÞ wðIÞ ak lk j¼1 kj kj k j k¼1
6 ðx1 ; x2 ; . . . ; xn Þ½M 0 wðIÞ wðIÞ ¼ ½qðM 0 Þ 1ðx1 ; x2 ; . . . ; xn ÞwðIÞ 6 0;
if qðM0 Þ 6 1:
Here w(I) = (w1(I1), w2(I2), . . . , wn(In)). If q(M0) < 1, then V0 = 0 if and only if I = 0. If q(M0) = 1, then V0 = 0 implies
"
#
n n X bkj C kj ðSk ÞIj c X xk k xk wk ðIk Þ: wj ðIj Þ ¼ ak lk j¼1 wj ðIj Þ k¼1 k¼1
n X
If S – S0 ðS01 ; S02 ; . . . ; S0n Þ, then
"
#
"
ð6Þ
#
n n n X bkj C kj ðSk ÞIj bkj C kj ðS0k ÞIj c X c X xk k xk k wj ðIj Þ < wj ðIj Þ 6 ðx1 ; x2 ; . . . ; xn ÞM0 wðIÞ ak lk j¼1 wj ðIj Þ ak lk j¼1 wj ðIj Þ k¼1 k¼1
n X
¼ ðx1 ; x2 ; . . . ; xn ÞqðM 0 ÞwðIÞ ¼ ðx1 ; x2 ; . . . ; xn ÞwðIÞ; which implies that (6) has only the trivial solution I = 0. Therefore, V0 = 0 if and only if I = 0 or S = S0 provided q(M0) 6 1. It can be verified that the only compact invariant subset of the set where V0 = 0 is the singleton {P0}. By LaSalle’s Invariance Principle, P0 is globally asymptotically stable in C if q(M0) 6 1.
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283
If R0 = q(M0) > 1 and I – 0, then
ðx1 ; x2 ; . . . ; xn ÞM 0 ðx1 ; x2 ; . . . ; xn Þ ¼ ½qðM0 Þ 1ðx1 ; x2 ; . . . ; xn Þ > 0 and thus, by continuity, n X
" Pn
ck
xk
j¼1 bkj fkj ðSk ; Ij Þ
a k lk
k¼1
# wk ðIk Þ > 0;
. This implies that P0 is unstable. Using a uniform persistence result from [20] and a similar arguin a neighborhood of P0 in C ment as in the Proof of Proposition 3.3 of [4], we can show that the instability of P0 implies the uniform persistence of system (2) when R0 > 1. This completes the Proof of Theorem 2.1. h Next we show that the endemic equilibrium P⁄ of system (2) is unique and globally asymptotically stable when R0 > 1. Note that the system (2) is uniformly persistent if R0 > 1 from the Theorem 2.1, together with the uniform boundedness , then system (2) admits at least one endemic equilibrium of solution of (2) in C
P ¼ ðS1 ; I1 ; E1 ; S2 ; I2 ; E2 ; . . . ; Sn ; In ; En Þ;
Si ; Ii
and Ei > 0 for 1 6 i 6 n:
To obtain our main global stability result, we further make the following assumptions: (G5) ðg k ðEk Þ g k ðEk ÞÞðEk Ek Þ > 0 for Ek – Ek , Ek P 0; (G6) ðwk ðIk Þ wk ðIk ÞÞðIk Ik Þ > 0 for Ik – Ik , Ik P 0; (G7) ðuk ðSk Þ uk ðSk ÞÞðSk Sk Þ 6 0 for Sk P 0. We notice that if gk (or wk) are increasing, then (G5) (or (G6)) holds; and if uk is decreasing, then (G7) holds. For examples given earlier, all these conditions are easily satisfied. However the monotonicity of functions gk, wk and uk are not necessary for (G5)–(G7) to hold. In Section 3, we give an example of non-monotone uk but (G7) is satisfied. Our main global stability result is: Theorem 2.2. Assume that the functions gk, wk, uk and fij satisfy (G1)–(G7) and (H1)–(H3), and the matrix B = (bij) is irreducible. If R0 > 1 and fkj(Sk, Ij) also satisfy the following conditions (H4) For Sk – Sk ,
½uk ðSk Þ uk ðSk Þ ½fkk ðSk ; Ik Þ fkk ðSk ; Ik Þ < 0; (H5) For Sk, Ij > 0,
ðfkk ðSk ; Ik Þfkj ðSk ; Ij Þ
fkj ðSk ; Ij Þfkk ðSk ; Ik ÞÞ
! fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkj ðSk ; Ij Þfkk ðSk ; Ik Þ 6 0; wj ðIj Þ wj ðIj Þ
. then there exists a unique endemic equilibrium P⁄ for system (2), and P⁄ is globally asymptotically stable in C , which implies that the endemic equilibrium is unique. Let Proof. We prove that P⁄ is globally asymptotically stable in C
Vk ¼
Z
Sk
Sk
fkk ðn; Ik Þ fkk ðSk ; Ik Þ dn þ fkk ðn; Ik Þ
Z
Ek
Ek
g k ðsÞ g k ðEk Þ l ds þ k g k ðsÞ ck
Z Ik
Ik
wk ðsÞ wk ðEk Þ ds: wk ðsÞ
Using equilibrium Eqs. (4) and (5), one obtains that
V 0k
# " # " n n X fkk ðSk ; Ik Þ g k ðEk Þ g k ðEk Þ X ¼ 1 uk ðSk Þ bkj fkj ðSk ; Ij Þ þ bkj fkj ðSk ; Ij Þ lk g k ðEk Þ fkk ðSk ; Ik Þ g k ðEk Þ j¼1 j¼1 þ
n X lk wk ðIk Þ wk ðIk Þ fkk ðSk ; Ik Þ fkk ðSk ; Ik Þ bkj fkj ðSk ; Ij Þ ½ck g k ðEk Þ ak wk ðIk Þ ¼ uk ðSk Þð1 Þþ þ lk g k ðEk Þ wk ðIk Þ fkk ðSk ; Ik Þ f ck ðS ; I Þ kk k k j¼1 n X
bkj fkj ðSk ; Ij Þ
j¼1
¼
n X j¼1
" bkj fkj ðSk ; Ij Þ
g k ðEk Þ ak lk w ðI Þ ak lk wk ðIk Þ lk g k ðEk Þ k k þ w ðI Þ g k ðEk Þ wk ðIk Þ ck ck k k
# fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þ a k lk w ðI Þ þ uk ðSk Þð1 2þ wk ðIk Þ lk g k ðEk Þ k k Þ fkk ðSk ; Ik Þ wk ðIk Þ ck fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þfkj ðSk ; Ij Þ
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# fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þ þ ½uk ðSk Þ uk ðSk Þ 1 ¼ 2þ fkk ðSk ; Ik Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þfkj ðSk ; Ij Þ j¼1 fkk ðSk ; Ik Þ ak lk w ðI Þ wk ðIk Þ lk g k ðEk Þ k k þ uk ðSk Þ 1 f"kk ðSk ; Ik Þ wk ðIk Þ ck # n X fkk ðSk ; Ik Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ wk ðIk Þ wk ðIk Þ g k ðEk Þ ¼ bkj fkj ðSk ; Ij Þ 3 þ þ ½uk ðSk Þ fkk ðSk ; Ik Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ wk ðIk Þ wk ðIk Þ g k ðEk Þ j¼1 fkk ðSk ; Ik Þ : uk ðSk Þ 1 fkk ðSk ; Ik Þ n X
"
bkj fkj ðSk ; Ij Þ
Let akj ¼ bkj fkj ðSk ; Ij Þ, and
F kj ðSk ; Ek ; Ik ; Ij Þ ¼ 3
fkk ðSk ; Ik Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ wk ðIk Þ wk ðIk Þ g k ðEk Þ þ : fkk ðSk ; Ik Þ g k ðEk Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ wk ðIk Þ wk ðIk Þ g k ðEk Þ
Then, by (H4),
V 0k 6
n X
akj F kj ðSk ; Ik ; Ij Þ:
ð7Þ
j¼1
Let U(a) = 1 a + lna, then U(a) 6 0 for a > 0 and equality hold only at a = 1. Furthermore, under (H5),
! wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þ wk ðIk Þ g k ðEk Þ þU þU F kj ðSk ; Ek ; Ik ; Ij Þ ¼ Gk ðIk Þ Gj ðIj Þ þ U fkk ðSk ; Ik Þ wk ðIk Þ g k ðEk Þ wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ ! " # # " wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ g k ðEk Þ fkj ðSk ; Ij Þ þ þU 1 1 g k ðEk Þ fkj ðSk ; Ij Þ wj ðIj Þ fkk ðSk ; Ij Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ 6 Gk ðIk Þ Gj ðIj Þ;
ð8Þ
where Gk ðIk Þ ¼ wwk ðIðIk ÞÞ þ ln wwk ðIðIk ÞÞ. k
k
k
k
Obviously, the equalities in (7) and (8) hold if and only if
fkk ðSk ; Ik Þ ¼ 1; fkk ðSk ; Ik Þ
fkk ðSk ; Ik Þ ½uk ðSk Þ uk ðSk Þ 1 ¼0 fkk ðSk ; Ik Þ
and
! ! wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ 1 1 ¼ 0; wj ðIj Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ fkk ðSk ; Ik Þfkj ðSk ; Ij Þ i.e., Sk ¼ Sk ; Ik ¼ Ik ; k ¼ 1; 2; . . . ; n. We can show that Vk, Fkj, Gk and akj satisfy the assumptions of Theorem 3.1 and Corollary 3.3 P in [10]. Therefore, the function V ¼ nk¼1 ck V k as defined in the Theorem 3.1 of [10] is a Lyapunov function for system (2), . One can only show that the largest invariant subset where V0 = 0 is namely, V0 6 0 for all (S1, I1, E1, S2, I2, E2, . . . , Sn, In, En) 2 C ⁄ the singleton {P } using the same argument as in [8,10]. By LaSalle’s Invariance Principle, P⁄ is globally asymptotically stable . This completes the Proof of Theorem 2.2. h in C Remark 1. We present a complete proof for global asymptotic stability of unique equilibrium P⁄ of system (2). The paper [12] gives part of the proof for that problem when fkj(Sk, Ij) = Ck(Sk)gj(Ij) in system (2). 3. Numerical example Consider the system (2) when k = 2, one has a two-group model as follows:
8 h i 0 S1 I 1 S1 I2 > > > S1 ¼ u1 ðS1 Þ b11 1þI21 þ b12 1þI22 ; > > h i > > > S1 I 1 S1 I2 > E01 ¼ b11 1þI > 2 þ b12 2 l1 g 1 ðE1 Þ; > 1þI > 1 2 > > < I0 ¼ c g ðE Þ a w ðI Þ; 1 1 1 1 1 1 1 h i > > S02 ¼ u2 ðS2 Þ b21 S2 I12 þ b22 S2 I22 ; > > 1þI1 1þI2 > > h i > > 0 S2 I 1 S2 I2 > > > E2 ¼ b21 1þI21 þ b22 1þI22 l2 g 2 ðE2 Þ; > > > : 0 I2 ¼ c2 g 2 ðE2 Þ a2 w2 ðI2 Þ;
ð9Þ
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where
fkj ðSk ; Ij Þ ¼
Sk I j 1 þ I2j
k; j ¼ 1; 2;
;
ð10Þ
u1 ðuÞ ¼ 2 u; u2 ðuÞ ¼ 3 u; g 1 ðuÞ ¼ g 2 ðuÞ ¼ w1 ðuÞ ¼ w2 ðuÞ ¼ u and
1 2
1 4
1 3
ð11Þ
5 ; 36
ð12Þ
c1 ¼ ; a1 ¼ 2; l1 ¼ ; c2 ¼ 1; a2 ¼ ; l2 ¼ 3: If bij are chosen as
b11 ¼
5 ; 24
b12 ¼
1 ; 24
b21 ¼
1 ; 36
b22 ¼
then we have R0 = 0.5 < 1, hence P0 = (2, 0, 0, 3, 0, 0) is the unique equilibrium of system (9) and it is globally stable in C from Theorem 2.1 (see Fig. 1 left panel). On the other hand, if bij are chosen as
b11 ¼
1 ; 2
b12 ¼ 1;
b21 ¼
2 ; 3
b22 ¼
1 ; 3
ð13Þ
then we have R0 = 3 > 1, hence P⁄ = (1.145, 3.421, 0.855, 2.005, 0.332, 0.995) is a unique endemic equilibrium for system (9) from Theorem 2.2 (see Fig. 1 right panel). and it is globally asymptotically stable in C As a second example, we still consider (9) but with
f1j ðS1 ; Ij Þ ¼ S21 Ij ; f 2j ðS2 ; Ij Þ ¼ S32 Ij ( 2 500 þ u þ 13 u 73 ; 27 u1 ðuÞ ¼ ð4 uÞð1 þ uÞ2 ; ( 2 256 þ u þ 16 u 23 ; 27 u2 ðuÞ ¼ ð2 uÞð2 þ uÞ2 ; g 1 ðuÞ ¼ g 2 ðuÞ ¼ u2 ;
j ¼ 1; 2; 0 6 u 6 73 ; u > 73 ;
ð14Þ
0 6 u 6 23 ; u > 23 ;
w1 ðuÞ ¼ w2 ðuÞ ¼ u
and
1 8
1 4
1 2
c1 ¼ ; a1 ¼ 1; l1 ¼ 2; c2 ¼ ; a2 ¼ 4; l2 ¼ :
ð15Þ
Notice here the growth rates of susceptible class uk are not monotone decreasing, but one can check that the condition (G7) is satisfied. If bij are chosen as
b11 ¼
3 ; 4
b12 ¼ 2;
b21 ¼ 0;
b22 ¼
2 ; 3
ð16Þ
8
8
7
7
6
6
5
5
4
4
3
1
I
1
0
10
S
1
I
1
E
0
E2 5
2
I2
1
I
1
S
2
E1
2
0 −1
3
S
2
E
1
S2
15
20
time t
25
30
35
40
−1
2
0
5
10
15
20
25
30
35
40
time t
Fig. 1. Numerical simulation of (9) with functions and parameters in (10) and (11). (Left) bij as in (12), and P0 is globally asymptotically stable; (Right) bij as in (13), and P⁄ is globally asymptotically stable. Initial condition in both graphs: S1(0) = 7, E1(0) = 3, I1(0) = 1.5, S2(0) = 5, E2(0) = 2, I2(0) = 1.
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R. Sun, J. Shi / Applied Mathematics and Computation 218 (2011) 280–286 14
14
12
12
10
10
8
8
6
6 S1
4
E
2
4
E
S
1
2
2 I1
0
I
2
E
E2
1
2
0
10
20
30
I1
40
50
time t
60
70
80
0
S
1
S
2
I
2
0
1
2
3
4
5
6
time t
Fig. 2. Numerical simulation of (9) with functions and parameters in (14) and (15). (Left) bij as in (16), and P0 is globally asymptotically stable; (Right) bij as in (17), and P⁄ is globally asymptotically stable. Initial condition in both graphs: S1(0) = 8, E1(0) = 4, I1(0) = 2.3, S2(0) = 10, E2(0) = 5.5, I2(0) = 3.
then we have R0 = 0.75 < 1, hence P0 = (4, 0, 0, 2, 0, 0) is the unique equilibrium of system (9) and it is globally stable in C from Theorem 2.1 (see Fig. 2 left panel). On the other hand, if bij are chosen as
b11 ¼ 1;
b12 ¼ 2;
b21 ¼ 1;
b22 ¼ 3;
ð17Þ
then we have R0 = 3.732 > 1, hence P⁄ = (2.422, 3.040, 1.159, 1.243, 3.990, 0.995) is a unique endemic equilibrium for system from Theorem 2.2 (see Fig. 2 right panel). (9) and it is globally asymptotically stable in C Acknowledgement This project is partially supported by a William and Mary Charles Center Biomathematics Summer Scholarship, NSF Grants DMS-1022648, DMS-0703532, EF-0436318 and William and Mary HHMI travel award. References [1] A. Lajmanovich, J.A. York, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976) 221–236. [2] R.M. Anderson, R.M. May, Population biology of infectious diseases I, Nature 280 (1979) 361–367. [3] W. Huang, K.L. Cooke, C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math. 52 (1992) 835–854. [4] M.Y. Li, J.R. Graef, L. Wang, J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci. 160 (1999) 191–213. [5] H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [6] D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci. 208 (2007) 419–429. [7] H. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q 14 (2006) 259–284. [8] H. Guo, M.Y. Li, Z. Shuai, A graphtheoretic approach to the method of global Lyapunov functions, Proc. Am. Math. Soc. 136 (2008) 2793–2802. [9] M.Y. Li, Z.S. Shuai, C.C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl. 361 (2010) 38–47. [10] M.Y. Li, Z.S. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Eqn. 248 (2010) 1–20. [11] C.C. McCluskey, Complete global stability for an SIR epidemic model with delay – distributed or discrete, Nonlinear Anal. Real World Appl. 11 (2010) 55–59. [12] Z.H. Yuan, L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. Real World Appl. 11 (2010) 995–1004. [13] Z.H. Yuan, X.F. Zou, Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population, Nonlinear Anal. Real World Appl. 11 (2010) 3479–3490. [14] A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol. 71 (2009) 75–83. [15] P. Georgescu, Y.H. Hsieh, H. Zhang, A Lyapunov functional for a stage structured predator–prey model with nonlinear predation rate, Nonlinear Anal. Real World Appl. 11 (2010) 3653–3665. [16] S.J. Gao, Z.D. Teng, D.H. Xie, The effects of pulse vaccination on SEIR model with two time delays, Appl. Math. Comput. 201 (2008) 282–292. [17] Y.N. Kyrychko, K.B. Blyuss, Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal. Real World Appl. 6 (2005) 495–507. [18] X. Wang, Y.D. Tao, X.Y. Song, Pulse vaccination on SEIR epidemic model with nonlinear incidence rate, Appl. Math. Comput. 210 (2009) 398–404. [19] L.M. Cai, X.Z. Lin, Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl. Math. Model. 33 (2009) 2919–2926. [20] H.I. Freedman, M.X. Tang, S.G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dyn. Differ. Eqn. 6 (1994) 583–600.
