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Global dynamics and bifurcation in delayed SIR epidemic model
Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Global dynamics and bifurcation in delayed SIR epidemic model T.K. Kar a,∗ , Prasanta Kumar Mondal b a
Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-3, India
b
Department of Mathematics, Shibpur Sri Ramkrishna Vidyalaya, Howrah-2, India
article
info
Article history: Received 1 November 2010 Accepted 22 December 2010 Keywords: SIR epidemic model Transcritical bifurcation Hopf bifurcation Global stability
abstract An SIR epidemic model with time delay, information variable and saturated incidence rate, where the susceptibles are assumed to satisfy the logistic equation and the incidence term, is of saturated form with the susceptibles. This model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. The local and global stability of endemic equilibrium is also discussed. Finally, numerical simulations are carried out to explain the mathematical conclusions. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Mathematical epidemiology, i.e. the building and analysis of mathematical models describing the spread and control of infectious diseases is one of the major areas of biology. In many epidemic models the population can be divided into three distinct classes, which are defined with respect to disease status. Kermack and Mckendrick [1] described the simplest SIR model which computes the theoretical number of people infected with a contagious illness in a closed population over time. Transmission of a disease is a dynamical process driven by the interaction between susceptible and infective. The behaviour of the SIR models are greatly affected by the way in which transmission between infected and susceptible individuals are modelled. Most of the models of epidemiology are based on the so-called ‘‘mass action’’ [2] assumption has faced some questions and consequently a number of realistic transmission functions have become the focus of considerable attention [3,4]. A delay differential equation has been successfully used to model varying infectious period in a range of SIR, SIS and SIRS epidemic models. Hethcote and Van den Driessche [5] have considered an SIS epidemic model with constant time delay, which accounts for duration of infectiousness. Beretta et al. [6] have studied global stability in an SIR epidemic model with distributed delay that describes the time it takes for an individual to lose infectiousness. Song and Cheng [7] have studied the effect of time delay on the stability of the endemic equilibrium. They were given some conditions for which the endemic equilibrium is asymptotically stable for all delays and also discussed the existence of orbitally asymptotically stable periodic solutions. The mathematical analysis of epidemiological modelling is often used for the assessment of the global asymptotic stability of both the disease free and endemic equilibrium. The modelling and analysis of infectious diseases have been done also by some other workers; see for example, [8–12] and the references therein. Here we will study SIR models with nonlinear incidence [13] and information variable. In most of the epidemiological models, the incidence rate is assumed of the form β SI, where β is the transmission rate, S and I are the susceptible and infected population respectively. Cai et al. [14] used a saturated incidence rate of the form bβ SI /(1 + α I ) where b, β, α > 0. The paper is organized as follows. In the next section, we present the model and its equilibria. In Section 3, we carry out a qualitative analysis of the model. Stability conditions for the disease free equilibrium and the endemic equilibrium are
∗
Corresponding author. Tel.: +91 33 26684561. E-mail addresses: [email protected] (T.K. Kar), [email protected] (P.K. Mondal).
1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.12.021
T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
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derived. Bifurcations also given in this section. Section 4 describes the global stability of the endemic equilibrium. Finally, some numerical simulations are given in Section 5. 2. The model and its equilibria We consider the following SIR epidemic model: dS dt dI dt dR dt
S = rS 1 − − k
=
β IS 1 + αS
β SI 1 + αS
,
− µ1 I − γ I ,
(1)
= γ I − µ2 R ,
where S (0), I (0), R(0) ≥ 0 and S is the density of susceptibles within the population, I is the density of infected within the population, R is the density of recovered within the population, and the parameters are r is the intrinsic growth rate of susceptibles, k is the carrying capacity of susceptibles, α is the saturation factor that measures the inhibitory effect, β is the transmission or contact rate, µ1 is the death rate, γ is the rate of recovery from infection, µ2 is the death rate. Consider the new variable Z , called information variable which summarizes information about the current state of the disease i.e. depends on current values of state variables and also summarizes information about past values of state variables. Many authors used this variable in their model (see e.g. [15–17]). We take up the formula Z (t ) =
∫
t
g (S (τ ), I (τ ))K (t − τ )dτ ,
(2)
−∞
where K (t − τ ) is the delaying kernel [18], τ is the distributed delay, which means that at time t susceptible individuals, S and infective individuals, I are affected by the state variables S and I at possibly all previous times τ ≤ t. In this paper we considered g (S , I ) = S, and K (t − τ ) = T1 exp − T1 (t − τ ) , where T is the average delay of the collected information on the disease, as well as the average length of the historical memory concerning the disease in study i.e. T is the ‘‘measure of the influence of the past’’. From the above assumptions we consider the following model: dS dt
S = rS 1 − − k
β SI , 1 + αS
β IZ = − µ1 I − γ I , dt 1 + αZ ∫ t 1 1 Z (t ) = S . exp − (t − τ ) dτ , dI
−∞
T
(3)
T
dR
= γ I − µ2 R . dt Then the nonlinear integro-differential system (3) can be transformed into the following set of nonlinear ordinary differential equations: dS dt
S = rS 1 − − k
β SI , 1 + αS
β IZ = − µ1 I − γ I , dt 1 + αZ dI
dZ dt dR dt
=
1 T
(S − Z ),
= γ I − µ2 R .
(4)
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Since the dynamical behaviour of the last equation of the system (4) i.e. the dynamics of R depends only the dynamics of I, so we do not consider that equation in our discussion. Here we will study the following nonlinear ordinary differential equations: dS dt
S = rS 1 − − k
β SI , 1 + αS
β IZ = − µ1 I − γ I , dt 1 + αZ dI
dZ dt
=
1 T
(5)
(S − Z ),
where r , k, α, β, µ1 , γ , T > 0. Theorem 1. (a) The system (5) has a trivial equilibrium E0 = (0, 0, 0) and the disease free equilibrium E1 = (k, 0, k). kβ
(b) Also, if R0 > 1, the system (5) has one endemic equilibrium point E = (S ∗ , I ∗ , Z ∗ ) where R0 = (µ +γ )(kα+1) is the basic 1 µ1 +γ reproduction number and S ∗ = Z ∗ = β−α(µ = m1 , I ∗ = βrk (1 + m1 α)(k − m1 ) = m2 . +γ ) 1
Proof. Proof is omitted here.
3. Stability of equilibrium points and bifurcation The Jacobian matrix of the system (5) is
1−
r
S
βI r αβ I +S − + 1 + αS k (1 + α S )2
−
k
J (S , I , Z ) =
0
− βZ
βS 1 + αS
1 + αZ
1
− µ1 − γ 0
T
0
βI . (1 + α Z )2 1 − T
Theorem 2. The equilibrium point E0 is unstable for all T > 0. Proof. The Jacobian matrix at E0 is
r 0 J (0, 0, 0) = 1
0
0 0 . 1
−(µ1 + γ )
−
0
T
The eigen values are r , −(µ1 + γ ), − Therefore E0 is unstable.
T
1 . T
Theorem 3. (i) If R0 > 1, then E1 is unstable for all T > 0, (ii) If R0 < 1, then E1 is locally asymptotically stable for all T > 0. Proof. The Jacobian matrix at E1 is
−r
J (k, 0, k) = 0
1
βk 1 + αk βk − µ1 − γ 1 + αk −
βk 1+α k
βk 1+α k
0 . 1 −
0
T The eigen values are −r ,
0
T
− µ1 − γ , − > µ1 + γ i.e. if β > (µ1 + γ ) α + 1k , i.e. if R0 > 1, the equilibrium point E1 is unstable. 1 . T
If If R0 < 1, then the equilibrium point E1 is locally asymptotically stable.
T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
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Note. From the above theorem, we can see that the stability of disease free equilibrium point E1 changes from stable to unstable when R0 increases though 1. Therefore, we use R0 as bifurcation parameter. For simplicity, let
β SI k 1 + αS β IZ − (µ1 + γ )I 1 + αZ 1 (S − Z )
rS
f1 (X , β, T ) f2 (X , β, T ) f3 (X , β, T )
fW (X , β, T ) =
=
S
1−
−
and X = [S , I , Z ]T .
T
Theorem 4. The system (5) undergoes transcritical bifurcation at the equilibrium point E1 when bifurcation parameter R0 = 1. Proof. When R0 = 1, the Jacobian matrix at E1 is
−(µ1 + γ )
−r
0 J (E1 ) = 1
0 0 . 1
0 0
T
−
T
Then J (E1 ) has a geometrically simple zero eigen value with right eigen vector ϕ = vector ψ = 0 Now
D β fW
r
1
−
µ1 + γ
T 1
and left eigen
1
0 .
SI − 1 + αS IZ = 1 + αZ 0
and
ψ(DX Dβ fW )ϕ
rk
̸= 0, (1 + α k)(µ1 + γ ) − ψ((DXX fW )(ϕ, ϕ)) = ψ (ei ϕ T DX (DX fi )T ϕ)
E1
=−
= −
E1
2r k(1 + α k)
̸= 0.
According to [19], the system (5) undergoes transcritical bifurcation at the disease free equilibrium point E1 . Hence the theorem. Stability and bifurcation analysis of endemic equilibrium The Jacobian matrix at E is
∗ S
r
− + k
J (E ) =
αβ I ∗ (1 + α S ∗ )2
−(µ1 + γ )
0
0
βI∗ . (1 + α Z ∗ )2
0
1
−
0
T
1
T
The characteristic equation is
λ3 + Aλ2 + Bλ + C = 0,
(6)
where A=
1 T
+
Now AB − C =
rm1 k 1 T2
−
αβ m1 m2 , (1 + α m1 )2
(P + QT ), where P =
B= rm1 k
−
1 T
[
rm1 k
−
] αβ m1 m2 , ( 1 + α m1 ) 2
αβ m1 m2 ,Q (1+α m1 )2
= P2 −
C =
β m2 (µ1 +γ ) . T (1+α m1 )2
β m2 (µ1 + γ ) . T ( 1 + α m1 ) 2
(7)
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β k(k−m1 ) , then the endemic equilibrium E m1 [1+(2m1 −k)α]2 P T∗ . Q
Theorem 5. If kα < 1 < R0 and r < and E is unstable for T > T , where ∗
Proof. Since P =
rm1 [1+α(2m1 −k)] , k(1+α m1 )
is locally asymptotically stable for T < T ∗
=−
therefore P > 0. Clearly Q < 0, AB − C > 0 for T < T ∗ and AB − C < 0 for T > T ∗ .
Hence by Routh–Hurwitz criterion, the theorem is trivially proved.
Note. From the above theorem, we can see that the stability of the equilibrium point E changes from stable to unstable when T increases through T ∗ . Consider f (T ) = TP2 + QT . When T = T ∗ i.e. AB = C , then from (6) there exists a pair of purely imaginary eigen values of the system (5). Therefore, the system (5) goes through Hopf bifurcation. β k(k−m1 ) , m1 [1+(2m1 −k)α]2
Theorem 6. If kα < 1 < R0 and r < ∗
then the system (5) undergoes Hopf bifurcation at E when the bifurcation
parameter T = T . Proof. For T = T ∗ , we have the following eigen values
λ1 =
√
√ λ1 = − Bi,
Bi,
λ2 = −A.
Let us assume λ1 = ξ1 (T ) + iη1 (T ), λ1 = ξ1 (T ) − iη1 (T ), λ2 = ξ2 (T ). Then the transversality condition is
dξ1 (T )
dT Since A =
=
2(A2 + B)
T•
+ P, B =
1 T
C ′ − B′ A − BA′
P T
and C =
. T•
β m2 (µ1 +γ ) , T (1+α m1 )2
from the above relation, we get
dξ1 (T )
Q3
dT
=− T•
2(
P4
− 3P 2 Q + Q 2 )
> 0.
Thus the system (5) undergoes a Hopf bifurcation at E.
4. Global stability of endemic equilibrium The analysis of the global asymptotic stability (GAS) of the endemic equilibria may be usefully approached by means of the Poincare–Bendixson trichotomy. If the endemic equilibrium is globally asymptotically stable, then the disease will permanently be present in the population in case of infinitesimal initial prevalence. Here we will provide an analytical proof of global stability of E by giving sufficient conditions. Global stability analysis for the endemic equilibrium will be performed through the approach due to Li and Muldowney [20]. The instability of E0 implies the uniform persistence [21] i.e. there exists a constant a > 0 such that any solution (S (t ), I (t ), Z (t )) with (S (0), I (0), Z (0)) in the orbit of the system (2) satisfies min{ lim inf S (t ), lim inf I (t ), lim inf Z (t )} > a. t →∞
t →∞
t →∞
Consider the following assumptions: a>
k 2
,
m1 = S ∗ > a,
ω > µ1 + γ ,
(8)
where
ω = min µ1 + γ +
2ar k
− r,
aβ(2 + aα)
(1 + aα)2
+
2ar k
− r , µ1 + γ −
aβ 1 + aα
.
Before starting our theorem, we give the following lemma. Lemma (Li and Muldowney [20]). If the system x˙ = f (x), where f : D → Rn , has a unique equilibrium x∗ and there exists ∗ a compact absorbing set, then the equilibrium tx is globally asymptotically stable provided that a function P (x) and a Lozinskii measure £ exist such that limt →∞ sup supx 1t 0 £ (B(x(s, x)))ds < 0. Where the symbols P , B and £ are stated in the next theorem. Theorem 7. Under the assumptions R0 > 1 and (8), the endemic equilibrium E of the system (5) is globally asymptotically stable.
T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
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Proof. The second additive compound matrix J [2] [17] of J (S , I , Z ) is
r
J
[2]
βI rS − k 1 + αS k βZ αβ IS + − µ1 − γ + (1 + α S )2 1 + αZ S
1−
=
−
βI (1 + α Z )2 r
1−
0
+
−
I
I
, ,
S S S
˙ Pf = diag
I
S
βI
k
1 + αS αβ IS 1
−
(1 + α S )2
1
−
−
βS . − 1 + αS βZ − µ1 1 + αZ
rS k
T
0
−γ −
S I
1
T
, SI , SI .
,
˙ ˙ ˙I , S − S ˙I , S − S ˙I . 2 2 2
S
−
I
0
S
T
Consider the function P = P (S , I , Z ) = diag Then P −1 = diag
I
I
I
I
I
Therefore, Pf P −1 = diag
˙
S S
˙I S˙ ˙I S˙ ˙I − , − , − . I S
I S
I
Also PJ [2] P −1 = J [2] . Therefore, B = Pf P −1 + PJ [2] P −1 =
B11 B21
B12 B22
,
where B11 =
S˙ S
−
˙I I
+r −
βI βZ + − µ1 − γ , 2 k (1 + α S ) 1 + αZ T 1 0 , B21 = 0 − ,
2rS
βI (1 + α Z )2 T ˙ ˙I 1 2rS βI S S − I + r − k − (1 + α S )2 − T = S˙
B12 =
B22
−
0
S
− −
˙I I
+
βS 1 + αS
βZ 1 + αZ
− µ1 − γ −
. 1 T
3
Consider the norm in R as:
|(u, v, w)| = max{|u|, |v| + |w|}, where (u, v, w) denotes the vector R3 and denoted by £ the Lozinskii measure with respect to this norm [22]. £(B) ≤ Sup{g1 , g2 }
≡ Sup{£1 (B11 ) + |B12 |, £1 (B22 ) + |B21 |}, where |B12 |, |B21 | are matrix norms with respect to the L1 vector norm and £1 denotes the Lozinskii measure with respect to the L1 norm.1 ∴ £1 (B11 ) =
S˙ S
−
˙I I
+r −
2rS k
−
βI βZ + − µ1 − γ , (1 + α S )2 1 + αZ
∑n ∑n 1 That is, for the generic matrix A = (a ), |A| = max ij 1≤k≤n j=1 |ajk | and £(A) = max1≤k≤n akk + j=1(j̸=k) |ajk | .
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T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
Fig. 1. For r = 2, k = 5, α = 0.01, β = 0.5, µ1 = 0.3, γ = 0.2 and T = 0.58 < T ∗ , the positive equilibrium of system (5) is asymptotically stable.
βI 1 , |B21 | = , and 2 (1 + α Z ) T ˙I S˙ 1 2rS £1 (B22 ) = − − + max r − − |B12 | =
S
I
T
k
βI βZ βS , − µ1 − γ + (1 + α S )2 1 + α Z 1 + αS
2rS βI βZ βI +r − − + − µ1 − γ + and k (1 + α S )2 1 + αZ (1 + α Z )2 ˙I S˙ 2rS βI βZ βS − , − µ1 − γ + . g2 = − + max r − S I k (1 + α S )2 1 + α Z 1 + αS ∴ g1 =
S˙ S
−
˙I I
From the system (5), ∴ g1 =
= ∴ g2 =
=
S˙ S S˙ S S˙ S S˙ S
˙I I
=
βZ 1+α Z
− µ1 − γ .
βI βI + (1 + α S )2 (1 + α Z )2 2rS βI βI + µ1 + γ + r − − + − µ1 − γ k (1 + α S )2 (1 + α Z )2 βZ 2rS βI βZ βS − + µ1 + γ + max r − − , − µ1 − γ + 1 + αZ k (1 + α S )2 1 + α Z 1 + αS 2rS βI βZ βS + µ1 + γ + max r − − − , − µ1 − γ . k (1 + α S )2 1 + αZ 1 + αS +r −
2rS k
−
Hence,
2rS βI βI 2rS βI + µ1 + γ + max r − − + − µ1 − γ , r − − 2 2 S k (1 + α S ) (1 + α Z ) k (1 + α S )2 βZ βS − , − µ1 − γ , 1 + αZ 1 + αS S˙ 2ar 2ar aβ(2 + aα) aβ i.e. £(B) ≤ + µ1 + γ + max r − − µ1 − γ , r − − , − µ − γ , 1 S k k (1 + aα)2 1 + aα £(B) ≤
S˙
where a is the constant of uniform persistence. i.e. £(B) ≤ i.e. £(B) ≤
S˙ S S˙ S
+ µ1 + γ − min µ1 + γ + + µ1 + γ − ω,
ω>0
2ar k
− r,
aβ(2 + aα)
(1 + aα)2
+
2ar k
− r , µ1 + γ −
aβ 1 + aα
,
T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
Fig. 2. Phase portrait of system (5) with T = 0.58 < T ∗ .
Fig. 3. The positive equilibrium of system (5) is unstable when T = 0.65 > T ∗ .
Fig. 4. Phase portrait of system (5) with T = 0.65 > T ∗ .
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T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
Fig. 5. Variation of infected population with reproduction number. It is showing that for R0 < 1, the disease dies out and for R0 > 1, the infection is maintained in the population.
Fig. 6. Variation of reproduction number with saturation factor. It is observed that for saturation factor greater than 0.8, reproduction rate falls below 1 and disease dies out.
i.e. £(B) ≤ i.e.
1 t
S˙ S
− (ω − µ1 − γ ),
t
∫
£(B)dS ≤ 0
i.e. lim sup sup t →∞
Hence the theorem.
1 t
1 t
log
S (t ) S (0)
− (ω − µ1 − γ ),
t
∫
£(B)dS < −(ω − µ1 − γ ) < 0. 0
T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
2067
Fig. 7. For time delay.
5. Numerical simulations For the arbitrary parameter values r = 2, k = 5, α = 0.01, β = 0.5, µ1 = 0.3, γ = 0.2, the given system has a positive equilibrium E (1.0101, 3.22416, 1.0101) and for T = 0.58, the eigen values are −2.1022, −0.0050 ± 0.8049i and for T = 0.65, the eigen values are −1.9415, 0.0075 ± 0.7912i (see Figs. 1–6). Also T ∗ = 0.606952. 6. Conclusions In this paper, we have studied the special type of delay SIR epidemic model with information variable, which depends on the current values of state variables. This model displays two codimension-1 bifurcations viz. transcritical and Hopf bifurcation. By analyzing the model, we have found a threshold parameter R0 . It is noted that when R0 < 1 then disease dies out and when R0 > 1 the disease becomes endemic. Two bifurcation parameters R0 and T plays an important role. R0 changes the stability of the disease free equilibrium and the delay parameter T changes the stability of the endemic equilibrium. We have shown the unique endemic equilibrium globally asymptotically stable under some conditions of the parameters of the system (5). Lastly, numerical simulation provided this model admits limit cycles. From Fig. 5 we see that when the reproduction number R0 is less than 1, the disease free equilibrium is stable and when R0 is greater than 1, the disease free equilibrium is unstable i.e. the endemic equilibrium exists. In Fig. 7, we found that if T is greater than 0.6, the endemic equilibrium becomes unstable and if T is less than 0.6, the endemic equilibrium is asymptotically stable. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Global dynamics and bifurcation in delayed SIR epidemic model T.K. Kar a,∗ , Prasanta Kumar Mondal b a
Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-3, India
b
Department of Mathematics, Shibpur Sri Ramkrishna Vidyalaya, Howrah-2, India
article
info
Article history: Received 1 November 2010 Accepted 22 December 2010 Keywords: SIR epidemic model Transcritical bifurcation Hopf bifurcation Global stability
abstract An SIR epidemic model with time delay, information variable and saturated incidence rate, where the susceptibles are assumed to satisfy the logistic equation and the incidence term, is of saturated form with the susceptibles. This model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. The local and global stability of endemic equilibrium is also discussed. Finally, numerical simulations are carried out to explain the mathematical conclusions. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Mathematical epidemiology, i.e. the building and analysis of mathematical models describing the spread and control of infectious diseases is one of the major areas of biology. In many epidemic models the population can be divided into three distinct classes, which are defined with respect to disease status. Kermack and Mckendrick [1] described the simplest SIR model which computes the theoretical number of people infected with a contagious illness in a closed population over time. Transmission of a disease is a dynamical process driven by the interaction between susceptible and infective. The behaviour of the SIR models are greatly affected by the way in which transmission between infected and susceptible individuals are modelled. Most of the models of epidemiology are based on the so-called ‘‘mass action’’ [2] assumption has faced some questions and consequently a number of realistic transmission functions have become the focus of considerable attention [3,4]. A delay differential equation has been successfully used to model varying infectious period in a range of SIR, SIS and SIRS epidemic models. Hethcote and Van den Driessche [5] have considered an SIS epidemic model with constant time delay, which accounts for duration of infectiousness. Beretta et al. [6] have studied global stability in an SIR epidemic model with distributed delay that describes the time it takes for an individual to lose infectiousness. Song and Cheng [7] have studied the effect of time delay on the stability of the endemic equilibrium. They were given some conditions for which the endemic equilibrium is asymptotically stable for all delays and also discussed the existence of orbitally asymptotically stable periodic solutions. The mathematical analysis of epidemiological modelling is often used for the assessment of the global asymptotic stability of both the disease free and endemic equilibrium. The modelling and analysis of infectious diseases have been done also by some other workers; see for example, [8–12] and the references therein. Here we will study SIR models with nonlinear incidence [13] and information variable. In most of the epidemiological models, the incidence rate is assumed of the form β SI, where β is the transmission rate, S and I are the susceptible and infected population respectively. Cai et al. [14] used a saturated incidence rate of the form bβ SI /(1 + α I ) where b, β, α > 0. The paper is organized as follows. In the next section, we present the model and its equilibria. In Section 3, we carry out a qualitative analysis of the model. Stability conditions for the disease free equilibrium and the endemic equilibrium are
∗
Corresponding author. Tel.: +91 33 26684561. E-mail addresses: [email protected] (T.K. Kar), [email protected] (P.K. Mondal).
1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.12.021
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derived. Bifurcations also given in this section. Section 4 describes the global stability of the endemic equilibrium. Finally, some numerical simulations are given in Section 5. 2. The model and its equilibria We consider the following SIR epidemic model: dS dt dI dt dR dt
S = rS 1 − − k
=
β IS 1 + αS
β SI 1 + αS
,
− µ1 I − γ I ,
(1)
= γ I − µ2 R ,
where S (0), I (0), R(0) ≥ 0 and S is the density of susceptibles within the population, I is the density of infected within the population, R is the density of recovered within the population, and the parameters are r is the intrinsic growth rate of susceptibles, k is the carrying capacity of susceptibles, α is the saturation factor that measures the inhibitory effect, β is the transmission or contact rate, µ1 is the death rate, γ is the rate of recovery from infection, µ2 is the death rate. Consider the new variable Z , called information variable which summarizes information about the current state of the disease i.e. depends on current values of state variables and also summarizes information about past values of state variables. Many authors used this variable in their model (see e.g. [15–17]). We take up the formula Z (t ) =
∫
t
g (S (τ ), I (τ ))K (t − τ )dτ ,
(2)
−∞
where K (t − τ ) is the delaying kernel [18], τ is the distributed delay, which means that at time t susceptible individuals, S and infective individuals, I are affected by the state variables S and I at possibly all previous times τ ≤ t. In this paper we considered g (S , I ) = S, and K (t − τ ) = T1 exp − T1 (t − τ ) , where T is the average delay of the collected information on the disease, as well as the average length of the historical memory concerning the disease in study i.e. T is the ‘‘measure of the influence of the past’’. From the above assumptions we consider the following model: dS dt
S = rS 1 − − k
β SI , 1 + αS
β IZ = − µ1 I − γ I , dt 1 + αZ ∫ t 1 1 Z (t ) = S . exp − (t − τ ) dτ , dI
−∞
T
(3)
T
dR
= γ I − µ2 R . dt Then the nonlinear integro-differential system (3) can be transformed into the following set of nonlinear ordinary differential equations: dS dt
S = rS 1 − − k
β SI , 1 + αS
β IZ = − µ1 I − γ I , dt 1 + αZ dI
dZ dt dR dt
=
1 T
(S − Z ),
= γ I − µ2 R .
(4)
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Since the dynamical behaviour of the last equation of the system (4) i.e. the dynamics of R depends only the dynamics of I, so we do not consider that equation in our discussion. Here we will study the following nonlinear ordinary differential equations: dS dt
S = rS 1 − − k
β SI , 1 + αS
β IZ = − µ1 I − γ I , dt 1 + αZ dI
dZ dt
=
1 T
(5)
(S − Z ),
where r , k, α, β, µ1 , γ , T > 0. Theorem 1. (a) The system (5) has a trivial equilibrium E0 = (0, 0, 0) and the disease free equilibrium E1 = (k, 0, k). kβ
(b) Also, if R0 > 1, the system (5) has one endemic equilibrium point E = (S ∗ , I ∗ , Z ∗ ) where R0 = (µ +γ )(kα+1) is the basic 1 µ1 +γ reproduction number and S ∗ = Z ∗ = β−α(µ = m1 , I ∗ = βrk (1 + m1 α)(k − m1 ) = m2 . +γ ) 1
Proof. Proof is omitted here.
3. Stability of equilibrium points and bifurcation The Jacobian matrix of the system (5) is
1−
r
S
βI r αβ I +S − + 1 + αS k (1 + α S )2
−
k
J (S , I , Z ) =
0
− βZ
βS 1 + αS
1 + αZ
1
− µ1 − γ 0
T
0
βI . (1 + α Z )2 1 − T
Theorem 2. The equilibrium point E0 is unstable for all T > 0. Proof. The Jacobian matrix at E0 is
r 0 J (0, 0, 0) = 1
0
0 0 . 1
−(µ1 + γ )
−
0
T
The eigen values are r , −(µ1 + γ ), − Therefore E0 is unstable.
T
1 . T
Theorem 3. (i) If R0 > 1, then E1 is unstable for all T > 0, (ii) If R0 < 1, then E1 is locally asymptotically stable for all T > 0. Proof. The Jacobian matrix at E1 is
−r
J (k, 0, k) = 0
1
βk 1 + αk βk − µ1 − γ 1 + αk −
βk 1+α k
βk 1+α k
0 . 1 −
0
T The eigen values are −r ,
0
T
− µ1 − γ , − > µ1 + γ i.e. if β > (µ1 + γ ) α + 1k , i.e. if R0 > 1, the equilibrium point E1 is unstable. 1 . T
If If R0 < 1, then the equilibrium point E1 is locally asymptotically stable.
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Note. From the above theorem, we can see that the stability of disease free equilibrium point E1 changes from stable to unstable when R0 increases though 1. Therefore, we use R0 as bifurcation parameter. For simplicity, let
β SI k 1 + αS β IZ − (µ1 + γ )I 1 + αZ 1 (S − Z )
rS
f1 (X , β, T ) f2 (X , β, T ) f3 (X , β, T )
fW (X , β, T ) =
=
S
1−
−
and X = [S , I , Z ]T .
T
Theorem 4. The system (5) undergoes transcritical bifurcation at the equilibrium point E1 when bifurcation parameter R0 = 1. Proof. When R0 = 1, the Jacobian matrix at E1 is
−(µ1 + γ )
−r
0 J (E1 ) = 1
0 0 . 1
0 0
T
−
T
Then J (E1 ) has a geometrically simple zero eigen value with right eigen vector ϕ = vector ψ = 0 Now
D β fW
r
1
−
µ1 + γ
T 1
and left eigen
1
0 .
SI − 1 + αS IZ = 1 + αZ 0
and
ψ(DX Dβ fW )ϕ
rk
̸= 0, (1 + α k)(µ1 + γ ) − ψ((DXX fW )(ϕ, ϕ)) = ψ (ei ϕ T DX (DX fi )T ϕ)
E1
=−
= −
E1
2r k(1 + α k)
̸= 0.
According to [19], the system (5) undergoes transcritical bifurcation at the disease free equilibrium point E1 . Hence the theorem. Stability and bifurcation analysis of endemic equilibrium The Jacobian matrix at E is
∗ S
r
− + k
J (E ) =
αβ I ∗ (1 + α S ∗ )2
−(µ1 + γ )
0
0
βI∗ . (1 + α Z ∗ )2
0
1
−
0
T
1
T
The characteristic equation is
λ3 + Aλ2 + Bλ + C = 0,
(6)
where A=
1 T
+
Now AB − C =
rm1 k 1 T2
−
αβ m1 m2 , (1 + α m1 )2
(P + QT ), where P =
B= rm1 k
−
1 T
[
rm1 k
−
] αβ m1 m2 , ( 1 + α m1 ) 2
αβ m1 m2 ,Q (1+α m1 )2
= P2 −
C =
β m2 (µ1 +γ ) . T (1+α m1 )2
β m2 (µ1 + γ ) . T ( 1 + α m1 ) 2
(7)
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β k(k−m1 ) , then the endemic equilibrium E m1 [1+(2m1 −k)α]2 P T∗ . Q
Theorem 5. If kα < 1 < R0 and r < and E is unstable for T > T , where ∗
Proof. Since P =
rm1 [1+α(2m1 −k)] , k(1+α m1 )
is locally asymptotically stable for T < T ∗
=−
therefore P > 0. Clearly Q < 0, AB − C > 0 for T < T ∗ and AB − C < 0 for T > T ∗ .
Hence by Routh–Hurwitz criterion, the theorem is trivially proved.
Note. From the above theorem, we can see that the stability of the equilibrium point E changes from stable to unstable when T increases through T ∗ . Consider f (T ) = TP2 + QT . When T = T ∗ i.e. AB = C , then from (6) there exists a pair of purely imaginary eigen values of the system (5). Therefore, the system (5) goes through Hopf bifurcation. β k(k−m1 ) , m1 [1+(2m1 −k)α]2
Theorem 6. If kα < 1 < R0 and r < ∗
then the system (5) undergoes Hopf bifurcation at E when the bifurcation
parameter T = T . Proof. For T = T ∗ , we have the following eigen values
λ1 =
√
√ λ1 = − Bi,
Bi,
λ2 = −A.
Let us assume λ1 = ξ1 (T ) + iη1 (T ), λ1 = ξ1 (T ) − iη1 (T ), λ2 = ξ2 (T ). Then the transversality condition is
dξ1 (T )
dT Since A =
=
2(A2 + B)
T•
+ P, B =
1 T
C ′ − B′ A − BA′
P T
and C =
. T•
β m2 (µ1 +γ ) , T (1+α m1 )2
from the above relation, we get
dξ1 (T )
Q3
dT
=− T•
2(
P4
− 3P 2 Q + Q 2 )
> 0.
Thus the system (5) undergoes a Hopf bifurcation at E.
4. Global stability of endemic equilibrium The analysis of the global asymptotic stability (GAS) of the endemic equilibria may be usefully approached by means of the Poincare–Bendixson trichotomy. If the endemic equilibrium is globally asymptotically stable, then the disease will permanently be present in the population in case of infinitesimal initial prevalence. Here we will provide an analytical proof of global stability of E by giving sufficient conditions. Global stability analysis for the endemic equilibrium will be performed through the approach due to Li and Muldowney [20]. The instability of E0 implies the uniform persistence [21] i.e. there exists a constant a > 0 such that any solution (S (t ), I (t ), Z (t )) with (S (0), I (0), Z (0)) in the orbit of the system (2) satisfies min{ lim inf S (t ), lim inf I (t ), lim inf Z (t )} > a. t →∞
t →∞
t →∞
Consider the following assumptions: a>
k 2
,
m1 = S ∗ > a,
ω > µ1 + γ ,
(8)
where
ω = min µ1 + γ +
2ar k
− r,
aβ(2 + aα)
(1 + aα)2
+
2ar k
− r , µ1 + γ −
aβ 1 + aα
.
Before starting our theorem, we give the following lemma. Lemma (Li and Muldowney [20]). If the system x˙ = f (x), where f : D → Rn , has a unique equilibrium x∗ and there exists ∗ a compact absorbing set, then the equilibrium tx is globally asymptotically stable provided that a function P (x) and a Lozinskii measure £ exist such that limt →∞ sup supx 1t 0 £ (B(x(s, x)))ds < 0. Where the symbols P , B and £ are stated in the next theorem. Theorem 7. Under the assumptions R0 > 1 and (8), the endemic equilibrium E of the system (5) is globally asymptotically stable.
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Proof. The second additive compound matrix J [2] [17] of J (S , I , Z ) is
r
J
[2]
βI rS − k 1 + αS k βZ αβ IS + − µ1 − γ + (1 + α S )2 1 + αZ S
1−
=
−
βI (1 + α Z )2 r
1−
0
+
−
I
I
, ,
S S S
˙ Pf = diag
I
S
βI
k
1 + αS αβ IS 1
−
(1 + α S )2
1
−
−
βS . − 1 + αS βZ − µ1 1 + αZ
rS k
T
0
−γ −
S I
1
T
, SI , SI .
,
˙ ˙ ˙I , S − S ˙I , S − S ˙I . 2 2 2
S
−
I
0
S
T
Consider the function P = P (S , I , Z ) = diag Then P −1 = diag
I
I
I
I
I
Therefore, Pf P −1 = diag
˙
S S
˙I S˙ ˙I S˙ ˙I − , − , − . I S
I S
I
Also PJ [2] P −1 = J [2] . Therefore, B = Pf P −1 + PJ [2] P −1 =
B11 B21
B12 B22
,
where B11 =
S˙ S
−
˙I I
+r −
βI βZ + − µ1 − γ , 2 k (1 + α S ) 1 + αZ T 1 0 , B21 = 0 − ,
2rS
βI (1 + α Z )2 T ˙ ˙I 1 2rS βI S S − I + r − k − (1 + α S )2 − T = S˙
B12 =
B22
−
0
S
− −
˙I I
+
βS 1 + αS
βZ 1 + αZ
− µ1 − γ −
. 1 T
3
Consider the norm in R as:
|(u, v, w)| = max{|u|, |v| + |w|}, where (u, v, w) denotes the vector R3 and denoted by £ the Lozinskii measure with respect to this norm [22]. £(B) ≤ Sup{g1 , g2 }
≡ Sup{£1 (B11 ) + |B12 |, £1 (B22 ) + |B21 |}, where |B12 |, |B21 | are matrix norms with respect to the L1 vector norm and £1 denotes the Lozinskii measure with respect to the L1 norm.1 ∴ £1 (B11 ) =
S˙ S
−
˙I I
+r −
2rS k
−
βI βZ + − µ1 − γ , (1 + α S )2 1 + αZ
∑n ∑n 1 That is, for the generic matrix A = (a ), |A| = max ij 1≤k≤n j=1 |ajk | and £(A) = max1≤k≤n akk + j=1(j̸=k) |ajk | .
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Fig. 1. For r = 2, k = 5, α = 0.01, β = 0.5, µ1 = 0.3, γ = 0.2 and T = 0.58 < T ∗ , the positive equilibrium of system (5) is asymptotically stable.
βI 1 , |B21 | = , and 2 (1 + α Z ) T ˙I S˙ 1 2rS £1 (B22 ) = − − + max r − − |B12 | =
S
I
T
k
βI βZ βS , − µ1 − γ + (1 + α S )2 1 + α Z 1 + αS
2rS βI βZ βI +r − − + − µ1 − γ + and k (1 + α S )2 1 + αZ (1 + α Z )2 ˙I S˙ 2rS βI βZ βS − , − µ1 − γ + . g2 = − + max r − S I k (1 + α S )2 1 + α Z 1 + αS ∴ g1 =
S˙ S
−
˙I I
From the system (5), ∴ g1 =
= ∴ g2 =
=
S˙ S S˙ S S˙ S S˙ S
˙I I
=
βZ 1+α Z
− µ1 − γ .
βI βI + (1 + α S )2 (1 + α Z )2 2rS βI βI + µ1 + γ + r − − + − µ1 − γ k (1 + α S )2 (1 + α Z )2 βZ 2rS βI βZ βS − + µ1 + γ + max r − − , − µ1 − γ + 1 + αZ k (1 + α S )2 1 + α Z 1 + αS 2rS βI βZ βS + µ1 + γ + max r − − − , − µ1 − γ . k (1 + α S )2 1 + αZ 1 + αS +r −
2rS k
−
Hence,
2rS βI βI 2rS βI + µ1 + γ + max r − − + − µ1 − γ , r − − 2 2 S k (1 + α S ) (1 + α Z ) k (1 + α S )2 βZ βS − , − µ1 − γ , 1 + αZ 1 + αS S˙ 2ar 2ar aβ(2 + aα) aβ i.e. £(B) ≤ + µ1 + γ + max r − − µ1 − γ , r − − , − µ − γ , 1 S k k (1 + aα)2 1 + aα £(B) ≤
S˙
where a is the constant of uniform persistence. i.e. £(B) ≤ i.e. £(B) ≤
S˙ S S˙ S
+ µ1 + γ − min µ1 + γ + + µ1 + γ − ω,
ω>0
2ar k
− r,
aβ(2 + aα)
(1 + aα)2
+
2ar k
− r , µ1 + γ −
aβ 1 + aα
,
T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
Fig. 2. Phase portrait of system (5) with T = 0.58 < T ∗ .
Fig. 3. The positive equilibrium of system (5) is unstable when T = 0.65 > T ∗ .
Fig. 4. Phase portrait of system (5) with T = 0.65 > T ∗ .
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Fig. 5. Variation of infected population with reproduction number. It is showing that for R0 < 1, the disease dies out and for R0 > 1, the infection is maintained in the population.
Fig. 6. Variation of reproduction number with saturation factor. It is observed that for saturation factor greater than 0.8, reproduction rate falls below 1 and disease dies out.
i.e. £(B) ≤ i.e.
1 t
S˙ S
− (ω − µ1 − γ ),
t
∫
£(B)dS ≤ 0
i.e. lim sup sup t →∞
Hence the theorem.
1 t
1 t
log
S (t ) S (0)
− (ω − µ1 − γ ),
t
∫
£(B)dS < −(ω − µ1 − γ ) < 0. 0
T.K. Kar, P.K. Mondal / Nonlinear Analysis: Real World Applications 12 (2011) 2058–2068
2067
Fig. 7. For time delay.
5. Numerical simulations For the arbitrary parameter values r = 2, k = 5, α = 0.01, β = 0.5, µ1 = 0.3, γ = 0.2, the given system has a positive equilibrium E (1.0101, 3.22416, 1.0101) and for T = 0.58, the eigen values are −2.1022, −0.0050 ± 0.8049i and for T = 0.65, the eigen values are −1.9415, 0.0075 ± 0.7912i (see Figs. 1–6). Also T ∗ = 0.606952. 6. Conclusions In this paper, we have studied the special type of delay SIR epidemic model with information variable, which depends on the current values of state variables. This model displays two codimension-1 bifurcations viz. transcritical and Hopf bifurcation. By analyzing the model, we have found a threshold parameter R0 . It is noted that when R0 < 1 then disease dies out and when R0 > 1 the disease becomes endemic. Two bifurcation parameters R0 and T plays an important role. R0 changes the stability of the disease free equilibrium and the delay parameter T changes the stability of the endemic equilibrium. We have shown the unique endemic equilibrium globally asymptotically stable under some conditions of the parameters of the system (5). Lastly, numerical simulation provided this model admits limit cycles. From Fig. 5 we see that when the reproduction number R0 is less than 1, the disease free equilibrium is stable and when R0 is greater than 1, the disease free equilibrium is unstable i.e. the endemic equilibrium exists. In Fig. 7, we found that if T is greater than 0.6, the endemic equilibrium becomes unstable and if T is less than 0.6, the endemic equilibrium is asymptotically stable. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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