Stability of epidemic model with time delays influenced by stochastic perturbations 1

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1 Mathematics and Computers in Simulation 45 (1998) 269±277 Stability of epidemic model with time delays influenced by stochastic perturbations 1 Edoardo Beretta a,*, Vladimir Kolmanovskii b, Leonid Shaikhet c a Istituto di Biomatematica, Universita di Urbino, I-6129 Urbino, Italy b Moscow Institute of Electronics and Mathematics, Department of Cybernetics, Bolshoy Vusovskii, 3/12, Moscow 1928, Russia c Donetsk State Academy of Management, Department of Mathematics, Informatics and Computering, Chelyuskintsev, 163a, Donetsk 3415, Ukraine Abstract Many processes in automatic regulation, physics, mechanics, biology, economy, ecology etc. can be modelled by hereditary equations (see, e.g. [1±6]). One of the main problems for the theory of stochastic hereditary equations and their applications is connected with stability. Many stability results were obtained by the construction of appropriate Lyapunov functionals. In [7± 11], the procedure is proposed, allowing, in some sense, to formalize the algorithm of the corresponding Lyapunov functionals construction for stochastic functional differential equations, for stochastic difference equations. In this paper, stability conditions are obtained by using this procedure for the mathematical model of the spread of infections diseases with delays influenced by stochastic perturbations. # 1998 IMACS/Elsevier Science B.V. Keywords: SIR-model; Stochastic perturbations; Stability conditions; Lyapunov functionals; Construction method 1. Problem statement Consider the mathematical model of the spread of infections diseases. Let S(t) be the number of members of a population susceptible to the disease, I(t) be the number of infective members and R(t)be the number of members who have been removed from the possibility of infection through full immunity. Then the epidemic model can be described by the system [4,13] _S t ˆ S t f s I t s ds 1 S t b ÐÐÐÐ * Corresponding author. 1 This paper was written during a visit of V. Kolmanovskii and L. Shaikhet in Italy (Napoli, Urbino) /98/$19. # 1998 IMACS/Elsevier Science B.V. All rights reserved PII S ( 9 7 ) 1 6-7

2 27 E. Beretta et al. / Mathematics and Computers in Simulation 45 (1998) 269±277 _I t ˆ S t f s I t s ds 2 I t (1) _R t ˆ I t 3 R t It is assumed that, b,, 1, 2, 3 are positive constants, 1ˆmin( 1, 2, 3 ), f(s) is nonnegative function, such that f s ds ˆ 1; sf s ds < 1 It is easy to see that positive point of equilibrium for system (1) is given by E ˆ(S *, I *, R * ), where S ˆ 2 ; I ˆ b 1S S ; R ˆ I (2) 3 provided that S ˆ 2 < b (3) 1 Remark that models of type (1) were considered in numerous papers [4,12,13]. A particular case of this model with fixed delay was proposed first in [12]. The stability properties of the system (1) by 1ˆ 2ˆ 3ˆbˆ were considered in [13]. Here we assume that stochastic perturbations are of white noise type, which are directly proportional to distances S(t), I(t), R(t) from values of S *, I *, R *, influence on the _S t ; _I t ; _R t, respectively. By this way, the system (1) will be reduced to the form _S t ˆ S t _I t ˆ S t f s I t s ds 1 S t b 1 S t S _w 1 t f s I t s ds 2 I t 2 I t I _w 2 t (4) _R t ˆ I t 3 R t 3 R t R _w 3 t Here 1, 2, 3 are constants, w 1 (t), w 2 (t), w 3 (t) are independent from each other standard Wiener processes [14]. Let us centre the system (4) on the positive equilibrium E by the change of variables u 1ˆS S *, u 2ˆI I *, u 3ˆR R *. By this way, we obtain _u 1 ˆ I 1 u 1 S f s u 2 t s ds u 1 f s u 2 t s ds 1 u 1 _w 1

3 E. Beretta et al. / Mathematics and Computers in Simulation 45 (1998) 269± _u 2 ˆ I u 1 S u 2 S f s u 2 t s ds u 1 f s u 2 t s ds 2 u 2 _w 2 ; (5) _u 3 ˆ u 2 3 u 3 3 u 3 _w 3 It is easy to see that the stability of the system (4) equilibrium is equivalent to the stability of zero solution of Eq. (5). Below we will obtain the sufficient conditions for stability in a probability sense of zero solution of system (5). Along with the system (5) we will consider the linear part of the system (5) _z 1 ˆ I 1 z 1 S _z 2 ˆ I z 1 S z 2 S f s z 2 t s ds 1 z 1 _w 1 f s z 2 t s ds 2 z 2 _w 2 (6) _z 3 ˆ z 2 3 z 3 3 z 3 _w 3 and the auxiliary system without delays _y 1 ˆ I 1 y 1 1 y 1 _w 1 _y 2 ˆ I y 1 S y 2 2 y 2 _w 2 (7) _y 3 ˆ y 2 3 y 3 3 y 3 _w 3 2. Definitions, auxiliary statements Consider the stochastic differential equation [14] dx t ˆ a t; x t dt b t; x t dw t ; x ˆ ' 2 H: (8) Let {,, P} be the probability space, {f t, t} be the family of -algebras, f t 2, H be the space of f - adapted functions '(s)2r n, s, k'k ˆ sup s j' s j; k'k 2 1 ˆ sup smj' s j 2, M is the mathematical expectation, x tˆx(t s), s, w(t) is m-dimensional f t -adapted Wiener process, n-dimensional vector a(t, ') and nm-dimensional matrix b(t, ') are defined by t, '2H, a(t, )ˆ, b(t, )ˆ. Generating operator L of Eq. (8) is defined [14] by formula M t;' V t ; y t V t; ' LV t; ' ˆ lim! Here a scalar functional V(t, ') is defined by t, '2H and x(s) is the solution of Eq. (8) by st with initial function x tˆ'2h. Let us describe one class of functionals V(t, ') for which the operator L can be calculated in final form. We reduce the arbitrary functional V(t, '), t, '2H, to the form V(t, ')ˆV(t, '(), '(s)), s<

4 272 E. Beretta et al. / Mathematics and Computers in Simulation 45 (1998) 269±277 and define the function V ' t; x ˆ V t; ' ˆ V t; x t ˆ V t; x; x t s ; s < ; ' ˆ x t ; x ˆ ' ˆ x t Let D be the class of functionals V(t, ') for which function V ' (t, x) has two continuous derivations with respect to x and one bounded derivative with respect to t for almost all t. For functionals from D the generating operator L of Eq. (8) is defined and is equal to LV t; x t ' t; a t; x ' t; x 2 Tr b t; x V ' t; 2 b t; x t Definition 1 The zero solution of Eq. (8) is called mean square stable if for any > there exists a > such that Mjx t j 2 < for any t provided that k'k 2 1 <. Definition 2 The zero solution of Eq. (8) is called asymptotically mean square stable if it is mean square stable and lim t!1 Mjx t j 2 ˆ. Definition 3 The zero solution of Eq. (8) is called stable in probability if for any 1 > and 2 > there exists > such that solution x(t)ˆx(t, ') of Eq. (8) satisfies Pfjx t; ' j > 1 g < 2 for any initial function '2H such that Pfj'j g ˆ 1. Here P{} is the probability of the event enclosed in braces. Theorem 4 Let there exists the functional V(t, ')2D such that c 1 Mjx t j 2 MV t; x t c 2 kx t k 2 1 ; MLV t; x t c 3 Mjx t j 2 c i >. Then the zero solution of Eq. (8) is asymptotically mean square stable. Theorem 5 Let there exists the functional V(t, ')2D such that c 1 jx t j 2 V t; x t c 2 kx t k 2 ; LV t; x t c i >, for any function '2H such that Pfj'j g ˆ 1, where > is sufficiently small. Then the zero solution of Eq. (8) is stable in probability. The proofs of these theorems can be found in [1,2]. By this way, the construction of stability conditions is reduced to construction of Lyapunov functionals. Below we will use the general method of Lyapunov functionals construction [7±11] allowing to obtain stability conditions immediately in terms of parameters of system under consideration. 3. Formal description of the Lyapunov functional: construction procedure Let us consider the stochastic differential equation of neutral type d x t G t; x t ˆ a 1 t; x t dt a 2 t; x t d t ; t ; x t 2 R n (9)

5 E. Beretta et al. / Mathematics and Computers in Simulation 45 (1998) 269± Here x tˆx(t s), s, (t)2r m is a standard Wiener process, G(t,)ˆa i (t, )ˆ, iˆ1, 2, functionals a i are assumed satisfying usual conditions sufficient for the existence of the solution of Eq. (9) with initial conditions x ˆ'(s), where '2R n is a given function. The problem is to construct stability conditions of the trivial solution with respect to the disturbances of the initial condition. From Theorems 4 and 5, it follows that construction of the stability conditions can be reduced to the construction of special Lyapunov functionals V(t, x t ) satisfying the assumptions of these theorems. The proposed procedure of Lyapunov functionals construction consists of four steps. 1. Let us transform the Eq. (9) to the form dz t; x t ˆ b 1 t; x t c 1 t; x t dt b 2 t; x t c 2 t; x t d t (1) where z(t, x t ) is some functional on x t, z(t, )ˆ, functionals b i, iˆ1, 2, depend on t and x(t) only and do not depend on the previous values x(t s), s<, of the solution, b i (t, )ˆ. 2. Consider equation without memory dy t ˆ b 1 t; y t dt b 2 t; y t d t (11) Let us assume that the zero solution of Eq. (11) is uniformly asymptotically mean square stable and therefore there exists Lyapunov function v(t, y), for which the condition L v t; y jyj 2 hold. Here L is generating operator of Eq. (11). 3. We'll construct the Lyapunov functional V(t, x t ) in the form VˆV 1 V 2. Let us replace argument y of the function v(t, y) on the functional z(t, x t ) from left-hand part of Eq. (1). As a result we obtain the main component V 1 (t, x t )ˆv(t, z(t, x t )) of the functional V(t, x t ). 4. Usually the functional V 1 almost satisfies the requirements of stability theorem for Eq. (1). In order that these conditions would be completely satisfied the auxiliary component V 2 can be easily chosen by standard way. Remark that the representation (1) is not unique. This fact allows using different representations (1) to construct different Lyapunov functionals and as a result to get different sufficient stability conditions. 4. Main results In this section, as a result we will obtain conditions of stability in probability for the zero solution of nonlinear system (5). Preliminary to this, let us prove some auxiliary statements. Lemma 6 Let be 2 1 < 2 1; 2 2 < 2 2 ; 2 3 < 2 3 (12) Then the zero solution of the system (7) is asymptotic mean square stable. Proof Let us show that the function v ˆ py 2 1 y2 2 p2 y 2 3 q y 1 y 2 2 (13)

6 274 E. Beretta et al./ Mathematics and Computers in Simulation 45 (1998) is Lyapunov function for the system (7) for some q > and p >. Let L be the generating operator [14] of the system (7). Then L v = 2(py 1 + q(y 1 + y 2 ))(βi + µ 1 )y 1 + 2β(y 2 + q(y 1 + y 2 ))(I y 1 S y 2 ) +2p 2 y 3 (λy 2 µ 3 y 3 ) + (p + q)σ 2 1y (1 + q)σ 2 2y p 2 σ 2 3y 2 3 2p(βI + µ 1 )y 2 1 2q(y 1 + y 2 )(µ 1 y 1 + βs y 2 ) + 2βI y 1 y 2 2βS y 2 2 +p 2 λ(y2/p 2 + py3) 2 2p 2 µ 3 y3 2 + (p + q)σ1y (1 + q)σ2y p 2 σ3y y1(p 2 + q)(σ1 2 2µ 1 ) + y2((1 2 + q)(σ2 2 2βS ) + pλ) +p 2 y 2 3(pλ 2µ 3 + σ 2 3) + 2y 1 y 2 (βi q(µ 1 + βs )) Let be By this way we obtain q = βi µ 1 + βs (14) L v y1(p 2 + q)(2µ 1 σ1) 2 y2((1 2 + q)(2βs σ2) 2 pλ) p 2 y3(2µ 2 3 σ3 2 pλ) Let us choose p such that p < 1 λ min [ ] (1 + q)(2βs σ2), 2 2µ 3 σ3 2 From Eq.(12) it follows that there exists c > such that L v c y 2, where y = (y 1, y 2, y 3 ). It means (Theorem 4) that the zero solution of the system (7) is asymptotic mean square stable. Lemma is proved. Theorem 7 Let be σ1 2 < 2µ 1, σ2 2 < 2q(µ 2 + λ), σ3 2 < 2µ 3 (15) 1 + q where q is described by Eq.(14). Then the zero solution of the system (6) is asymptotic mean square stable. Proof Following of the formal procedure of Lyapunov functionals construction we will construct Lyapunov functional V for the system (6) in the form V = V 1 + V 2, where V 1 is Lyapunov function (13) for the auxiliary system (7) without delays: V 1 = pz z p 2 z q(z 1 + z 2 ) 2 (16)

7 E. Beretta et al. / Mathematics and Computers in Simulation 45 (1998) 269± Let L 1 be the generating operator [14] of the system (6). Then L 1 V 1 ˆ 2 pz 1 q z 1 z 2 I 1 z 1 S 2 z 2 q z 1 z 2 I z 1 S z 2 S f s z 2 t s ds f s z 2 t s ds 2p 2 z 3 z 2 3 z 3 p q 2 1 z2 1 1 q 2 2 z2 2 p2 2 3 z2 3 2q 1 2 z 1 z 2 z 2 1 p q pI Š z q 2 2 2S pš p 2 z p 2z 1 z 2 I q S 2S z 2 pz 1 f s z 2 t s ds Using Eq. (14) we obtain L 1 V 1 z 2 1 q z q 2 2 2S pš p 2 z p ps z 2 1 f s z 2 2 t s ds Š S z 2 2 f s z 2 2 t s ds Š ˆ z 2 1 q ps Š z q 2 2 2S p S Š p 2 z p S 1 p Following [7±11] let us choose V 2 in the form V 2 ˆ S 1 p Z t f s Using Eq. (15), for functional VˆV 1 V 2 we obtain t s f s z 2 2 t s ds z 2 2 d ds (17) L 1 V z 2 1 q ps Š z 2 2 2qS 1 q 2 2 p S Š p 2 z p (18) From Eq. (15) it follows that there exists p> such that p < min q S ; 2qS 1 q 2 2 S ; Therefore, there exists c> such that L 1 V cjzj 2, where zˆ(z 1, z 2, z 3 ). It means (Theorem 4) that the zero solution of the system (6) is asymptotic mean square stable. Theorem is proved.

8 276 E. Beretta et al. / Mathematics and Computers in Simulation 45 (1998) 269±277 Remark 8 It is known [15] that if the initial nonlinear system has a nonlinearity order more than one, then the conditions providing the asymptotic mean square stability of linear part of the initial system, in the same time provide the stability in probability of the initial system. Let us show it for the system (5). Theorem 9 Let the conditions of Theorem 7 hold. Then the zero solution of the system (5) is stable in probability. Proof Let L be the generating operator of the system (5). Consider the functional VˆV 1 V 2, where V 1 and V 2 are defined by Eq. (16) and Eq. (17), i.e. V ˆ pu 2 1 u2 2 p2 u 2 3 q u 1 u 2 2 S 1 p Z t f s t s u 2 2 d ds Then analogously to Eq. (18) we obtain 2 3 LV ˆ 2 pu 1 q u 1 u 2 4 I 1 u 1 S f s u 2 t s ds u 1 f s u 2 t s ds u 2 q u 1 u 2 4I u 1 S u 2 S f s u 2 t s ds u 1 f s u 2 t s ds5 2p 2 u 3 u 2 3 u 3 1 p S u p S 1 q 2 2 u2 2 p2 2 3 u2 3 2q 1 2 u 1 u 2 u 2 1 q ps Š u 2 2 2qS 1 q 2 2 p S Š p 2 u p 2u 1 u 2 pu 1 f s u 2 t s ds Let us suppose that Pfju 2 s j < g ˆ 1. Then Therefore, 2ju 1 u 2 pu 1 f s u 2 t s dsj u p u2 2 f s u 2 2 t s ds p q 2 1 u2 1 LV u 2 1 q ps 1 2p Š u 2 2 2qS 1 q 2 2 p S Š p 2 u p Hence, for sufficiently small > we obtain LV. It means (Theorem 5) that the zero solution of the system (5) is stable in probability. Theorem is proved.

9 E. Beretta et al. / Mathematics and Computers in Simulation 45 (1998) 269± References [1] V.B. Kolmanovskii, V.R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, [2] V.B. Kolmanovskii, A.D. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Boston, [3] V.B. Kolmanovskii, L.E. Shaikhet, Control of Systems with Aftereffect, Translations of mathematical monographs, 157, American Mathematical Society, Providence, RI, [4] E. Beretta, V. Capasso, F. Rinaldi, Global stability results for a generalized Lotka±Volterra system with distributed delays, J. Math. Biol. 26 (1988) 661±688. [5] E. Beretta, Y. Takeuchi, Qualitative properties of chemostat equations with time delays: Boundedness local and global asymptotic stability, Diff. Eqs. and Dynamical Systems 2 (1994) 19±4. [6] E. Beretta, A. Fasano, Yu. Hosono, V.B. Kolmanovskii, Stability Analysis of the Phytoplankton Vertical Steady States in a Laboratory Test tube, Math. Methods in the Applied Sciences 17 (1994) 551±575. [7] V.B. Kolmanovskii, L.E. Shaikhet, Stability of stochastic hereditary systems, Avtomatika i telemekhanika (in Russian) 7 (1993) 66±85. [8] V.B. Kolmanovskii, On stability of some hereditary systems, Avtomatika i telemekhanika (in Russian) 11 (1993) 45±59. [9] V.B. Kolmanovskii, L.E. Shaikhet, On one method of Lyapunov functional construction for stochastic hereditary systems, Differentsialniye uravneniya (in Russian) 11 (1993) 199±192. [1] V.B. Kolmanovskii, L.E. Shaikhet, New results in stability theory for stochastic functional differential equations (SFDEs) and their applications, Dynamical Systems and Applications, Dynamic Publishers Inc., N.Y. 1 (1994) 167±171. [11] V.B. Kolmanovskii, L.E. Shaikhet, General method of Lyapunov functionals construction for stability investigations of stochastic difference equations, Dynamical Systems and Applications, World Scientific Series in Applicable Analysis, Singapore 4 (1995) 397±439. [12] K.L. Cooke, Stability analysis for a vector disease model, Rocky Mount. J. Math. 7 (1979) 253±263. [13] E. Beretta, Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol. 33 (1995) 25±26. [14] I.I. Gihman, A.V. Skorokhod, The Theory of Stochastic Processes, I,II,III, Springer, Berlin, 1974, 1975, [15] L.E. Shaikhet, Stability in probability of nonlinear stochastic hereditary systems, Dynamic Systems and Applications 4(2) (1995) 199±24.

STABILITY OF A POSITIVE POINT OF EQUILIBRIUM OF ONE NONLINEAR SYSTEM WITH AFTEREFFECT AND STOCHASTIC PERTURBATIONS

Dynamic Systems and Applications 17 28 235-253 STABILITY OF A POSITIVE POINT OF EQUILIBRIUM OF ONE NONLINEAR SYSTEM WITH AFTEREFFECT AND STOCHASTIC PERTURBATIONS LEONID SHAIKHET Department of Higher Mathematics,