The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time

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1 Applied Mathematics, 05, 6, Published Online September 05 in SciRes The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time Xiaodan Liao, Hongbo Wang, Xiaohua Huang, Wenbo Zeng, Xiaoliang Zhou * Department of Mathematics, Lingnan Normal University, Zhanjiang, China * zxlmath@6com Received May 05; accepted 5 September 05; published 8 September 05 Copyright 05 by authors and Scientific Research Publishing Inc This work is licensed under the Creative Commons Attribution International License (CC BY) Abstract In this paper, a class of discrete deterministic SIR epidemic model with vertical and horizontal transmission is studied Based on the population assumed to be a constant size, we transform the discrete SIR epidemic model into a planar map Then we find out its equilibrium points and eigenvalues From discussing the influence of the coefficient parameters effected on the eigenvalues, we give the hyperbolicity of equilibrium points and determine which point is saddle, node or focus as well as their stability Further, by deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the non-hyperbolic equilibrium point Finally, we give some numerical simulation examples for illustrating the theoretical analysis and the biological explanation of our theorem Keywords Epidemic Model, Equilibrium Point, Transcritical Bifurcation, Center Manifold, Hyperbolicity Introduction Since Kermack and McKendrick [] proposed the Susceptible-Infective-Recovered model (or SIR for short) in 97, a lot of glorious studies on the dynamics of the epidemic models have been presented (see []-[9]) The basic and important research subjects for these systems are local and global stability of the disease-free equilibrium and the endemic equilibrium, existence of periodic solutions, persistence and extinction of the disease, etc According to the dependence on variable (ie, time), these systems were classified into two types: continuoustime system and discrete-time system For the epidemic models, there have been a lot of researches focusing on the case of continuous-time (see []-[6] and that cited therein) However, discrete-time models (or called difference equation are also useful for * Corresponding author How to cite this paper: Liao, XD, Wang, HB, Huang, XH, Zeng, WB and Zhou, XL (05) The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time Applied Mathematics, 6,

2 X D Liao et al modeling situations of epidemic They can not only have the basic features of the corresponding continuoustime models but also provide a substantial reduction of computer time (see [0]) What is more, a lot of discretetime models are not trivial analogues of their continuous ones and simple models can even exhibit complex behavior (see [5] [0]) In 989, Hethcote [7] considered a class of continuous epidemic model with vertical and horizontal transmission S = β SI bs + pdi + b( S + R), I = β SI di ri + qdi, R = ri br, () where S represents the proportion of individuals susceptible to the disease, who are born (with b) and die (with d) at the same rate b (b = d), and have mean life expectancy /b The susceptible becomes infectious at a bilinear rate βi, where I is the proportion of infectious individuals and β is the contact rate The infectious recover (ie, acquire lifelong immunity) at a rate r, so that /r is the mean infectious period The constant p, q, 0< p <, 0< q <, and p+ q =, where p is the proportion of the offspring of infective parents that are susceptible individuals, and q is the proportion of the offspring of infective parents that are infective individuals Because of biological meanings, a natural constraint is b pd > 0 A similarly detailed description of the model and its dynamics may be found in [7] In recent, Meng and Chen [8] have also studied the epidemic system () In their work, the basic reproductive rate determining the stability of disease-free equilibrium point and endemic equilibrium point was found out and the local and global stability of the equilibrium points have been researched by using Lyapunov function and Dulac function In this paper, we pay attention to the discrete situation of () From discussing the influence of the coefficient parameters effected on the eigenvalues, we give the hyperbolicity of equilibrium points and determine which point is saddle, node or focus as well as their stability Further, by deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the non-hyperbolic equilibrium point and research how does small perturbation of coefficient parameters affect the number and stability of equilibrium points Moreover, we give some numerical simulation examples for illustrating the theoretical analysis and explain the biological meaning of our theorem Discrete SIR Epidemic Model with Vertical and Horizontal Transmission In this section, we consider the discrete SIR epidemic model with vertical and horizontal transmission: Sn+ = β SnIn + Sn + pdin + brn, In+ = β SnIn + ( d r) In + qdin, Rn+ = rin + ( b) Rn, where S n, I n and R n represent susceptible, infective and removed (or isolated) subgroups respectively, n represents a fixed time, n = 0,,, It is assumed that S 0 > 0, I0 0, R0 0 and S0 + I0 + R0 = In view of assumption that population is a constant size in [], ie, system () can be changed into S0 + I0 + R0 = () ( ) Sn+ = β SI n n+ bsn+ b p In+ b, In+ = β SnIn + pb r In () (4) Rewrite (4) as a planar map F: ( ) ( ) SI + ( pb r) I S β SI + b S + b p I + b I β (5) 666

3 X D Liao et al It is obvious that this map has a disease-free equilibrium point : (,0) * * * * b( w ) β Q: ( S, I ), where S =, I = and w = w β + b( p) w pb + r P and an endemic equilibrium point The organization of this paper is as follows In next section, we identify all cases of non- and hyperbolic equilibria, which is a fundament for all succeeding studies In Section 4, we discuss the transcritical bifurcation at the disease-free equilibrium of (), the direction and stability of the transcritical bifurcation is investigated by computing a center manifold In Section 5, some simulations are made to demonstrate our results and the biologic explanation of the theorem is also given Hyperbolic and Non-Hyperbolic Cases In this section, we will discuss the hyperbolic and non-hyperbolic cases in a two parameters space parameter,0 wb, lies on the lines: Theorem The equilibrium point P is non-hyperbolic if and only if : {( wb, ) w= β ( + β ), 0 < b< } and Otherwise, the equilibrium point (,0) Remark By Theorem the domain {( wb ) w= < b< } :,, 0 P is in one of the following types (see Table ) { wb, 0< w,0< b < } is divided by line and into three districts, and for equilibrium point P (see Figure ) P,0 is Proof The Jacobian matrix of (5) at and its eigenvalues are b β b + pb DF ( P) = 0 + β pb r = b, = β w Table Types of hyperbolic equilibrium point P (,0) Cases Conditions Eigenvalues Properties 0 w β ( β) β ( β) < < + 0< <, < Saddle + < w < 0< <, < < w > 0< <, > Saddle Figure Districts for equilibrium point P 667

4 X D Liao et al From the assumption 0< b <, we see that 0< < Then non-hyperbolicity will be happened in the case = ± From =, we know w = β ( + β) which means ( wb, ) lies on ; Also, From =, we get that w = and ( wb, ) lies on When 0< w < β( + β) (referred to the case ), the eigenvalue satisfies <, then the equilibrium point P is a stable node When β( + β) < w < (referred to the case ), the equilibrium point P is a node since < < and meanwhile when < w, the equilibrium P is a saddle since > The proof is complete * * Theorem There does not exist non-hyperbolic case for equilibrium point Q( S, I ) But the hyperbolicity can be divided into the following cases: (I) When b < β, there exist six types for hyperbolic equilibrium point Q (see Table ) Where s and s satisfy ( s, r) : and ( s, r) : (II) When b Where {( s, r) r = ( s b) ( 4, b = ( b+ ) s b} { s, r r s b 4 s, b s β} = <, respectively < β b, there exist four types for hyperbolic equilibrium point Q (see Table ) { s, r r = s b 4 s, b < s < β} { sr, b < s,0 < r < } s satisfies ( s, r) : < β, the domain Remark By Theorem, when b is divided by line, and into three districts 4, 4, 6 and 7 for equilibrium point Q (see Figure ) When b < β b, the domain ( sr, ) b < s,0 < r < is divided by line 4 and into three dis- { } tricts 8, 9 and 9 for equilibrium point Q (see Figure ) Proof Performing a coordinate shift as follows: * * I = I I, R = R R, * * Table Types of hyperbolic equilibrium (, ) Q S I Cases Conditions Eigenvalues Properties =, b s b < = < r ( s b) ( 4 r = ( s β ) ( 4, b< s < β < = < 4 b s 0 < < >, 0 < < Saddle 0 s s 5 < < 0 < <, 0 < < s s s 6 < < are complex, <,, Stable focus s 7 < s 0 < <, 0 < < * * Table Types of hyperbolic equilibrium (, ) Q S I Cases Conditions Eigenvalues Properties = 4, b s β 0< = < r ( s b) ( 8 b s 0 < < >, 0 < < Saddle 0 s s 9 < < 0 < <, 0 < < 0 s < s< β are complex, <,, Stable focus 668

5 X D Liao et al Figure Districts for equilibrium point Q when b < β Figure Districts for equilibrium point Q when b < β b and letting F * * denote the transformed F, we translate equilibrium (, ) 0, 0 and discuss equili- Q S I into brium point ( 0,0 ) of the map F The matrix of linearization of F at ( 0,0 ) is b( β pb r) b b r r a s a r DF β + (( 0,0 )) = =, b( β pb r) s where a = b and It is known that ( β ) b pb r s = β + r = = β + r, and its eigenvalues are ( s+ a) + ( s+ a) 4( a+ ( r a) ( s+ a) ( s+ a) 4( a+ ( r a), 0,0 is hyperbolic if and only if none of eigenvalues, lies on the unit circle S In the following we discuss the eigenvalues in two cases, ie, b < β and b < β b Case (I) When discriminant = ( s+ a) 4( a+ ( r a) 0, then and are both real Because nonhyperbolicity happens if and only if or is For whether = or =, we can get a r = However, for positive equilibrium point Q, we have 0 < a < and 0 < r < Therefore, neither = nor = is possible Next, let s examine = and = From whether = or =, we can get γ = ( a+ )( By condition 0< a <, b< s < β, we see that ( a+ )( > This is a contradiction with r <, so = and = are impossible (6) 669

6 X D Liao et al In the case of < 0, and are a pair of conjugate complex Since = = ( s+ a) + 4 ( a+ ( r a) ( s+ a) 4 = a+ r a s = s b+ r s+ <, S and the equilibrium point Q is a stable focus referred to the case and lie inside of 6 If = 0, ie, 4rs = ( s a), the matrix has a double real eigenvalue = = ( s+ a) From the constraint condition b< s < β of s, it is obvious that 0< = < Therefore, equilibrium Q is stable node in the cases of and If > 0, ie 4rs < ( s a) and b< s < β, the eigenvalues and are different real numbers We first discuss the case that b s 0 sr, In this case we have Since we have 0 since and < <, ie 4 r 0+ r 0+ < < for (, ) 4 we have For the case 0 s s and s = < d dr s a 4 a r a s ( + ) ( + ( ) ) ( s+ a) ( s+ a) 4( a+ ( r a) lim = lim = a <, sr On the other hand, there also exists 0 r 0+ r 0+ ( s+ a) + ( s+ a) 4( a+ ( r a) lim = lim = s > s = > d dr s a 4 a r a s ( + ) ( + ( ) ) > Therefore, the equilibrium Q is a stable node as (, ) 4 < <, ie, ( sr, ) 5, we have dr r , sr d s+ a > 0, lim = a > 0, = <, d < 0, lim = s < dγ γ 0+ Therefore, the equilibrium Q is a stable node as (, ) 5 s < s, ie (, ) 7 Finally, we study the case of Then, we have 0 sr sr We have ( + ) 4( + ( ) ) < < for (, ) 4 d s s+ a = < 0, lim = a <, > > 0 dr r 0+ s a a r a s < < for (, ) 7 sr Moreover, there also has 0 ( + ) 4( + ( ) ) < < for (, ) 7 d s s+ a = > 0, lim = s > 0, < < dr r 0+ s a a r a s This means that the equilibrium Q is a stable node for ( sr, ) 7 sr In fact, sr In fact that, 670

7 Case (II) When discriminant ( s a) ( a ( r a) X D Liao et al = , then and are both real Because non-hyperbolicity happens if and only if or is Similar to the proof in case (I), neither = nor = is possible When < 0, and are a pair of conjugate complex Since and = = a+ r a s <, S and the equilibrium point Q is a stable focus referred to the case lie inside of 0 If = 0, the matrix has a double real eigenvalue = = ( s+ a) From the constraint condition b< s < β of s, it is obvious that 0< = < Therefore, equilibrium Q is stable node in the cases of If > 0, We first discuss the case that b s 0 sr, In this case we have We know 0 < < for (, ) 8 < <, ie 8 d < 0, lim = a < dr r 0+ sr On the other hand, there exists 0 d > 0, lim = s > dr r 0+ < > for (, ) 8 sr In fact, since Therefore, the equilibrium Q is a saddle as ( sr, ) 8 Finally, we study the case of 0 < s < s, ie ( sr, ) 9 We easily prove 0< < < for (, ) 9 same methods as in case () This means that the equilibrium Q is a stable node for ( sr, ) 9 The proof is complete 4 Transcritical Bifurcation In this section we consider the case that (, ) (,0) sr by wb, where the transcritical bifurcation at equilibrium point P will happen The following lemmas were be derived from reference [] Lemma ([], Theorem 4) The map x Ax + f x, y, y Bx + g x, y, ( xy) c s,, (7) satisfies that A is c c matrix with eigenvalues of modulus one, and B is s s matrix with eigenvalues of modulus less than one, and r where f and g are Df Dg f g 0,0 = 0,0 = 0, 0,0 = 0,0 = 0, C r in some neighborhood of the origin Then there exists a (7) which can be locally represented as a graph as follows { } δ c c s W 0 = x, y y = h x, x <, h 0 = 0, Dh 0 = 0 (8) r C center manifold for for δ sufficiently small Moreover, the dynamics of (7) restricted to the center manifold is, for u sufficiently small, given by the c-dimensional map ( ) c u Au + f u, h u, u r Lemma ([], in page 65) A one-parameter family of having a nonhyperbolic fixed point, ie, (, µ ),, µ C r one-dimensional maps u f x x (9) f f 0,0 = 0, 0,0 =, x 67

8 X D Liao et al undergoes a transcritical bifurcation at ( x, µ ) = ( 0,0) if f f f ( 0,0) = 0, ( 0,0) 0, ( 0,0) 0 µ x µ x Theorem A transcritical bifurcation occurs at the equilibrium P when w = More concretely, for w < slightly there are two equilibriums: a stable point P and an unstable negative equilibrium which coalesce at w = and for w > slightly there are also two equilibriums: an unstable equilibrium P and a stable positive equilibrium Q Thus an exchange of stability has occurred at w = Proof For ( wb, ), we have = and 0< < Consider w as the bifurcation parameter and write F as DF 0,0 is F to emphasize the dependence on w One can easily see that the matrix w and it has eigenvectors b r b 0 T b, 0,, r+ b corresponding to and respectively, where T means the transpose of matrices First, we put the matrix DF w ( 0,0) into a diagonal form Using the eigenvectors (0), we obtain the transformation with inverse which transform system (5) into S u b I = 0 v r+ b r+ b u b S v = r b I + 0 b ( + ) ( β ) β ( + ) rβ pb r rβ rβ wv v uv + b u b 0 u r b pb r r b r b red v 0 v + () pb r wv v + β + β uv β pb r Rewrite system () in the suspended form with assumption w w ( β pb r) ( pb r) where T w = = +, u b 0 0 u avw av auv + b v 0 0 v + avw + av + auv (4) w 0 0 w 0 rβ ( pb + r) rβ β ( pb + r) a =, a = a =, a ( r + b)( β pb r) r + = b β pb r Thus, from Lemma 4, the stability of equilibrium (, ) ( 0,0), a = a = β uv = near w = 0 can be determined by studying a one-parameter family of map on a center manifold which can be represented as follows, { } c W 0,0 = uvw,, u= hvw,, h0,0 = 0, Dh0,0 = 0 (0) () () 67

9 X D Liao et al for sufficiently small v and w We now want to compute the center manifold and derive the mapping on the center manifold We assume near the origin, where ( ) (, ) ( ) h v w = Av + Bvw + Cw + (5) means terms of order By Lemma 4, those coefficients A, B and C can be determined by the equation (( b) h( v w) avw av avh( v w) b) ( ) = h v, w : h v a vw a v a vh v, w, w Substituting (5) into (6) and comparing coefficients of from which we solve, +, + = 0 v, vw and A+ Ab + a = 0, B + B b a = 0, C + C b = 0, = a a,, 0 b = b = A B C w in (6), we get Therefore the expression of (5) is approximately determined Substituting (5) into (4), we obtain a one dimensional map reduced to the center manifold It is easy to check that a a a a v φ w v = v + avw + av v v w ( 4 ) b + b + (7) φw φw φw ( 0,0) = 0, ( 0,0) 0, ( 0,0) 0 w vw w The condition (8) implies that in the study of the orbit structure near the bifurcation point terms of ( ) do not qualitatively affect the nature of the bifurcation, namely they do not affect the geometry of the curves of equilibriums passing through the bifurcation point Thus, (8) shows that the orbit structure of (7) near, 0,0 vw, = 0,0 of the map ( vw ) = is qualitatively the same as the orbit structure near (6) (8) v v + a vw + a v (9) Map (9) can be viewed as truncated normal form for the transcritical bifurcation (see Lemma 4) The stability of the two branches of equilibriums lying on both sides of s = 0 are easily verified 5 Simulations In this section, we will give a simulation to illustrate the result obtained in the above section Example Let p = 08, b = 0, 00 follows: r = and choose three groups of initial values for (,, ) ( 0,09,0 ), ( 04,06,0 ), ( 0,0,06 ) S I R as If let β = 005, we see that w < 05 < and two equilibrium points of system () occurs, where disease-free equilibrium point P (,0,0) is stable and other negative equilibrium point (no meaning in biology) is unstable The simulation result is presented in Figure 4 If let β = 05, then w = 5> and and two equilibrium points of system () occurs, where disease-free equilibrium point P (,0,0) is unstable and other positive equilibrium point Q ( 0, 06, 07) is unstable The simulation result is presented in Figure 5 67

10 X D Liao et al Figure 4 P (,0,0) is stable for case of w = 05 < Figure 5 Q ( 0,06,07) is stable for case of w = 5> 674

11 X D Liao et al 6 Biological Explanation The conclusion in Theorem 4 reveals a fact that the topological structure changes at disease-free equilibrium point will take place when system () encounters small perturbation for coefficient parameters Concretely, when parameter w =, system () has only one equilibrium point (,0,0 ) which is structural unstable With the change of the parameter w, the number and the stability of equilibrium point will be changed If let w <, we see that two equilibrium points of system () occur, where disease-free equilibrium point P (,0,0) is stable and other negative equilibrium points (no meaning in biology) are unstable No matter how to select the initial value, the system tends to the disease-free equilibrium point after multiple iterations Therefore, in this case, whether selection of initial value or small perturbation for coefficient parameters, epidemic disease will tend to be eliminated, that is, the population will be in disease-free state finally If let w >, we also see that two equilibrium points of system () occur, where disease-free equilibrium * * * point P (,0,0) is unstable and other endemic disease equilibrium points ( S, I, R ) are stable No matter how to select the initial value, the system tends to the endemic disease equilibrium point after multiple iterations Therefore, in this case, whether selection of initial value or small perturbation for coefficient parameters, disease does not tend to be eliminated, that is, the population will be diseased state finally Therefore, in reality we may control the factors of contact rate, birth rate, recovery rate, etc, to achieve the aim of prevention and treatment of disease Acknowledgements We thank the Editor and the referee for their comments This work has been supported by the Science Innovation Project (Grant 0KJCX05) and the Innovation and Developing School Project (Grant 04KZDXM065) of Department of Education of Guangdong province, the NSF of Guangdong province (Grant S000085) and the NSFP of Lingnan Normal University (Grant ZL0) References [] Kermack, WO and McKendrick, AG (97) Contributions to the Mathematical Theory of Epidemics Proceedings of the Royal Society of London, Series A, 5, [] Chen, LS (988) Models for Mathematical Ecology and Research Method Science Press, Beijing (In Chinese) [] Ma, Z, Zhou, Y, Wang, W and Jin, Z (004) Mathematical Modelling and Research of Epidemic Dynamical Systems Science Press, Beijing (In Chinese) [4] Allen, LJS (994) Some Discrete-Time SI, SIR and SIS Epidemic Models Mathematical Biosciences, 4, [5] Anderson, RM and May, RM (99) Infections Diseases of Humans: Dynamics and Control Oxford University Press, Oxford [6] Diekmann, O and Heersterbeek, JAP (000) Mathematical Epidemiology of Infectious Diseases: Model Building Analysis, and Interpretation John Wiley & Sons Ltd, Chichester [7] Hethcote, M (989) Three Basic Epidemiological Models In: Levin, S, et al, Eds, Applied Mathematical Ecology, Springer, New York, [8] Meng, XZ and Chen, LS (008) The Dynamics of a New SIR Epidemic Model Concerning Pulse Vaccination Strategy Applied Mathematics and Computation, 97, [9] Zhou, XL, Li, XP and Wang, WS (04) Bifurcations for a Deterministic SIR Epidemicmodel in Discrete Time Advances in Difference Equations, 04, 68 [0] Robinson, RC (004) An Introduction to Dynamical Systems: Continuous and Discrete Pearson Prentice Hall, Upper Saddle River [] Wiggins, S (990) Introduction to Applied Nonlinear Dynamical Systems and Chaos Springer, New York 675