c 2016 Society for Industrial and Applied Mathematics

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1 SIAM J. APPL. MATH. Vol. 76, No. 2, pp c 2016 Society for Inustrial an Applie Mathematics TURNING POINTS AND RELAXATION OSCILLATION CYCLES IN SIMPLE EPIDEMIC MODELS MICHAEL Y. LI, WEISHI LIU, CHUNHUA SHAN, AND YINGFEI YI Abstract. We stuy the interplay between effects of isease buren on the host population an the effects of population growth on the isease incience, in an epiemic moel of SIR type with emography an isease-cause eath. We revisit the classical problem of perioicity in inciences of certain autonomous iseases. Uner the assumption that the host population has a small intrinsic growth rate, using singular perturbation techniques an the phenomenon of the elay of stability loss ue to turning points, we prove that large-amplitue relaxation oscillation cycles exist for an open set of moel parameters. Simulations are provie to support our theoretical results. Our results offer new insight into the classical perioicity problem in epiemiology. Our approach relies on analysis far away from the enemic equilibrium an contrasts sharply with the metho of Hopf bifurcations. Key wors. epiemic moels, perioicity in isease incience, interepiemic perio, turning point, elay of stability loss, relaxation oscillation cycles AMS subject classifications. 34C26, 92D25 DOI /15M Introuction. Investigation of oscillations in isease incience is of funamental importance in mathematical epiemiology. Empirical ata of isease incience has shown clearly ientifiable cyclic patterns in many common iseases, incluing iseases for which environmental influences o not appear to play an important role, such as measles, pertussis, chicken pox, an mumps [2, 20]. Mechanisms for this type of autonomous oscillation have been extensively stuie in the mathematical epiemiology literature. These inclue, together with papers that introuce them, time elays in the transmission process [14, 20], varying total population size with ensity epenent emography an transmission [1, 22], nonlinear incience forms [16], iscrete age-structures with a nonsymmetric contact matrix among age groups [8], an seasonality in the transmission process in both eterministic an stochastic moels [3, 4, 12, 20]. The mathematical approach for these earlier works was bifurcation analysis (e.g., Hopf bifurcation theory, which analyzes moel behaviors in a neighborhoo of an enemic equilibrium. In the case of Hopf bifurcation, a certain egree of complexity nees to be introuce into the transmission process to prouce instability of the enemic equilibrium, an the bifurcation may occur in parameter regimes Receive by the eitors September 8, 2015; accepte for publication (in revise form January 19, 2016; publishe electronically March 24, Department of Mathematical an Statistical Sciences, University of Alberta, Emonton, AB, T6G 2G1, Canaa (mli@math.ualberta.ca, chunhuashan@gmail.com. The work of these authors was partially supporte by the Natural Science an Engineering Research Council of Canaa. The first author was also supporte by the Canaa Founation for Innovation. The thir author was also supporte by the National Science Founation. Department of Mathematics, University of Kansas, Lawrence, KS (wsliu@ku.eu. This author s work was partially supporte by University of Kansas GRF awar an a Jilin University Tang Ao-Qing Professorship. Department of Mathematical an Statistical Sciences, University of Alberta, Emonton, AB, T6G 261, Canaa, an School of Mathematics, Jilin University, Changchun, , People s Republic of China, an School of Mathematics, Georgia Institute of Technology, Atlanta, GA (yingfei@ualberta.ca. This author s work was partially supporte by the National Science Founation an the Natural Science an Engineering Research Council of Canaa. 663

2 664 M. Y. LI, W. LIU, C. SHAN, AND Y. YI that are not biologically realistic. For more complete reviews of relate work, we refer the reaer to [2, 13]. In the present paper, we apply a singular perturbation approach to this investigation. Our goal is to reveal a simple an biologically soun mechanism that can prouce large-amplitue oscillations in isease incience. Our basic assumption is that the host population has a small intrinsic growth rate ε > 0, the ifference between the natural birth rate an the natural eath rate. This slow-growth assumption is not biologically unrealistic. Demographic ata has shown that annual population growth rates in many inustrialize countries have been only slightly above zero, in the range of per year, for a long perio of time [28]. The slow-growth assumption may also apply to animal populations on livestock farms, where, for economic reasons, population may be kept near its carrying capacity, where the growth is close to zero. Using the intrinsic growth rate ε as a perturbation parameter, we show that a stanar SIR epiemic moel can be reformulate as a singularly perturbe problem. Applying techniques from geometric singular perturbation an global center manifol theory, we prove that for an open an biologically realistic parameter regime, stable perioic oscillations exist in rather simple SIR moels. Furthermore, our analysis emonstrates that the perioic solution has a large amplitue of orer O(1. This overcomes a common rawback of Hopf bifurcation analysis where the bifurcating perioic solutions are of small amplitue. Relaxation oscillations emonstrate istinguishe an robust cyclic patterns that consist a graual (slow change in the state variables over a long perio of time followe by a suen (fast change. A istinction between relaxation oscillations an harmonic oscillations was first mae by van er Pol [29]. Relaxation oscillation cycles have been use to explain fast-slow ynamics frequently observe in electrical circuits, mechanics, an many other physical an natural systems. In the present paper, for a simple epiemic moel with a slowly growing host population, we show that the perioic solutions are of relaxation oscillation type. An important characteristic of the moel uner the slow-growth assumption is the existence of a turning point. This is a point on the slow manifol with the population size at the critical community size to support an epiemic [2]. In the presence of turning points, the ynamics uner general perturbations are extremely rich an complicate (see [23, 24, 25, 27]. For the specific moel problem at han, the isease-free subspace is invariant for ɛ 0 since the isease will not evelop if it is not present at the initial time. Uner the invariance of the isease-free subspace, the turning point yiels a critical phenomenon calle elay of stability loss, in which a solution starts with a fast motion to approach a vicinity of the slow manifol, moves slowly along the slow manifol, passes through the turning point, an continues the slow motion along the slow manifol, then, up to some point, moves away from the slow manifol in a fast motion (see, e.g., [7, 17, 18, 23, 24, 25, 27]. In our moel, the slow manifol is in the isease-free region, an the time perio a solution spens in the vicinity of the slow manifol correspons to the interepiemic perio (IEP with low isease incience: the perio between epiemics (fast ynamics away from the slow manifol. The fast-slow oscillations characterize the global ynamics of the moel an capture the qualitative nature of the oscillatory behaviors in empirical isease ata; see Figure 1 for a set of ata on reporte cases of rubella in Canaa uring the perio Our analysis of the simple SIR moels has emonstrate that the existence of turning points an the associate elay of stability loss ue to the slow growth of the population offers a simple an robust mechanism for sustaine oscillations of isease incience.

3 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS 665 x Reporte Cases Year Fig. 1. The number of reporte cases of rubella in Canaa uring the perio The ata shows a long perio between 4 to 6 years of low incience followe by a sharp increase within 2 4 months. The ata was obtaine from the Notifiable Disease Surveillance System of Canaa [26]. Mathematically, our singular perturbation analysis is one for a three-imensional system, an the presence of turning points leas to a significant challenge. At a turning point, two eigenvalues are zero. This results in the loss of normal hyperbolicity of the one-imensional slow manifol, an the stanar geometric singular perturbation theory of Fenichel [9, 10] no longer applies. Another ifficulty we encounter in the analysis is having to eal with the nonlinear ynamics in a large neighborhoo of the slow manifol. Such a ifficulty oes not seem to appear in the analysis of many other biological moels, e.g., in the analysis of relaxation oscillation of a preator-prey moel [19]. The primary objective of our paper is to establish the mathematical framework an carry out etaile mathematical analysis for the singular perturbation approach to the stuy of epiemic moels. We have chosen a simple SIR moel to keep the mathematical technicality to its minimum, an the analysis is applicable to more complex moels. In a subsequent paper, we will investigate relaxation oscillations in a SEIR moel an give a more in-epth iscussion of biological implications of the mathematical results. The singular perturbation approach an associate asymptotic analysis have been successfully applie to the analysis of relaxation oscillation phenomena in many mechanical, physical, chemical, an biological systems. We hope that our stuy will lea to more applications of singular perturbation analysis to the stuy of isease transmission processes. 2. The moel an statements of main results The moel problem. Consier the sprea of an infectious isease in a host population of size N. Partition the population into susceptible, infectious, an recovere classes, an enote the sizes by S, I, an R, respectively, so that N = S + I + R. In the absence of the isease, we assume that N satisfies N = εg(n, where constant ε > 0 is assume to be small. A typical example of g(n is the quaratic form N(1 N/N, such that N has the logistic growth with carrying capacity N an intrinsic growth rate ε. It is natural to require the following. (A1 The function g(n satisfies g (N < 0, g(0 = g(n = 0 for some N > 0.

4 666 M. Y. LI, W. LIU, C. SHAN, AND Y. YI As a consequence, we have the following properties. Lemma 1. Assume (A1. Then, N is unique, an g(n > 0 for N (0, N an g(n < 0 for N > N. We further assume that the per capita natural eath rate is a constant > 0, an newborns b(n has a ensity epenent form b(n = N +εg(n. For simplicity, we assume that all newborns are susceptible to the isease. We consier the type of iseases that sprea through irect contact of hosts, an incience is given by h(s, NI, where h(s, N is a smooth function. We will assume the following basic properties on h(s, N: (A2 The function h(s, N is increasing in S an h(0, N = 0. A specific form of h(s, N that is commonly use is h(s, N = β(nsq, q 1, K 0. K + S This incience form h(s, NI inclues the bilinear incience βsi (with K = 0, q = 2, nonlinear incience βs q 1 I (with K = 0, q > 2, stanar incience λsi N (with β(n = λ/n, K = 0, q = 2, an saturation incience βsi K+S. The transmission process is emonstrate in the following iagram: ps b(n S h(s,ni I γi R S I+αI R The parameter γ enotes the recovery rate, an p enotes vaccination rate for a simple vaccination strategy. We assume that the infectious iniviuals suffer a iseasecause eath αi with a constant rate α. It is assume that isease confers permanent immunity, an all parameters are assume to be positive. The transfer iagram leas to the following system of ifferential equations: S = b(n h(s, NI ( + ps, (1 I = h(s, NI ( + γ + αi, R = ps + γi R. As a consequence, the total population size N satisfies (2 N = εg(n αi. It follows that for ε > 0 an α > 0, N varies with time, an moel (1 is a threeimensional system. Using b(n = N + εg(n an replacing the R equation by (2, we rewrite the moel (1 as the following equivalent system: S = N + εg(n h(s, NI ( + ps, (3 I = h(s, NI ai, N = εg(n αi,

5 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS 667 where a = + α + γ. We stuy system (3 for ε 0 in the feasible region D = {(S, I, N R 3 : S 0, I 0, N 0 an S + I N N }. From Lemma 1 an (2 we know that N < 0 if N > N. If follows that the region D is positively invariant with respect to system (3 an globally attracts all nonnegative solutions of (3. Global ynamics of moel (3 for the case ε = 0 were stuie in [11]. It was shown that the essential ynamics consist of a local, stable, two-imensional invariant manifol an, on the invariant manifol, a line of equilibria exists an all other solutions are heteroclinic orbits each connecting a pair of equilibria. This is a highly unstable structure an small perturbations can ramatically change the nature of the global ynamic. We will stuy the global ynamics of (3 for the case ε > 0 an show that, uner certain conitions, there exists a stable relaxation perioic cycle for small ε. In the rest of this section, we escribe the structure of the equilibria an their stability an state our main result on relaxation oscillations Structure of equilibria an statement of the main result. For ε 0, (0, 0, 0 an (S, 0, N, with N efine in (A1 an S = N /(+p, are equilibria of system (3. Proposition 2. There are no other equilibria for ε > 0 if an only if for all N, where h(s(n, N < a = + γ + α S(N = + p N ε a α α( + p g(n. Furthermore, if h(s, N < a for all S an N, then the equilibrium (S, 0, N attracts all solutions except (0, 0, 0. The global ynamics are trivial. Proof. The first statement can be checke irectly. Assume that h(s, N < a for all S an N. Then, for any initial conition other than (0, 0, 0, the solution (S(t, I(t, N(t satisfies that I(t 0 as t + an, on the plane {I = 0}, (S(t, N(t (S, N as t. In this work, we will focus on the cases where nontrivial ynamics are possible. In view of the statements in Proposition 2, we assume the following: (A3 The function h(n/( + p, N is nonecreasing for N (0, N. There is a unique N 0 (0, N such that h(s 0, N 0 = a, where S 0 = +p N 0. Furthermore, +p h S(S 0, N 0 + h N (S 0, N 0 > 0. Assumption (A3 is biologically intuitive since the force of infection h(s, N shoul increase as the population size N increases. We note that (S 0, 0, N 0 in (A3 has both ynamical an biological significance. In the case when h(s, N = βs, the equation h(s 0, N 0 = a becomes βs 0 = + γ + α an thus S 0 = ( + γ + α/β. In the classical SIR moel with no emography (b = = 0 an no isease-cause eath (α = 0, we have S 0 = γ/β, which is known as the critical size of susceptible population to sustain an epiemic [2, 12]. The ynamical significance of point (S 0, 0, N 0 is that it is a turning point, whose existence is the founation of the relaxation oscillation phenomenon. Lemma 3. Assume that (A3 hols. For ε > 0 small, there is a unique equilibrium E ε = (S ε, I ε, N ε with S ε, I ε, N ε > 0, an E ε (S 0, 0, N 0 as ε 0.

6 668 M. Y. LI, W. LIU, C. SHAN, AND Y. YI Proof. In aition to (0, 0, 0 an (S, 0, N, other equilibria of system (3 are etermine by h(s, N = a, I = ε α g(n, S = + p N ε a α α( + p g(n. The N coorinates are roots of ( f(n; ε := h + p N ε a α α( + p g(n, N a = 0. It follows from assumption (A3 that f(n 0 ; 0 = 0, f N (N 0 ; 0 = + p h S(S 0, N 0 + h N (S 0, N 0 > 0. An application of the implicit function theorem gives that for ε > 0 small, there is N ε such that f(n ε ; ε = 0 an N ε N 0 as ε 0. Note that the corresponing I-coorinate is I ε = ε α g(n ε > 0 for ε > 0 small. Stability of equilibria of system (3 is escribe in the next result, whose proof is given in Appenix I. Denote (4 ( a 0 = α h S (S 0, N 0 g(n 0 ( + pg N (N 0. + p Theorem 4. Assume that (A1, (A2, an (A3 hol. Then, for ε > 0 small, (i the equilibria (0, 0, 0 an (S, 0, N are sales each with two negative eigenvalues an one positive eigenvalue; (ii the equilibrium E ε always has a real negative eigenvalue an a pair of complex conjugate eigenvalues. If 0 > 0, then the complex eigenvalues have a negative real part an E ε is locally stable; if 0 < 0, then the complex eigenvalues have a positive real part an E ε is a sale. A rough statement of our main result is given in the following. A more technical statement (Theorem 10 of this result an its proof will be given in section 4. Theorem 5. Assume that (A1, (A2, an (A3 hol. Then, for system (3 with ε > 0 small, one of the following hols: (i the equilibrium E ε is a sink an it attracts all orbits except equilibria (0, 0, 0 an (S, 0, N ; (ii there exists an invariant annulus-like or isk-like two-imensional region that attracts all but equilibria orbits an contains at least one stable perioic orbit. We note that for fixe ε > 0 small, as 0 varies from positive to negative, in view of statement (ii in Theorem 4, it is possible that a perioic solution can be create through a supercritical Hopf bifurcation of E ε. This has been extensively stuie for many biological moels in the literature. We will not pursue this irection. Instea, we will investigate the existence of a relaxation oscillation using a global approach. More precisely, we will treat ε as a parameter, first unerstan the limiting global behaviors when ε = 0, an then examine how a relaxation oscillation is create for ε > 0, far from the enemic equilibrium E ε. In particular, the example in section 4.3 shows that a stable relaxation oscillation may exist even if E ε is stable.

7 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS Global ynamics of system (3 for ε = 0. In this section, we give a complete escription of the ynamics for the limiting system (3 at ε = 0. The result extens the work in [11] for a semilocal escription of the ynamics. We recall that system (3 for ε = 0 is (5 S = N h(s, NI ( + ps, I = (h(s, N ai, N = αi with feasible region D = {(S, I, N R 3 : S 0, I 0, N 0, S + I N N }, which is positively invariant for (5. It can be verifie that the isease-free plane {I = 0} an the half-line Z 0 := { S = N + p, I = 0, N 0 are both invariant uner system (5. In particular, Z 0 consists of equilibria of ( A complete characterization of ynamics of (5. On the invariant plane {I = 0}, all solutions (S(t, I(t, N(t satisfy that I(t 0, N(t N(0, an S(t N(0 as t. + p The set Z 0 of equilibria attracts all solutions within {I = 0}. The linearization at each point (N/( + p, 0, N Z 0 is ( + p h(n/( + p, N 0 h(n/( + p, N a 0 0 α 0 with eigenvalues λ 1 = 0, λ 2 = ( + p < 0, an λ 3 = h(n/( + p, N a. The eigenvectors associate with λ 1 an λ 2 span the plane {I = 0} an that associate with λ 3 is transversal to the plane {I = 0}. The eigenvalue λ 3 = h(n/( + p, N a changes sign across the point (S 0, 0, N 0 Z 0, where S 0 an N 0 are efine in (A3. The complete ynamics for the case ε = 0 are escribe in the following result an epicte in Figure 2. The proof is given in Appenix I. Theorem 6. Assume that (A2 an (A3 are satisfie. Then the following statements hol: (i Every solution of system (5 is boune for t 0 an the set Z 0 is the global attractor. (ii The unstable manifol of each equilibrium (N/( + p, 0, N Z 0 with N > N 0 is a heteroclinic orbit to an equilibrium ( S, 0, N Z 0 with 0 < N < N 0. The relationship N 1 < N 2 < N 0 if N 1 > N 2 > N 0 hols. Furthermore, lim N N = N (0, N 0. We enote by M(Z 0 the two-imensional invariant manifol that consists of heteroclinic orbits establishe in Theorem 6(ii, an efine a map } H : (N 0, (0, N 0, H(N = N, where N is efine by the heteroclinic orbits in Theorem 6(ii. The invariant manifol M(Z 0 an the map H will play important roles in our results on relaxation oscillations for moel (3 with ε > 0.

8 670 M. Y. LI, W. LIU, C. SHAN, AND Y. YI I O S N 0 N Z 0 Fig. 2. Heteroclinic structure of (3 with ε = Persistence of M(Z 0 for ε > 0 small. We are intereste in whether the invariant manifol M(Z 0 will persist for ε > 0 small, that is, for ε > 0 small, whether there is an invariant manifol M ε for system (3 so that M ε M(Z 0 as ε 0. Recall that when ε = 0, for each equilibrium w = ( +p N, 0, N Z 0, the eigenvalues of the linearization at w are ( λ 1 = 0, λ 2 = ( + p, λ 3 = h + p N, N a. Base on the relative size of eigenvalues, the consieration can be ivie into two cases. Case 1: a = + α + γ < + p. It follows that h( +pn, N a > ( + p for all N 0. At each point w Z 0, we have λ 1 > λ 2 an λ 3 > λ 2. Applying a center manifol theorem in [5, 6] to the invariant set Z 0, we obtain the existence of a two-imensional center manifol W c (Z 0. The center manifol W c (Z 0 is invariant uner (5 an contains Z 0 an all orbits boune in the vicinity of Z 0. At each w Z 0, the tangent space T w W c (Z 0 is spanne by the eigenvectors associate with λ 1 an λ 3 (both are larger than λ 2. Most importantly, the center manifol theorem guarantees the persistence of W c (Z 0 for ε > 0 small. In general, a center manifol may not be unique but any center manifol will contain all orbits that are boune in the vicinity of Z 0. Therefore, for this moel problem, W c (Z 0 coincies with M(Z 0 an is unique, an M(Z 0 persists for ε > 0. Case 2: a = +α+γ +p. In this case, there exists a unique ˆN [0, N 0 such that h( +p N, N a > (+p for N > ˆN but h( +pn, N a (+p for N ˆN. The general results on center manifols in [5, 6] cannot be applie to the whole set Z 0 to obtain a two-imensional center manifol. For any fixe δ > 0, the results in [5, 6] can be applie to the subset Z0 δ := Z 0 {N ˆN + δ} but the corresponing center manifol W c (Z0 δ will only be a proper subset of M(Z 0. It turns out, for ε > 0, that parts of relaxation oscillations coul occur outsie W c (Z0 δ for all δ > 0. We take the avantage of a crucial property that the set {I = 0} is invariant uner system (3 for all ε 0 an show that M(Z 0 persists for ε > 0 small even though it is not normally hyperbolic. This is establishe in Appenix II. This persistence result appears to be contraictory to Mãné s result that an invariant manifol is persistent if an only if it is normally hyperbolic [21]. It is not, since the persistence in Mãné s result is with

9 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS 671 respect to all small perturbations, while the perturbations, in our system are special: they leave the set {I = 0} invariant. As mentione above, it is possible that a portion of a relaxation oscillation occurs over the region where N < ˆN. In the limit as ε 0, this portion approaches Z 0 along the eigenvector associate with (p + q in general The map H near N 0. The map H : (N 0, (0, N 0 efine in Theorem 6 will be a key ingreient for our main result on relaxation oscillations. Detaile global properties of H seem to be not achievable. On the other han, it is possible to examine properties of H near N 0 base on an approximation of W c (Z 0 near (S 0, 0, N 0 or, simply, a center manifol W c (S 0, 0, N 0 of the equilibrium (S 0, 0, N 0. Note that the eigenvalues at (S 0, 0, N 0 are λ 1 = λ 3 = 0 > λ 2 = ( + p. Thus, for an equilibrium w Z 0 near (S 0, 0, N 0, the corresponing eigenvalues satisfy λ 1 > λ 2 an λ 3 > λ 2. As a consequence, W c (S 0, 0, N 0 M(Z 0 an hence is unique. It shoul be pointe out that, in general, a center manifol may not be unique An approximation of the center manifol W c (S 0, 0, N 0. We look for an approximation of the center manifol W c (S 0, 0, N 0 in the vicinity of (S 0, 0, N 0 as the graph of a function S = N + U(N, II = + p + p N + a 0(NI + a 1 (N, II 2. The form is justifie by the fact that {I = 0} is invariant an W c (S 0, 0, N 0 {I = 0} Z 0. Taking the erivative of S = +pn + U(N, II with respect to t, we have S = + p N + a 0IN + a 1,N I 2 N + a 0 I + 2a 1 II + a 1,I I 2 I. From (5 we have ( N h + p N + a 0(NI + a 1 (N, II 2, N I ( ( + p + p N + a 0(NI + a 1 (N, II 2 ( = α + p + a 0I + a 1,N I 2 I ( ( + (a 0 + 2a 1 I + a 1,I I 2 I h + p N + a 0(NI + a 1 (N, II 2, N a I. Expaning h at the point (bn/(b + p, N we get ( ( N h + p N, N I ( + p + p N + a 0(NI + O(I 2 = α ( ( + p I + a 0 h + p N, N a I + O(I 2. Comparing coefficients of I 0 an I 1 we obtain ( α +p h +p N, N a 0 (N = (. + p + h +p N, N a Note that we restrict the approximation of W c (S 0, 0, N 0 near (S 0, 0, N 0. Thus, N is close to N 0, an hence, the enominator in the above expression is close to + p > 0.

10 672 M. Y. LI, W. LIU, C. SHAN, AND Y. YI Near equilibrium (S 0, 0, N 0, the center manifol W c (S 0, 0, N 0 is given as the graph of the function (6 S = + p N + a 0(NI + O(I 2 ( = α + p N + +p h +p ( N, N I + O(I 2. + p + h +p N, N a On the center manifol W c (S 0, 0, N 0 an near (S 0, 0, N 0, system (5 is reuce to a two-imensional system, ( N I = h + p + a 0(NI + O(I 2, N I ai, (7 N = αi Properties of the map H near N 0. Proposition 7. The map H satisfies H(N 0 = N 0, H (N 0 = 1, an H (N 0 = 2 ( α a 0(N 0 h S S 0, N 0. Proof. Set v(t = N(t N 0. In terms of (I, v, system (7 becomes ( I = h S 0 + v + p + a 0(N 0 + vi + O(I 2, N 0 + v I ai, v = αi. Since {I = 0} is invariant an the map H is efine through the ynamics where I > 0, we ivie the two equations above to get I v = 1 ( ( h S 0 + v (8 α + p + a 0(N 0 + vi + O(I 2, N 0 + v a. Expaning the right-han sie at v = 0 leas to ( h S 0 + v + p + a 0(N 0 + vi + O(I 2, N 0 + v a ( v = h S + p + (a 0 + a 0vI + h N v h SS ( v + h SN + p + (a 0 + a 0vI ( v 2 + p + (a 0 + a 0vI v h NN v 2 + O(I 2, v 2 I, I 3, where the partial erivatives of h are all evaluate at (S 0, N 0, an a 0 = a 0 (N 0 an a 0 = a 0(N 0. Denote (9 L = + p h 2 S + h N an Q = ( + p 2 h SS p h SN + h NN. Equation (8 becomes (10 I v = 1 1 α 2α (2Lv + Qv2 ( h S + ( + p h SS + h SN v (a 0 + a 0vI + O(I 2, v 2 I, v 3.

11 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS 673 By the existence an smoothness of solutions an smooth epenence of solutions on parameters, for v small, we look for solutions of the form I(v = c 0 + c 1 v + c 2 v 2 + O(v 3. Substituting I(v into (10 an comparing terms of like powers in v we get (11 c 1 = 1 α a 0h S c 0 + O(c 2 0, c 2 = 1 2α L 1 2α ( a 0h S 1 α a2 0h 2 S + + p a 0h SS + a 0 h SN c 0 + O(c 2 0. Thus, near v = 0, the solution of (10 is (12 I(v =c 0 a 0h S v α c 0 L 2α v2 + O(c 0 v 2. To efine H, we nee the initial conition I(v = 0 at v = N N 0 for N > N 0 an N N 0 1. We can then etermine the value c 0 corresponing to this initial conition. From (12, or equivalently, Thus, (13 0 =c 0 a 0h S (N N 0 c 0 L α 2α (N N O(c 0 (N N 0 2, c 0 ( 1 1 α a 0h S (N N 0 + O(N N 0 2 = L 2α (N N 0 2. c 0 = L 2α (N N La 0h S 2α 2 (N N O(N N 0 4. The value of H(N satisfies I(H(N N 0 = 0. Note that H(N N 0 = H(N H(N 0 = H (N 0 (N N H (N 0 (N N 0 2 +O(N N 0 3. It then follows from I(H(N N 0 = 0, (11, (12, an (13 that 0 = L 2α (N N La 0h S 2α 2 (1 H (N 0 (N N 0 3 L ( H (N 0 (N N α 2 H (N 0 (N N O(N N 0 4. Comparing (N N 0 2 terms gives that H (N 0 = 1 (ue also to that H is ecreasing. The (N N 0 3 terms then yiel This completes the proof. La 0 h S α 2 + L 2α H (N 0 = A iscussion an the link to the main result. In this section, we summarize the results for system (5, iscuss the impact of the sign changing eigenvalue h(s, N a, an provie mathematical an biological motivations for our main result. For N < N 0, h(n/( + p, N a < 0, an it implies that for an initial state (S(0, I(0, N(0 near the region {I = 0, N < N 0 }, I(t ecreases an the solution

12 674 M. Y. LI, W. LIU, C. SHAN, AND Y. YI converges to an equilibrium in Z 0 with N < N 0. Biologically speaking, if the total population N is below the critical community size N 0 or, equivalently, the number of susceptibles S is below the critical size S 0 = N 0 /( + p, then the population can not sustain an epiemic an the isease ies out. We escribe the ynamics for solutions with initial conitions near the other region {I = 0, N > N 0 } in three stages. Stage I. For an initial state (S(0, I(0, N(0 with N > N 0, h(n/( + p, N a > 0 for small t > 0 an I(t increases initially. In biological terms, if the population size surpasses the critical community size N 0, then any initial infection will lea to a isease outbreak. Stage II. As I(t increases away from {I = 0}, the ynamics outsie {I = 0} become ominant; in particular, N(t ecreases. Once N(t < N 0 (or equivalently S(t < S 0, we know that h(s, N a < 0 an I(t begins to ecrease. The solution follows a heteroclinic orbit epicte in Figure 2. Stage III. As time goes on, I(t continues to ecrease. Eventually the solution will enter a vicinity of the region {I = 0, N < N 0 } an is attracte to an equilibrium in Z 0 with N < N 0. The isease outbreak leas to an epiemic but the isease eventually ies out. We see that when ε = 0, moel (5 only escribes epiemics of the isease; the isease eventually ies out. There is no mechanism for the recurrence of the isease if the population growth is zero. This is parallel to the classical SIR moel with no emography an isease-cause eath. When ε > 0, solutions of system (3 with N(0 > N 0 an I(0 small go through Stages I an II as escribe above, but Stage III will no longer be the terminal stage. In this case, the isease-free set {I = 0} remains invariant. The half-line Z 0 also remains invariant but is no longer a set of equilibria. Instea, Z 0 becomes an orbit for which N increases with t with spee of orer O(ε. For this reason, Z 0 is calle the slow manifol for small ε > 0. Stage IV. When ε > 0, for a solution in the vicinity of Z 0 with N < N 0 uring Stage III, it will follow an orbit on the slow manifol Z 0 by the continuous epenence on initial conitions. As N(t increases beyon the critical community size N 0, the solution enters the region {I = 0, N > N 0 }. As a consequence, I(t begins to increase an the solution repeats Stages I III, leaing to another epiemic. The perio uring which the solution moves along the slow manifol is the IEP. We see that when ε > 0, the fall of susceptible population uring an epiemic an the recovery of the susceptible population uring the IEP prouce an oscillating behavior. In summary, for ε > 0 small, all orbits, except for solutions on {I = 0} an E ε of system (3, will exhibit oscillating behaviors. Three key conitions are responsible for the mechanism of oscillation: (C0 the plane {I = 0} is invariant for ε 0, (C1 the assumption on the natural growth g(n of the total population in the absence of isease, an (C2 the sign changing assumption of the eigenvalue λ 3 = h(s, N a. In the language of singular perturbation theory, conition (C2 means that the point (S 0, 0, N 0 at which h(s 0, N 0 a = 0 is a turning point. This point marks the level of N or S that separates the region of isease ecline from that of isease rise. Conition (C1 implies that on Z 0, with the population growth an increase of susceptibles from newborns, all orbits move from a region of isease ecline where N < N 0 to a region of isease rise where N > N 0. Conitions (C0 (C2 imply that the turning point (S 0, 0, N 0 is associate with the elay of stability loss [7, 17, 18,

13 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS , 24, 25, 27]. Conition (C0 is the consequence of the biological fact that if the isease is not present at time t = 0, it remains absent from the population for t 0. We emphasize that while conition (C0 hols true naturally for the specific moel we consier, it is, however, highly egenerate in general when turning points are present. Without conition (C0, presence of turning points can make {I = 0} nonnormally hyperbolic [9, 15] an estroy the persistence of {I = 0} for ε > 0. The impact of turning points for ε > 0 can be extremely ifficult to investigate. It is important to note that though the oscillating behaviors near the positive equilibrium E ε when ε > 0 are apparent from the fact that E ε has a pair of complex eigenvalues, which bifurcate from the ouble zero eigenvalue of E 0 (Theorem 4, the heteroclinic orbit structure establishe in Theorem 6 an the elay of stability loss associate with the turning point etermine the oscillatory behaviors far away from E ε. While the oscillating behaviors of the SIR moel when ε > 0 as escribe above are biologically intuitive an mathematically verifiable, they can be ecaye oscillations. The important mathematical question with biological significance is whether there exists a stable perioic oscillation. Our main result in the next section characterizes, for the existence of stable perioic solutions, abstract conitions in general an verifiable sufficient conitions in particular. Those perioic oscillations, if they exist, will typically have a large perio of orer O(ε Global ynamics of (3 for ε > 0 small. Recall that the two-imensional invariant manifol M(Z 0 from Theorem 6 persists to M ε for ε > 0 small. We use the properties of M ε to establish an abstract result from geometric singular perturbations with turning point, focusing on results on relaxation oscillations. Due to the lack of explicit global representation of M(Z 0, not all abstract results can be transforme back to the concrete moel (3 in the sense that the corresponing conitions are not easy to verify. For some sufficient conitions on the existence of perioic oscillations, we are able to transform the conitions back to the original moel an they are verifiable Formulation of a singularly perturbe problem. For δ > 0 small, let M be the manifol consisting of all heteroclinic orbits from (S, 0, N with N 0 < N < N +δ, together with the point (S 0, 0, N 0. Then, M persists in the sense as iscusse in Case 2 of section 3.2 an prove in Appenix II. Let M ε be the perturbe manifol of M for ε > 0 small; that is, M ε is invariant an M ε M as ε 0. Due to the fact that {I = 0} is invariant for all ε an the set Z 0 is normally hyperbolic within {I = 0}, we have that Z 0 persists for ε > 0 small; that is, Z ε = M ε {I = 0} persists as a portion of the bounary of M ε. Let φ(u, v; ε for (u, v R be a parameterization of the center manifol M ε, where R is a boune omain in {u 0, v 0} to be further characterize later on. We require that (P1 for ε = 0, the heteroclinic orbits are etermine by v = const an the u- variable is ecreasing from the right branch of slow manifol v = T (u to the left branch as time increases (see Figure 3; (P2 for ε 0, the set Z ε correspons to the curve {v = T (u} for function T : (0, U (0, V with T (U = V, where (U, V correspons to the point (S, 0, N + δ Z, for arbitrarily fixe δ > 0 inepenent of ɛ, an hence {v = V } correspons to the heteroclinic orbit from (S, 0, N + δ Z 0 ; therefore, R = {(u, v : 0 < u < U, T (u v < V };

14 676 M. Y. LI, W. LIU, C. SHAN, AND Y. YI (P3 for ε 0, the point (u, v = (u 0, T (u 0 correspons to the point (S 0, 0, N 0, (u, v = (u 0, T (u 0 correspons to (S, 0, N. v v=t(u 0 u0 u 0 u Fig. 3. Heteroclinic structure of (15 with ε = 0. In terms of (u, v R, suppose that system (3 on the center manifol can be put into the form (14 u = F (u, v; ε, v = G(u, v; ε. We now examine the properties that the vector fiel of system (14 must satisfy. First of all, (P1 implies that G(u, v; 0 = 0, F (u, T (u; 0 = 0, an F (u, v; 0 < 0 for v > T (u. Thus, we can write G(u, v; ε = εg 1 (u, v; ε, F (u, v; ε = T (u v + εf 1 (u, v; ε. The property (P2 implies that G 1 (u, T (u; ε = T u F 1 (u, T (u; ε. System (14 can be rewritten as (15 u = T (u v + εf 1 (u, v; ε, v = εt u (uf 1 (u, T (u; ε + ε(v T (ug 2 (u, v; ε. System (15 is a singularly perturbe problem with ε as the singular parameter. As usual, the time t is calle the fast time, which is the physical time of our problem. In terms of the slow time τ = εt, system (15 becomes (16 ε u = T (u v + εf 1 (u, v; ε, v = T u (uf 1 (u, T (u; ε + (v T (ug 2 (u, v; ε, where the overot symbol inicates the erivative with respect to τ. The slow manifol is Z = {v = T (u}. On the slow manifol Z, the flow is given by u = εf 1 (u, T (u; ε. It has a global sink at u = u 0. We recall that the set Z is invariant uner system (15 (or equivalently uner system (16 for all ε 0. This property is crucial in creating oscillations in the system. In fact, one will see later that there is a turning point on Z an, ue to the invariance of Z for all ε, the turning point causes the elay of stability loss [7, 17, 18, 23, 24, 25, 27]. We believe that the elay of stability loss is one of the most important mechanisms for the oscillation structure in biological population systems.

15 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS 677 To escribe the elay of stability loss, we efine a map P : (0, u 0 (u 0, u 0 via (17 P (u u T u (ξ ξ = 0. F 1 (ξ, T (ξ Also, for any v > min{t (u : u (0, }, let l(v an r(v be the two solutions of v = T (u for u with l(v < r(v an set v 0 = T (u 0. Proposition 8 (elay of stability loss. Fix δ > 0 small, for ε > 0 small, let (u(τ; ε, v(τ; ε be the solution of system (16 with the initial conition (u(0, v(0, where u(0 < u 0 an v(0 = T (u(0+δ. Let τ(ε > 0 be the time such that v(τ(ε; ε = T (u(τ(ε + δ. Then, as ε 0, r(v(τ(ε P (l(v(0. Note that P (l(v 0 < u 0, an hence, T (l(v 0 = v 0 > T (P (l(v 0. Theorem 9. For ε > 0 small, either the equilibrium (u ε, T (u ε is a global attractor of R for system (15 or there is a stable perioic relaxation oscillation. Furthermore, (i if there exists u 1 (l(v 0, u 0 such that T (u 1 < T (P (u 1, then, for ε > 0 small, system (15 has a stable perioic relaxation oscillation whose limiting orbit, as ε 0, is the union of the heteroclinic orbit from (P (u c, T (P (u c to (u c, T (u c an the curve on {v = T (u} from (u c, T (u c to (P (u c, T (P (u c for some u c (l(v 0, u 1 satisfying T (u c = T (P (u c ; (ii if for every u (l(v 0, u 0, T (u > T (P (u, then, for ε > 0 small, the equilibrium (u ε, T (u ε is a global attractor of R for system (15. Proof. To prove statement (i, note that the unstable manifol W u (u 0, v 0 will approach the left branch of the slow manifol {v = T (u} almost horizontally near the set {v = v 0 } towar the point (l(v 0, v 0 an then follow the slow orbit through (l(v 0, v 0 up to near the point (P (l(v 0, T (P (l(v 0, an leave the slow manifol almost horizontally near the set {v = T (P (l(v 0 }. Due to the fact that T (l(v 0 = v 0 > T (P (l(v 0, upon leaving the slow manifol at near the point (P (l(v 0, T (P (l(v 0, the unstable manifol W u (u 0, v 0 stays below its initial portion. Therefore, the unstable manifol spirals inwar. By the same argument, the existence of u 1 with the property T (u 1 < T (P (u 1 implies that the forwar orbit starting from (u 1 +δ, T (u 1 for some δ > 0 small spirals outwar. This orbit together with the unstable manifol W u (u 0, v 0 encloses a positively invariant region. By the Poincaré Benixon theorem, there is a stable perioic orbit. The above argument also shows that between any numbers û 1, û 2 (l(v 0, u 0 with û 2 < û 1, T (û 1 < T (P (û 1, an T (û 2 > T (P (û 2, there is a perioic orbit strictly enclose by the two orbits through, respectively, the points (û 1 + δ, T (û 1 an (û 2 + δ, T (û 2 for some small δ. Therefore, the limiting position of a perioic orbit is exactly as escribe in the statement. The proof for the statement (ii follows from the above argument an we will omit the etails here Statement of the main results for system (3. To translate Theorem 9 in terms of the original system (3, we recall that H : (N 0, (0, N 0 is the function efine as follows: for ε = 0 an for (N/( + p, 0, N Z 0 with N > N 0, (H(N/(+p, 0, H(N Z 0 is the unique equilibrium so that there is a heteroclinic orbit from (N/( + p, 0, N to (H(N/( + p, 0, H(N. The map P efine in

16 678 M. Y. LI, W. LIU, C. SHAN, AND Y. YI (17 is given by P : (0, N 0 (N 0, by (18 P (N N h(ξ/( + p, ξ a g(ξ ξ = 0. Theorem 10. Let H(N an P (N be efine as above. For ε > 0 small, either the enemic equilibrium (S ε, I ε, N ε is a global attractor or there is a stable perioic relaxation oscillation. More precisely, (i if there exists N 1 (H(N, N 0 such that N 1 > H(P (N 1, then for ε > 0 small, system (3 has a stable perioic relaxation oscillation whose limiting orbit, as ε 0, is the union of the heteroclinic orbit from the point (P (N c /( + p, 0, P (N c to the point (N c /( + p, 0, N c an the segment on Z 0 from the point (N c /( + p, 0, N c to the point (P (N c /( + p, 0, P (N c for some N c (H(N, N 1 satisfying N c = H(P (N c ; (ii if for every N (H(N, N 0, N < H(P (N, then for ε > 0 small, the enemic equilibrium (S ε, I ε, N ε is a global attractor for system (3. Proof. It suffices to show that for ε > 0, M ε attracts all solutions except the equilibria (0, 0, 0 an (S, 0, N. Since M ε has a region attracting orbits on M ε an M ε is normally stable, there is neighborhoo U of M ε inepenent of ε such that for ε > 0 small enough, any solution entering U is attracte by the attracting region on M ε. Therefore, we only nee to show that any solution will enter U. First of all, we see that N (t < 0 if N(t > N. Thus, all solutions are attracte by the omain D an the omain D is positively invariant. It can be verifie that M ε attracts all solutions on {I = 0} except (0, 0, 0 an (S, 0, N. Now, for a solution (S(t, I(t, N(t with the initial conition (S(0, I(0, N(0 D an I(0 > 0, by continuity, for ε > 0 small inepenent of the solution starting in D, the solution will approach a point ( S, 0, N Z 0 with N N 0 an then follow the slow orbit through ( S, 0, N Z 0. Therefore, it enters a neighborhoo of (S 0, 0, N 0 an hence into U Concrete conitions for the existence of relaxation oscillations of system (3. Proposition 11. The map P satisfies P (N 0 = N 0, P (N 0 = 1, an where L an Q are efine in (9. P (N 0 = 4g (N 0 L 2g(N 0 Q, 3g(N 0 L Proof. It follows from the efinition of P that P (N 0 = N 0. Differentiating with respect to N on (18 we get (19 h(p (N/( + p, P (N a P (N = g(p (N h(n/( + p, N a. g(n Note that P (N =P (N 0 + P (N 0 (N N P (N 0 (N N O(N N 0 3 =N 0 + P (N 0 (N N P (N 0 (N N O(N N 0 3,

17 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS 679 P (N =P (N 0 + P (N 0 (N N 0 + O(N N 0 2, g(n =g(n 0 + g (N 0 (N N g (N 0 (N N O(N N 0 3, g(p (N =g(n 0 + g (N 0 (P (N N g (N 0 (P (N N 0 2 =g(n 0 + g (N 0 (P (N 0 (N N P (N 0 (N N g (N 0 (P (N 0 2 (N N O(N N 0 3, an h(n/( + p, N a =L(N N Q(N N O(N N 0 3, h(p (N/( + p, P (N a =L(P (N N Q(P (N N O(N N 0 3 =L (P (N 0 (N N P (N 0 (N N Q(P (N 0 2 (N N O(N N 0 3. Substituting these expansions into (19 an comparing the terms of like powers in (N N 0 we get for N N 0, gl(p 2 = gl = P = 1, for (N N 0 2, 1 2 g(lp + Q + g L glp = g L gq = P = 4g L 2gQ. 3gL This completes the proof. Combining Propositions 7 an 11 we obtain the following result. Proposition 12. The function F = H P satisfies F (N 0 = N 0, F (N 0 = 1, an F (N 0 = H (N 0 P (N 0 = 2 0 ( + pg(n where 0 is efine in (4, an L an Q are efine in (9. g (N 0 L + g(n 0 Q, g(n 0 L As a irect consequence of Theorem 10 an Proposition 12, we have the following. Corollary 13. If F (N 0 < 0, then, for ε > 0 small, there is at least one stable relaxation oscillation. Example. We establish the existence of a stable relaxation oscillation in the case that E ε is stable. More precisely, we take a special case of h that is biologically plausible an show that for any g satisfying (A1, there are parameter ranges for β an K, epenent on all other fixe parameters so that 0 > 0 in Theorem 4, for which the equilibrium E ε is stable an F (N 0 < 0 hols. This guarantees the existence of a stable relaxation oscillation. Thus, a stable relaxation oscillation may exist even when the equilibrium E ε is stable. In this case, there exists at least an unstable perioic

18 680 M. Y. LI, W. LIU, C. SHAN, AND Y. YI orbit between the stable relaxation an the equilibrium. In general, the unstable perioic orbit is not necessarily a relaxation oscillation but a small perioic orbit through a subcritical Hopf bifurcation. Consier h(s, N = with β > a; it can be verifie that We also note that ( a α + p βs K+S S 0 = ak β a > 0, N 0 = + p ak β a, L = βk + p (K + S 0 2, Q = 2 2βK ( + p 2 (K + S 0 3, ( a 0 = α βk + p (K + S 0 2 g(n 0 ( + pg N (N 0, F 2 0 (N 0 = ( + pg(n g (N 0 L + g(n 0 Q 3 g(n 0 L ( 2 ( a = ( + pg(n 0 α βk + p (K + S 0 2 g(n ( + pg N (N 0 2 K + S 0 3 (K + S 0 2 g(n 0. βk (K + S Choose β > a such that a α K + S ( 0 a (K + S 0 2 = α + p + p an choose K such that for N 0 = N 0 = +p ( a 0 = α F (N 0 = 2 + p 2 3(β a < 0, ak β K + p (K + S 0 2 g(n 0 > 0, ( a α + p 2 3(β a 2 3(β a β a, g N(N 0 = 0 hols. Then βk (K + S 0 2 < 0. βk (K + S 0 2. This accomplishes the goal of this example. We note that the construction of the above example strongly inicates that it may not be rare to have stable relaxation oscillations when the enemic equilibrium E ε is stable. It is also possible to give a more etaile analysis, for fixe forms of h an g, on the parameter ranges for such coexistence of stable structures. It may reveal a more comprehensive unerstaning of the global ynamics of this moel. 5. Numerical simulations an biological interpretations. In this section, we provie results from numerical simulations of moel (3 that emonstrate an support our theoretical results on the existence of stable perioic solutions of relaxation oscillation type. Unless otherwise state, we choose g(n = N (1 N N an h(s, N = βs K + S. It can be verifie that g(n an h(s, N satisfy assumptions (A1, (A2, an (A3.

19 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS Existence of relaxation oscillations. Case 1. Existence of relaxation oscillation when E ε is unstable. Choose = 0.2, p = 0.01, α = 0.048, β = 1, γ = 0.75, K = 0.1, N = 400, an ε = The enemic equilibrium E ε = (49.9, , is unstable. In Figure 4, we show that a trajectory starting from (35, , 67 approaches a stable relaxation oscillation cycle with IEP I IEP Time (a Time series plot x 10 5 I N N 10 0 I S (b Projection in the (N, I plane (c Plot in the 3D phase space Fig. 4. An orbit converging to a stable relaxation oscillation cycle when E ε is unstable. Case 2. Existence of relaxation oscillation when E ε is stable. Choose = 0.2, p = 0.01, α = 0.049, β = 1, γ = 0.75, K = 0.1, ε = 10 4, an N = 380. In Figure 5, we show that a trajectory starting from (197, 1.47, approaches a stable relaxation oscillation cycle. We moifie the function h(s, N in a small neighborhoo of E ε so that it becomes locally asymptotically stable. Such moification oes not change the relaxation oscillation cycle since it is far away from E ε. A trajectory starting from (150, 1, 160 is shown in Figure 5 to approach the stable equilibrium E ε. We note that there shoul be a secon perioic orbit that is unstable (not shown in Figure Depenence of IEP on physical parameters. 1. Depenence of IEP on the intrinsic growth rate ε. We emonstrate using numerical evience that the IEP is of orer 1/ε. We choose = 0.2, p = 0.01, α = 0.048, β = 1, γ = 0.75, K = 0.1, an N = 400, an we vary

20 682 M. Y. LI, W. LIU, C. SHAN, AND Y. YI N I S Fig. 5. Numerical simulations show the existence of a stable perioic solution when the enemic equilibrium is stable. An oscillatory orbit with a large amplitue is shown to converge to a stable relaxation oscillation cycle, an an orbit with a smaller amplitue converges to the stable enemic equilibrium E ε. 7 x x IEP 4 3 IEP (a Depenence of IEP on ε ε x /ε (b Depenence of IEP on 1/ε x 10 4 Fig. 6. The IEP increases as the intrinsic growth rate ε ecreases in (a, an the IEP is in proportion to 1/ε in (b. the values of ε in the interval [10 5, 10 4 ]. For the simulations, we assume that the isease is in the IEP if the number of the infecte iniviuals is less than Plots of IEP against the values of ε an 1/ε are shown in Figure Depenence of IEP on parameters α an β. In Figure 7, we show that the IEP ecreases as the transmission coefficient β in creases, an the IEP increases as the rate α of isease-cause eath increases. For the simulations, we choose = 0.2, p = 0.01, γ = 0.75, K = 0.1, N = 400, an ε = 10 4 an vary values of β when α = in Figure 7(a or vary the values of α when β = 1 in Figure 7(b.

21 RELAXATION OSCILLATIONS IN EPIDEMIC MODELS x x IEP (a Depenence of IEP on β β IEP (b Depenence of IEP on α α Fig. 7. Depenence of IEP on the transmission coefficient β an on the rate of isease-cause eath α. Appenix I: Technical proofs. Proof of Theorem 4. To show (i, note that the linearization of system (3 at (0, 0, 0 is ( + p 0 + εg N (0 J(0, 0, 0 = 0 a 0, 0 α εg N (0 whose eigenvalues are ( + p < 0, a < 0, εg N (0 > 0, where εg N (0 > 0 follows from (A1. Similarly, the linearization at (S, 0, N is J(S, 0, N = ( + p h + εg N (N 0 h a 0 0 α εg N (N with eigenvalues (+p < 0, h(s, N a > 0, an εg N (N < 0, where h(s, N a > 0 follows from (A3 an εg N (N < 0 follows from (A1. The linearization at E ε is J = J(S ε, I ε, N ε = whose characteristic polynomial is given by ( + p + h S I ε a h N I ε + εg N h S I ε 0 h N I ε 0 α εg N ( P ε (λ = λ p + εh ( Sg α εg N λ 2 ε ( + pg N ah Sg α h N g h S g N I ε λ Hence, + αh S I ε + α( + ph N I ε ε(a + αh S g N I ε., (20 tr(j = ( + p εh Sg α + εg N < 0, et(j = αh S I ε α( + ph N I ε + ε(a + αh S g N I ε

22 684 M. Y. LI, W. LIU, C. SHAN, AND Y. YI [ ] = ε( + p + p h S(S 0, N 0 + h N (S 0, N 0 g(n 0 + O(ε 2 < 0, ( tr(ja 2 et(a = ε ( + pg N ah Sg α h Ng h S g N I ε = ε( + p 0 + O(ε 2, where a 2 is the coefficient of λ in P ε (λ, namely, the sum of all 2 2 principal minors of J. When ε = 0, P 0 (λ = λ 3 + ( + pλ 2. It has a negative root, ( + p. Therefore, when ε > 0 small, P ε (λ has a negative root. We show that the remaining roots of P ε (λ are always complex conjugates. To see this, write P ε (λ as P ε (λ = λ 3 a 1 λ 2 + a 2 λ a 3, where a 1 = tr(a < 0, a 3 = et(a < 0, an a 2 is as above. The larger of the two critical points of P ε (λ is λ 1 = 1 ( a 1 + a a 2. Straightforwar calculation leas to P ε (λ 1 = 1 [ ] 2a a 1 a 2 27a 3 2(a 2 1 3a 2 a a 2. It can be verifie that P ε (λ 1 > 0 if an only if (21 27a a 3 1a 3 + 4a a 1 a 2 a 3 a 2 1a 2 2 > 0. When ε = 0, we have λ 1 = 0, which is a ouble root of P 0 (λ. Therefore, P ε (λ 1 = 0 when ε = 0, an thus the sign of the expression in (21 is etermine by the ε orer terms, which is given by [ ] 4( + p 4 g(n 0 + p h S(S 0, N 0 + h N (S 0, N 0 ε > 0, by assumption (A3 an continuity. Hence P ε (λ 1 > 0. This implies that P ε (λ has only one real root. The signs of the real parts of the complex roots can be etermine by the Routh Hurwitz conitions, which state that all roots of P ε (λ have negative real parts if an only if the following three conitions hol: a 1 = tr(a < 0, a 3 = et(a < 0, an a 1 a 2 a 3 < 0. From relations in (20, we see that for ε > 0 small, if 0 > 0, then all three eigenvalues have negative real parts, an if 0 < 0, then at least one eigenvalue has positive real parts. This establishes (ii. Proof of Theorem 6. To show (i, we note that for solution (S(t, I(t, N(t with initial conition (S, 0, N D, I(t 0, N(t N, an S(t N/(+p as t. Thus, (S(t, I(t, N(t (N/( + p, 0, N as t. Now let (S(t, I(t, N(t be the solution with the initial conition (S(0, I(0, N(0 D with I(0 > 0. From system (5, we have I(t > 0 for all t 0 an hence N(t monotonically ecreases. Therefore N(t N as t for some N epenent on the initial conition. We claim that N N 0. First of all, note that the equilibrium (S 0, 0, N 0 has two zero eigenvalues an one negative eigenvalue ( + p. Locally, there is a two-imensional center manifol