Stability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay

Transcription

1 Stability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay Ekkachai Kunnawuttipreechachan Abstract This paper deals with stability properties of the discrete numerical scheme for the SIS epidemic model with maturation delay We provide the sufficient conditions of the numerical step-size for the numerical solutions to be asymptotically stable These will be useful for choosing a suitable numerical stepsize when we simulate problems with the provided numerical scheme Keywords: SIS epidemic model, delay differential equations, discrete numerical scheme, asymptotic stability Introduction In this paper we aim to analyse a numerical scheme for a class of the SIS epidemic models In a general SIS model, the population size is divided into two groups: It the infected population size, and St the susceptible population size Here we continue further the results from Cooke et al [], who have developed the SIS epidemic model with maturation delay: I t = µ It It δ + ε + γit, S t = B Nt τ Nt τe δ τ δst µstit + γit, N t = B Nt τ Nt τe δ τ δ εit Here BN is a birth rate function, ε is the disease induced death constant rate, γ is the recovery constant rate ie /γ is the average infective time, µ > is the contact constant rate, δ is the death rate constant, and δ is the transfer rate constant for the life stage prior to the adult stage The standard incidence function µi/n refers to the average number of adequate contacts with infectives of one susceptible per unit time The delay τ is considered as the development or the maturation time In addition, model is based as follows on the assumptions by Cooke et al []: the transmission of the disease occurs due to a contact between the susceptible class St and the in- Department of Mathematics, King Mongkut s University of Technology North Bangkok KMUTNB, 58 Pibulsongkram Road, Bangkok, THAILAND 8; Tel: /Fax: ekc@kmutnbacth, or ekcmath@gmailcom fective class It; there is no vertical transmission; the disease presents no immunity against reinfection, ie, on recovery, an infective individual returns to the susceptible class Moreover, in Cooke et al [], they also identified the basic reproduction number = µ δ + ε + γ, which generates the average number of new infective individuals produced by one infective during the mean death adjusted infective Many authors have studied dynamics of the system by considering as a parameter [,, 3] Cooke et al [] showed the existence of the equilibria of They proved that if <, there exists a unique nontrivial equilibrium Ī, S, =,,, called diseasefree equilibrium, which is globally asymptotic stable see Cooke et al [], Theorems 4 and Theorem 43 When >, there also exists another nontrivial equilibrium Ī, S, = Ī+, S +, + or, so called, endemic equilibrium Analysis of the system in this case becomes harder For the non-delay problem τ =, Cooke et al [] showed that the reproduction number acts as a sharp threshold Their results show that, for nonnegative solutions, when <, the disease dies out, ie It Ī = as t On the contrary, if > and I >, then the disease remains endemic with It Ī+, St S + and + as t In addition, for the positive delay τ > with ε =, Cooke et al [] obtained the globally asymptotic stability for the nontrivial equilibrium in this case They also indicated that if > and τ is sufficiently large, the positive solutions of oscillate about the positive equilibrium, ie they are periodic solutions Later, Zhao and Zou [3] provided a proof where ε is very small using the perturbation technique They obtained sufficient conditions for the stability of Moreover, Wei and Zou [] determined the bifurcation analysis of with p as the bifurcation parameter when ε = However, the dynamics of the SIS model still largely remain undetermined, especially for ε > [3]

2 Our approach in this paper is to study a special class of Assume that the death rate in each stage prior to the adult stage is negligible compared with the death rate of the adult stage, then we can assume δ = Using the birth rate function B = pe a Hence becomes I t = µ It It δ + ε + γit, S t = pnt τe ant τ δst µstit + γit, N t = pnt τe ant τ δ εit Like with the model, the dynamical behaviour of the SIS model 3 is still largely undetermined because there are many open problems which are still unsolved at present [] In this paper we provide alternatively a numerical scheme for 3 Our aim is to analyse further the dynamical properties of a numerical scheme for 3 The main objective in this study is to determine the asymptotic stability of the equilibria of the given numerical scheme and illustrate numerical solutions of 3 We obtain the sufficient conditions for the numerical stepsize causing the numerical solutions are asymptotically stable 3 This paper is organised as follows In the next section we analyse the stability of 3 For any nonnegative solutions, our result in Theorem shows conditions for the existence of the equilibria, which are called disease-free equilibrium and endemic equilibrium Most results for the continuous system have already been studied, but we restate some related results with our own proof in Theorem 3 and Theorem 4 for the stability of 3 Later, in Section 3, we describe our main problems for this paper Here we develop the numerical scheme for 3, and then investigate the stability of the numerical solutions Our main contributions are presented in Theorem 6 and Theorem 7 for the stability of the disease-free equilibrium and the endemic equilibrium, respectively Finally, the numerical results are also illustrated in Section 4 Dynamics of the Equilibria Since the total number of population is the sum of the infective class It and the susceptible class St, we have = St + It It is adequate to reduce the dimension of the system to I t = µ It It δ + ε + γit, 4 N t = pe ant τ Nt τ δ εit 5 Note that ε =, then 5 becomes the Nicholson s blowflies equation: N t = pe ant τ Nt τ δ, 6 which proposed by Gurney et al [4] The dynamics of 6 have been studied by many authors see for examples [5, 6, 7, 8, 9] It is not difficult to show that 6 has the trivial equilibrium = and the positive equilibrium + = a lnp/δ Here we state the theorem for stability properties of 6 resulted by [9] Theorem Stability of the blowflies equation; [9] Let and + be the trivial equilibrium and positive equilibrium of 6, respectively If p < δ, then the trivial equilibrium is locally asymptotically stable If the ratio p/δ satisfies < p δ < e, 7 then the positive equilibrium + of 6 is locally asymptotically stable Proof See [4, 9] for more details on the proof The equilibria To analyse the dynamical behaviour of 4-5, we investigate in a first stage all equilibria Suppose that It Ī and Let I t = N t =, then the system 4-5 becomes µ Ī Ī δ + ε + γī =, 8 pe a δ ε Ī = 9 Here, we only consider the equilibria in the positive quadrant, ie > and Ī Note that the case that N = is not our interested We state the following theorem to show conditions for the existence of the equilibria with our proof Theorem The equilibria of 4-5 Suppose that ε, γ, a and µ > If and p > δ, there exists only a semi-trivial equilibrium Ī,, namely Ī = and = a ln p δ This equilibrium is called the disease-free equilibrium On the other hand, if > and p > δ + ε /, there exists also another positive equilibrium Ī+, +, namely Ī + = R called the endemic equilibrium + and + = a ln p δ + ε /, Proof Rearranging 8, we have Ī µ µī δ + ε + γ =

3 For >, then the solutions of Ī are divided into two cases: Ī =, or Ī = δ + ε + γ µ where is defined by Case I: For Ī =, then 9 becomes pe a δ = = R, Since >, there exists only the nonnegative equilibrium Ī, = Ī,, where Ī = and = a ln p δ Clearly, Ī, exists for all values of whenever p > δ Hence Ī, is the equilibrium of 4-5 satisfying 8-9 for both and > Case II: For Ī = /, it can be seen that there exists also a positive equilibrium if and only if > Then 9 becomes pe a δ ε = Since >, then pe a δ ε R = It yields that = a ln p δ + ε Clearly, when > it is easy to verify that there exists another equilibrium, Ī, = Ī+, +, defined by Ī + = R + and + = a ln p δ + ε / It can be seen that if p > δ+ε / and >, then Ī + and + are positive As the results, it concludes that when >, there exist two positive equilibria, which are the disease-free equilibrium Ī,, and the endemic equilibrium Ī+, + On the other hand, if, the disease-free equilibrium Ī, is only a nonnegative equilibrium of 4-5 Note that the disease-free equilibrium Ī, =, a lnp/δ is sometimes called the semi-trivial equilibrium It means that there will be no infective population in the system as time tends to infinity, and the number of population will reach a constant value Throughout this paper we use the words disease-free equilibrium to represent Ī, and endemic equilibrium to represent Ī+, + Stability analysis To analyse the stability properties of 4-5, we use as a parameter We also investigate how the reproduction number and the delay τ affect the behaviour of the system Our study is divided into two cases: and > First, we consider Cooke et al [] gave the result in this case for the general model For 4-5, the following theorem shows the sufficient conditions for the stability of the equilibria Theorem 3 Stability of the equilibria when Let < Consider the SIS model 4-5 with the positive initial values where It > on [ τ, ] Then It tends to zero as t tends to infinity If, in addition, the condition < p/δ < e holds, then = a lnp/δ is asymptotically stable Proof Let It = Ī +yt and = +xt Using the linearisation method The SIS system 4-5 becomes the linearised system y t = µ δ + ε + γ µ Ī yt + µ Ī xt, x t = εyt δxt + pe a a xt τ, where Ī, is the equilibrium of 4-5 For the disease-free equilibrium Ī, =, a lnp/δ, the linearised equations are y t = µ δ + ε + γ yt = µ R yt and x t = εyt δxt + pe a a xt τ 3 It can be seen that yt is asymptotically stable provided that µ R < Since µ >, it implies that < Hence, yt as t Note that if =, then becomes y t = and it is asymptotically stable Moreover, for a sufficiently large of time t, we can ignore the term of yt and 3 becomes x t = δxt + pe a a xt τ 4 From Theorem, when p > δ, the equilibrium Ī, is locally asymptotically stable provided that < p δ < e 5 As the result, if <, then It I = as t In addition, is locally asymptotically stable when the condition 5 holds

4 If there is no disease-related death ε =, then + = = a lnp/δ The system 4-5 becomes I t = µ It It δ + γit, 6 N t = pe ant τ Nt τ δ 7 Note that, in this case, in 7 satisfies the Nicholson s blowflies equation and its properties is provided in Theorem Moreover, since in 7 is independent from It, the system 6-7 is called a decoupled or non-interactive system The following theorem presents the dynamics of the equilibria in the decoupled system Theorem 4 Stability of the equilibria when > Let = µ δ+γ > Consider the decoupled SIS system 6-7 with the positive initial values It > on [ τ, ] If, in addition, the condition < p/δ < e holds, then + = = a lnp/δ and Ī+ are asymptotically stable, while Ī is unstable Proof Since 7 is the Nicholson s blowflies equation, we can use the results from Theorem In this case we know that if p > δ and < p/δ < e, then = a lnp/δ is asymptotically stable for all τ If as t, then the long term behaviour of It is governed by I t = µ Ī It δ + γit 8 Let It = Ī + yt The linearised equation of 8 is y t = µ δ + γ µ Ī yt = µ R µ Ī First, if Ī = Ī =, 9 becomes y t = µ R yt, yt 9 which is unstable if > Hence, Ī is unstable when > Next, when Ī = Ī+ = / +, then 9 yields y t = µ R yt, which is asymptotically stable provided that > Hence, this shows that Ī+ is asymptotically stable if and only if > Note that the dynamical analysis when ε has been studied by Zhao and Zou [3], but they can only provide a case when ε is sufficiently small As in our knowledge, dynamics of the SIS model 4-5 for general values of ε is still undermined In the next section we will show dynamics of the discretisation numerical scheme and compare them with the results for the continuous case 3 Numerical Scheme and Analysis This section contains our main contributions to the analysis of numerical solutions of the SIS model 4-5 Firstly, we introduce a numerical scheme for the problem, and then analyse its stability near the equilibria Our main target is to find sufficient conditions for the equilibria of the numerical scheme for 4-5 to be asymptotically stable or unstable 3 The numerical scheme From the forward difference formula: y t y n+ y n, h where n =,,,, y n = yt n, t n = t +nh and h is the numerical step-size of the derivative approximation Let I n = It n and N n = Nt n We construct the numerical scheme for the SIS model 4-5 as follows: I n+ I n h N n+ N n h = µn n I n I n N n δ + ε + γi n, = pn n k e an n k δn n εi n Here, we use the equal step-size h = τ/k, where k is a positive integer After rearranging the system above, we get the explicit scheme: I n+ = + hµ hδ + ε + γ I n hµ I n, N n N n+ = εhi n + hδn n +hpn n k e an n k The initial conditions become I = It and N i = ϕ i, where ϕ i = ϕt i for i = k, k +,, The equilibria of - can be calculated by putting I = Ī and N = So we have Ī = + hµ hδ + ε + γ Ī hµī, 3 = εhī + hδ + hp e a 4 Solving the nonlinear system 3-4, the equilibria are the same as in Theorem, ie if, there exists only the disease-free equilibrium Ī, : Ī = ; = a ln p δ 5 On the other hand, if >, there exist both the diseasefree equilibrium Ī, and also the endemic equilibrium Ī+, + which defined by Ī + = R + ; + = a ln p δ + ε 6

5 Note that the equilibria of the numerical scheme - are the same as in the continuous problem 4-5 We will next investigate sufficient conditions for the asymptotic stability of the equilibria Our analysis is focused on both the disease-free equilibrium and the endemic equilibrium 3 Dynamical analysis of the equilibria for the numerical scheme First, we use the linearisation method on the system 3-4 about the equilibrium Ī, Let It = Ī +ut and = + vt, the linearised system becomes u n+ = + hµ hµ Ī u n + hµ Ī v n, 7 v n+ = εhu n + hδv n + hpe a a v n k Suppose that x n = v n, and let x n = v n, x n = x n = v n, x k n = x k n = xk n = = x n k = v n k, then 8 becomes x n+ = εhu n + hδx n + hpe a a x k n 8 The system 7-8 can be rewritten in a system of k+ equations; u n+ = + hµ hµ Ī u n + hµ Ī x n, x n+ = εhu n + hδx n +hpe a a x k 9 n, x n+ = x n, x k n+ = x k n, for n =,, 3, ; and x n = u n, x n, x n,, x k n, x k n T The system 9 can be written in matrix form x n+ = A x x n, 3 where x is the equilibrium of the system, ie x = Ī,,,, T Here, A is the constant matrix defined by A x = where and A, hµ Ī εh hδ A,k+ A, = + hµ R hµ Ī A,k+ = hpe a a, 3 The matrix A has dimension k+ k+ The system 3 is a linear system, so the stability of the system can be determined from its eigenvalues To find the eigenvalues λ of A, we need to solve A λi =, where I is the identity matrix It is not difficult to simplify that the characteristic polynomial of A is Φλ = A, λ hδ λλ k + A,k+ +εh µ Ī λk = 3 For the system 9 to be stable at its equilibrium Ī,, we need all roots eigenvalues of 3 at the equilibrium to lie in the unit disk, ie λ < The following lemma gives the sufficient conditions so that all roots of a polynomial equation lie inside the open unit disk Lemma 5 [] Consider the polynomial of degree k λ k + p λ k + p λ k + + p k λ + p k = 33 If k i= p i, then all roots of 33 lie inside the open unit disk, ie λ < Proof See [] and references therein for more details Next, using advantages of Lemma 5, we show the theorem for sufficient conditions for the disease-free equilibrium, Ī,, to be asymptotically stable Theorem 6 Consider the numerical scheme -, with the positive initial condition in Suppose that < p/δ < e If, in addition, the condition { } h < min δ, δ + ε + γ µ 34 holds and <, then Ī and are asymptotically stable

6 Proof For Ī = and = a lnp/δ, the characteristic equation 3 becomes A, λ hδ λλ k + A,k+ =, 35 where A, = + hµ and A,k+ = hδ ln p δ Solving 35, we have two factors: λ = A, = + hµ R, 36 and hδ λλ k + hδ ln p = 37 δ For the stability of the numerical scheme, we need to show that all solutions of lie in the unit disk, ie λ < Case I: Consider 36 The solution is asymptotically stable when λ = + hµ < Solving the inequality above, it yields that < hµ R < Because <, so / <, we have the condition Hence, for all h > < hµ < h < = µ δ + ε + γ µ δ + ε + γ µ, 38 Case II: Consider 37 It can be rearranged into λ k+ hδλ k hδ ln p = 39 δ From Lemma 5, the system is asymptotically stable if hδ + hδ ln p δ < It provides that h < δ and < p δ < e 4 Combining the conditions 38 and 4, we conclude that if < p/δ < e, with <, and the condition 34 holds, then the disease-free equilibrium Ī, is asymptotically stable Note that, for the disease-free equilibrium, the sufficient conditions are depended only on the ratio p/δ and the numerical step-size h The results are similar as in Theorem 3 Next, we will consider the case of no disease related death ε =, and analyse dynamics of the selected SIS system at the endemic equilibrium, Ī+, + The following theorem shows the conditions for which Ī+ and + are asymptotically stable Theorem 7 Consider the numerical method - with the positive initial condition in Let ε =, then = λ/δ + γ Assume that < p/δ < e If > and the condition { } h < min δ, 4 µ δ + γ holds, then the endemic equilibrium Ī+, + of -, where ε =, is asymptotically stable Proof Let ε =, with + = = a lnp/δ, and Ī+ = / + or Ī+/ + = / Then 3 becomes hµ λ hδ λλ k + hδ ln p δ = 4 Solving for λ in 4, we have λ = hµ R, 43 or hδ λλ k + hδ ln p = 44 δ For the asymptotic stability of the equilibria of the numerical scheme, we need to show that all solutions of lie inside the unit disk, ie λ < Case I: in 43, the solution is asymptotically stable provided that hµ < Hence < hµ R < Since >, we have / > It yields or for all h > < hµ < µ = µ δ + γ, h < µ δ + γ, 45 Case II: consider 44, it can be rearranged to λ k+ hδλ k hδ ln p = 46 δ Like with 37, using Lemma 5, all roots of 46 lie inside the unit disk λ < provided that h < δ and < p δ < e 47 Combining the conditions 45 and 47, we conclude that if 4 holds for < p/δ < e and >, then Ī+ and + = are asymptotically stable

7 Compare the results for the numerical analysis in Theorem 7 with the analytical analysis in Theorem 4 We see that the conditions for stability are the same in general There is only one exception The conditions for the stability of the numerical solutions need an additional condition on the step-size h 4 Numerical Results The numerical experiments given in this part are divided into two cases: < and > This is to support our main contributions in Theorems 6 and Theorem 7 4 Case < the delay affects the dynamical behaviour of the numerical solutions We can see that the solution for It is attracted by Ī =, while the solution for is changed its stability from asymptotic stability Figure left to periodic behaviour Figure right 4 Case > For the case that >, Figure 3 shows the numerical experiments of the SIS epidemic model with p = e, δ = p/δ < e, a =, µ = 5, γ =, ε = when τ = and τ = 3 The results indicate that if >, I and N are attracted by the endemic equilibrium, ie Ī, Ī+, + Figure shows the numerical solutions of the SIS model with p = e, δ = p/δ < e, a =, µ =, γ = ε = ; hence < In this case Ī = and 953 In Figure, both I dot and N solid are attracted by the disease-free equilibrium, ie Ī, Ī, =, time t time t 5 Ni It Figure 3: The numerical solutions of the SIS model with p/δ < e and > ; where τ = left and τ = 3 right time t In addition, Figure 4 shows that when p/δ > e Here we set p = 5, δ = p/δ > e, a =, µ = 5, γ =, ε = The numerical solutions oscillate about the endemic equilibrium Figure 4left and Figure 4right represent the solutions when τ = 5 and τ = 65, respectively We can see that the behaviour of the solutions changes from a solution converging to the equilibrium to a periodic solution Hence, a Hopf bifurcation occurs when τ is sufficiently large and > Figure : The numerical solutions of the SIS model with p/δ < e and <, where τ = time t time t time t time t Figure 4: The numerical solutions of the SIS model with p/δ > e and > ; where τ = 5 left and τ = 65 right Figure : The numerical solutions of the SIS model with p/δ > e and < ; where τ = left and τ = 3 right Next, Figure compares the numerical solutions with different delay τ Here we set p = 5, δ = p/δ > e, a =, µ =, γ =, ε = We can see that Note that in Figure 4, we can see that the numerical solution undergoes a Hopf bifurcation when τ is sufficiently large Suppose that τ is the bifurcation point We can see that τ 5, 65 The study of Hopf bifurcation is an interesting topic, and it can be a further study for this problem

8 References [] K L Cooke, P V D Driessche, and X Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, Journal of Mathematical Biology, vol 39, no 4, pp 33 35, 999 [] J Wei and X Zou, Bifurcation analysis of a population model and the resulting SIS epidemic model with delay, Journal of Computational and Applied Mathematics, vol 97, no, pp 69 87, 6 [3] X Q Zhao and X Zou, Threshold dynamics in a delayed SIS epidemic model, Journal of Mathematical Analysis and Applications, vol 57, no, pp 8 9, [4] W S C Gurney, S P Blythe, and R M Nisbet, Nicholson s blowflies revisited, Nature, vol 87, pp 7, 98 [5] Q X Feng and J R Yan, Global attractivity and oscillation in a kind of Nicholson s blowflies, Journal of Biomathematics, vol 7, no, pp 6, [6] I Györi and S I Trofimchuk, On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation, Nonlinear Analysis: Theory, Methods and Applications, vol 48, no 7, pp 33 4, [7] E Kunnawuttipreechachan, The stability analysis of discrete numerical methods for the delay nicholson s blowflies equations and related problems, PhD Thesis, Brunel University, 9 [8] S H Saker and S Agarwal, Oscillation and global attractivity in a periodic Nicholson s blowflies model, Mathematical and Computer Modelling, vol 35, no 7-8, pp 79 73, [9] H L Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooporative Systems USA: American Mathematical Society, 995 [] V L Kocic and G Ladas, Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications Netherlands: Kluwer Academic Publishers, 993