Solution and α-path of uncertain SIS epidemic model with standard incidence and demography

Transcription

1 Journal of Inelligen & Fuzzy Sysems 35 (218) DOI:1.3233/JIFS IOS Press 927 Soluion and α-pah of uncerain SIS epidemic model wih sandard incidence and demography Jun Fang, Zhiming Li, Fan Yang and Mengyuan Zhou College of Mahemaics and Sysem Science, Xinjiang Universiy, Urumqi, Xinjiang, China Absrac. Li e al. [1] applied uncerain differenial equaion (UDE) o sudy a class of suscepible-infecious-suscepible (SIS) epidemic model wih he bilinear incidence rae and consan populaion size. In his paper, we sudy an uncerain SIS epidemic model wih sandard incidence and demography. By scale ransformaion, he deerminisic and uncerain proporion epidemic models are inroduced. Soluions of hese models and α-pahs of uncerain model are given. Furher, under hreshold condiions, we discuss exincion and permanence of he disease and reveal relaionship of he deerminisic and uncerain models. Some examples are provided o illusrae hese resuls. Keywords: Sandard incidence, SIS epidemic model, uncerain differenial equaion, Liu process, α-pah 1. Inroducion Mahemaical models play an imporan role in analysing he spread of infecious diseases and ge more and more aenions by many people, for example Hehcoe [2], Yang and Mao [3], Aralejo and Lopez-Herrero [4], Bai and Mu [5]. Le S and I be he numbers of suscepible and infeced individuals a ime, respecively. Hehcoe and Yorke [6] inroduced a classical suscepible-infecious-suscepible (SIS) epidemic model, described by he following ordinary differenial equaion (ODE) { ds = [μn βs I + γi μs ]d, (M1) di = [βs I (μ + γ)i ]d, where N is a consan and means he oal size of he populaion among whom he disease is spreading a ime, μ is he naural-deah rae of popula- Corresponding auhor. Zhiming Li, College of Mahemaics and Sysem Science, Xinjiang Universiy, Urumqi, Xinjiang, China. zmli@xju.edu.cn. ion per uni ime, γ is he rae a which infeced individuals become cured per uni ime, β is he disease ransmission coefficien per uni ime and βs I is of bilinear incidence. For model (M1), i has powerful qualiaive resuls such as globally asympoically sable under some condiions. However, mos of ODE epidemic models are ineviably affeced by environmen flucuaions. For beer undersanding of epidemic dynamics in realiy, many scholars inroduced whie noise ino deerminisic models and esablished sochasic versions wih he effec of noise. For example, Gray e al. [7] considered a sochasic differenial equaion (SDE) for SIS epidemic model as follows { ds = [μn βs I + γi μs ]d σs I db, (M2) di = [βs I (μ + γ)i ]d + σs I db. Here, B is a Wiener process wih B =, defined in probabiliy space (,{F },P) wih a filraion {F } saisfying he usual condiion ha conains all P-null ses. The inensiy of whie noise is σ (> ). Parameers N, μ, β and γ are defined as above. By he sochasic Lyapunov funcional mehod, hey obained asympoic behavior of he sochasic sysem around equilibria /18/$ IOS Press and he auhors. All righs reserved

2 928 J. Fang e al. / Uncerain SIS epidemic model wih sandard incidence and demography When no samples are available o esimae probabiliy properies of he model (M2), we have o use a means of handling uncerainy raher han o randomness. One of he mos imporan ools is fuzzy se heory, which he idea of fuzzy se was firs proposed by Zaheh in A fuzzy se is a class of objecs. Such a se is characerized by a se funcion (measure) which assigns o each objec ranging beween zero and one. In order o measure a fuzzy even, Liu [9] inroduced a se funcion of fuzzy se named uncerain measure based on normaliy, dualiy, subaddiiviy and produc axioms, and esablished uncerainy heory as a branch of mahemaics for sudying behavior of uncerain phenomena. For a non-empy fuzzy se Ɣ, L is a σ-algebra on he se Ɣ and uncerain measure denoed by M is a funcion from L o [,1], saisfying four axioms. The riple (Ɣ, L, M) is called an uncerainy space. Uncerain process C is a given one-dimensional Liu process defined on uncerainy space. For he model (M1), here exis some uncerain phenomena influencing he cured rae γ. Suppose he rae γ is described by Liu process C, ha is, γd in model (M1) is replaced by γd + σ dc.lieal. [1] firsly esablished an uncerain differenial equaion (UDE) of SIS epidemic model wih bilinear incidence, formed by { ds = [μn βs I + γi μs ]d + σ I dc, (M3) di = [βs I (μ + γ)i ]d σ I dc, where σ (> ) is he inensiy of Liu process C. Furher, hey obained he soluions, convergence properies of α-pahs of model (M3) and compared he resuls of ODE, SDE and UDE epidemic models, respecively. Comparing wih he resuls of ODE model, hose of UDE model refleced some pracical problems beer reasonably han hose of SDE case wihou sample daa. For model (M3), we have S + I = N (N N )e μ, where N = S + I. If +, hen S + I N. This means he model (M3) has consan populaion size. However, for he long ime he oal populaion size may vary and he disease can cause deahs during he spread of epidemic. In his work, we applied he uncerainy heory o sudy a class of uncerain SIS epidemic model wih sandard incidence and demography. The res is organized as follows. In Secion 2, some basic conceps and noaion of uncerainy heory are reviewed. Two modified SIS epidemic model are esablished in Secion 3. In Secion 4, α-pahs of he models are obained. Furher, by α-pahs, uncerainy disribuions and expeced values of uncerain SIS model are given. Some examples are provided o illusrae hese resuls. A brief conclusion is in Secion Preinaries This secion will briefly review some basic conceps and noaion of uncerainy heory, including uncerain measure, uncerain variable, uncerain process, uncerain differenial equaion and α-pah. More deails o see Liu [9 12], Yao and Chen [13], Yao [14]. A se funcion M : L [, 1] is called an uncerain measure if i is supposed o saisfy he following axioms: Axiom 1 (Normaliy Axiom) M{Ɣ} =1 for he universal se Ɣ. Axiom 2 (Dualiy Axiom) M{ }+M{ c }=1 for any even. Axiom 3 (Subaddiiviy Axiom) For every counable sequence of evens 1, 2,..., { } M i M{ i }. Axiom 4 (Produc Axiom) Le (Ɣ i, L i, M i )be uncerainy spaces and i be arbirarily chosen evens from L i for i = 1, 2,..., hen he produc uncerain measure M is an uncerain measure saisfying { } M i = M i { i }. Le T be an index se and (Ɣ, L, M) be an uncerainy space. An uncerain process X is a measurable funcion from T (Ɣ, L, M) o he se of real numbers, i.e., for each T and any Borel se B of real numbers, he se {X B} ={γ X (γ) B} is an even. The uncerain process X is said o have an uncerainy disribuion (x) if a each ime, uncerain variable X has uncerainy disribuion (x) and (x) = M{X x} for any real number x R. Definiion 1. An uncerain process C is said o be Liu process if (i) C = and almos all sample pahs are Lipschiz coninuous. (ii) C has saionary and independen incremens. (iii) Every incremen C +s C s is a normal uncerain variable wih an uncerainy disribuion

3 J. Fang e al. / Uncerain SIS epidemic model wih sandard incidence and demography 929 (x) = ( ( 1 + exp πx )) 1, x R. 3 Le X be an uncerain process and C be Liu process. For any pariion of a closed inerval [a, b] wih a = 1 < 2 < < k+1 = b, he mesh is wrien as = max i+1 i. 1 i k Then, Liu inegral of X is defined by b a X dc = k X i (C i+1 C i ) provided ha he i exiss almos surely and is finie. Definiion 2. Suppose C is Liu process, and f and g are wo given funcions. Then dx = f (, X )d + g(, X )dc is called an uncerain differenial equaion (UDE). An uncerain process X is called a soluion of he equaion, if X saisfies dx f (, X )d + g(, X )dc for any ime. For <α<1, α-pah of an uncerain differenial equaion dx = f (, X )d + g(, X )dc wih iniial value X is a deerminisic funcion X α wih respec o ha solves he ordinary differenial equaion (ODE) dx α = [f (, X α ) + g(, Xα ) 1 (α)]d, where 1 (α) is he inverse uncerainy disribuion of sandard normal uncerain variable, i.e., 3 1 (α) = π ln α, <α<1. (1) 1 α Le X be an uncerain process. If for each α (, 1), here exiss a real funcion X α such ha M{X X α, } =α, M{X >X α, } =1 α, hen X is called a conour process. X α is called α- pah of he process X. Expeced value of X is defined by + E[X ] = M{X x}dx M{X x}dx, provided ha a leas one of he wo inegrals is finie. If X has he regular uncerainy disribuion, hen E[X ] = 1 1 (α)dα. 3. Two SIS epidemic models and soluions Le N be he oal number of people a ime and N = S + I. The sudy of growh and change of populaions is called demography. Thus, for some epidemic diseases wihou permanen immuniy, i is appropriae o add he moraliy and populaion variable N o SIS model. A new SIS model is esablished as follows { ds = [μn βs I N + γi μs ]d, di = [ βs (2) I N (μ + γ)i θi ]d wih he iniial values S and I, where θ (θ <β)ishe moraliy rae of he disease per uni ime. μ, β and γ are defined in he model (M1). βs N I is he average number of infecion ransmissions by all infecious individuals per uni ime, called sandard incidence. The bilinear and sandard incidences agree when he oal populaion size is a consan, bu hey differ if he oal populaion size is variable. For he model (M1), N N as +. However, for model (2), we have dn = θi d. Thus, he modified SIS model (2) is an SIS ype wih sandard incidence rae and demography. For simpliciy, we nondimensionalize model (2) by defining X = S N, Y = I N, where X and Y are he proporions of suscepible and infeced individuals, respecively. Thus, he model (2) is equivalen o he ODE model { dx = [(μ + γ)(1 X ) (β θ)(1 X )X ]d, (3) dy = [(β θ)(1 Y )Y (μ + γ)y ]d wih he iniial values X = S /N,Y = I /N.For any ime, X + Y = 1. By (3), we have dy d = (β θ μ γ)y (β θ)y 2. Denoe a = β θ μ γ and b = θ β. Thus, he soluion of he model (3) is = a + by (a + b)y e a, X D Y D = a + by (1 e a ) ay e a a + by (1 e a ). For he ODE model (3), a basic reproducive number is defined as R D = μ + γ β θ. (4)

4 93 J. Fang e al. / Uncerain SIS epidemic model wih sandard incidence and demography Through he fundamenal heory of ordinary differenial equaion, i is easy o obain he following resuls. Theorem 1. (i) If R D 1, hen he disease freeequilibrium (X,Y ) = (1, ) of he model (3) is globally asympoically sable. (ii) If R D < 1, hen here exiss a unique endemic equilibrium (X,Y ) = ( μ + γ β θ, β θ μ γ ) β θ for model (3), which is globally asympoic sabiliy. Le (Ɣ, L, M) be an uncerainy space and C is Liu process defined on uncerainy space. Suppose he uncerain flucuaing environmen in he form of Liu process effecs he disease ransmission coefficien β, ha is, replace βd wih βd + σ dc, hen he model (3) becomes an uncerain differenial equaion (UDE) of SIS epidemic model in form of dx = [(μ + γ)(1 X ) (β θ)(1 X )X ]d σ (1 X )X dc, (5) dy = [(β θ)(1 Y )Y (μ + γ)y ]d +σ (1 Y )Y dc, where σ (> ) is an inegrable and differenial funcion for any, called he inensiy of environmenal disurbances. In order o provide soluions of he uncerain SIS model (5), we discuss he soluion of an uncerain differenial equaion and obain he following resul. Lemma 1. Le u i,v i be wo inegrable and differenial funcions abou ime for i = 1, 2 and v 1 /=. Then, he uncerain differenial equaion dy = (u 1 Y + u 2 Y 2 )d + (v 1Y + v 2 Y 2 )dc has a soluion Y = [Z 1 W v 2 v 1 ] 1, where and dw = Z ( u 1 ( ( ) Z = exp v 1s dc s Z 1 ( v2 ) ) u 2 + d v 1 W v ) 2 v 1 wih W = 1 Y + v 2 v 1. Proof. By Theorem 14.2 in Liu [9], we have d ln Y = (u 1 + u 2 Y )d + (v 1 + v 2 Y )dc. Denoe Ỹ = ln Y, i.e. Y = e Ỹ. Thus, dỹ = (u 1 + u 2 e Ỹ )d (v 1 + v 2 e Ỹ )dc. Furher, deỹ = (u 1 eỹ + u 2 )d (v 1 eỹ + v 2 )dc. Based on Theorem 1 of Liu [15], he above equaion has a soluion eỹ = Z 1 W v 2. v 1 Therefore, [ Y = e Ỹ = Z 1 W v ] 2 1. v 1 Since X + Y = 1, we only sudy he Y of he uncerain SIS model (5), ha is, dy = [(β θ)(1 Y )Y (μ + γ)y ]d (6) +σ (1 Y )Y dc. Theorem 2. The uncerain SIS model (5) has a soluion W 1 Z 1 X =, Y =, (7) W 1 + Z 1 W 1 + Z 1 where Z 1 = exp( σ sdc s ), and dw 1 = [(μ + γ + θ β)(w 1 + Z 1 ) + β θ]d wih W 1 = 1 Y Y. Proof. Denoe u 1 = β θ μ γ, u 2 = θ β, v 1 = σ and v 2 = σ. Then, for he equaion (6), we obain dy = (u 1 Y + u 2 Y 2 )d + (v 1Y + v 2 Y 2 )dc From Lemma 1 and Y = 1 X, he soluion of he model (5) can be obained. An examples is provided o illusrae he above resuls. Example 1. Take θ =.1,β =.5,μ =.3, γ =.27 and X =.9, Y =.1 for ODE SIS model (3) and UDE SIS model (5), respecively. Evidenly, he soluion of model (3) is

5 X D = Y D = J. Fang e al. / Uncerain SIS epidemic model wih sandard incidence and demography e.1.1.4(1 e.1 ),.1e.1.1.4(1 e.1 ). By he definiion of R D,wehaveRD =.75 < 1. From Theorem 1, here exiss an endemic equilibrium (X,Y ) = (.75,.25), which is globally asympoically sable. Comparing wih he model (3), by Theorem 2 he model (5) has he following soluion X = W 1 W 1 + e σc, Y e σc = W 1 + e σc, where dw 1 = [.1(W 1 + e σc ) +.4]d wih W 1 = 9. Nex we respecively provide wo differen inensiies of environmenal disurbances in model (5): (i) σ =.2; (ii) σ =.7. Fig. 1 shows curves of soluions for wo models (3) and (5) wih σ =.2 and σ =.7, respecively. Under uncerainy disurbance, rajecories of he model (5) is highly relevan o hose of he model (3) in a imely manner o reflec changes, and will no deviae from he long-erm value of model (3). On he oher hand, for model (5), he larger he value of disurbance inensiy is, he greaer he flucuaion is. 4. Main resuls of α-pahs In order o obain uncerainy disribuion, we need o sudy α-pah of he uncerain SIS model (5). Le he α-pahs be denoed by X α and Y α, respecively. Yao [14] verified ha a soluion of he UDE is a conour process. Thus, soluions X and Y of he uncerain SIS model (5) are conour processes. An uncerain process Z is a conour process if and only if for each α (, 1), here exiss α-pah Z α such ha M{Z <Z α, } =α, M{Z Z α, } =1 α. Then, for soluion (X,Y ) of he uncerain model (5), we have M{X <X α, } =α, M{X X α, } =1 α and M{Y <Y α, } =α, M{Y Y α, } =1 α. Fig. 1. Trajecories of soluions X,Y in wo models (3) and (5) wih σ =.2 and σ =.7. Based on equaion (6), α-pah Y α is o solve he following ODE dy α ={[(β θ)(1 Y α )Y α (μ + γ)y α ] (8) +σ (1 Y α )Y α 1 (α)}d wih Y α = Y, where 1 (α) is defined by (1). Theorem 3. The α-pahs of he model (5) are X α = 1 exp( g(, 1 α)) [ (β θ + σ s 1 (1 α)) ] 1, exp( g(s, 1 α))ds + Y 1 (9) [ Y α = exp( g(, α)) (β θ + σ s 1 (α)) ] 1, exp( g(s, α))ds + Y 1

6 932 J. Fang e al. / Uncerain SIS epidemic model wih sandard incidence and demography where g(, α) = (μ + γ + θ β) 1 (α) σ u du. Proof. From X = 1 Y,wehave {Y Y 1 α, } ={X 1 Y 1 α, }, {Y <Y 1 α, } ={X > 1 Y 1 α, }. Since Y is a conour process, i is o ge M{X 1 Y 1 α, } =M{Y Y 1 α, } =α, M{X >1 Y 1 α, } =M{Y <Y 1 α, }=1 α. Thus, X α = 1 Y 1 α, which is α-pah of X in he uncerain SIS model (5). For he ODE (8), we denoe u = Y α and ge du = [(β θ)(1 u )u (μ + γ)u ] d +σ (1 u )u 1 (α) = [β θ μ γ + σ 1 (α)]u [β θ + σ 1 (α)]u 2, which saisfies a Bernoulli differenial equaion. Due o he equaion X α = 1 Y 1 α, he resul follows. Le 1 Z (α)( <α<1) and Z (x) be he inverse uncerainy disribuion and uncerainy disribuion of he conour process Z, where Z = X or Y. Based on Theorem 2 in Yao [14] and Theorem 4, he resul below is obained. Theorem 4. For he model (5), he inverse uncerainy disribuions of X and Y are 1 X (α) = X α, and expeced values are E[X ] = 1 1 Y (α) = Y α, (1) X α dα, E[Y ] = 1 Y α dα, (11) where X α and Y α are defined in (9). By (1), we can obain uncerainy disribuions X (x) and Y (x) of X, Y, respecively. From Theorem 1, equilibria of he model (3) have globally asympoically sable. The following resuls show asympoic properies of α-pahs in he model (5) and reveal he relaionship of wo models. Theorem 5. Le σ = σ> and R U α = RD σ 1 (α) β θ wih <α<1. (i) If R U α 1, hen Xα = X, Y α = Y, where (X,Y ) = (1, ) is he disease free equilibrium of he model (3). (ii) If R U α < 1, hen Xα = Y α Especially, if α =.5, hen (β θ)x β θ + σ 1 (1 α), = (β θ)y + σ 1 (α) β θ + σ 1. (α) X.5 = X, Y.5 = Y. where (X,Y ) = ( μ+γ equilibrium of he model (3). β θ, β θ μ γ β θ ) is he endemic Proof. Since σ = σ, we have σ sds = σ and g(, α) = (μ + γ + θ β σ 1 (α)). (i) By Theorem 3, if R U α = 1, hen Y α = exp( g(, α)) [ (β θ + σ s 1 (α)) exp( g(s, α))ds + 1 Y ] 1 [ = (μ + γ) + 1 ] 1 =, Y Xα 1 α = (1 Y ) = 1. If R U α > 1, i.e. μ + γ + θ β σ 1 (α) >, we have e g(,α) =. Thus, Y α = exp( g(, α)) [ (β θ + σ s 1 (α)) exp( g(s, α))ds + 1 Y ] 1 =, Xα 1 α = (1 Y ) = 1. (ii) If R U α < 1, i.e. μ + γ + θ β σ 1 (α) <, hen eg(,α) =. Furher, Y α = exp( g(, α))

7 J. Fang e al. / Uncerain SIS epidemic model wih sandard incidence and demography 933 Table 1 The values of R U α wih σ =.2,.7 α R U α wih σ = R U α wih σ = Fig. 2. Trajecories of α-pahs X α wih σ =.2,.7 for α =.1,.2,...,.9. Fig. 3. Trajecories of α-pahs Y α wih σ =.2,.7 for α =.1,.2,...,.9. [ (β θ + σ s 1 (α)) exp( g(s, α))ds + 1 Y ] 1 [ θ β σ 1 (α) = μ + γ + θ β σ 1 (α) + eg(,α) Y = (β θ)y + σ 1 (α) β θ + σ 1, (α) Xα 1 α = (1 Y ) = (β θ)x β θ + σ 1 (1 α). ] 1 For α =.5, we have 1 (α) = 1 (1 α) =. Theorem 5 reflecs a fac: if α =.5, hen α-pahs of he uncerain model (5) are equivalen o model (3). Example 2. (Example 1 Coninue.) In he model (5), ake θ =.1,β =.5,μ=.3,γ =.27 and X =.9, Y =.1. From Theorem 3, we ge α- pahs as follows X α = 1 e (.1+σ 1 (1 α)) [.4+σ 1 (1 α).1+σ 1 (1 α) (e (.1+σ 1(1 α)) 1) + 1] 1, Y α = e (.1+σ 1 (α)) [.4+σ 1 (α).1+σ 1 (α) (e (.1+σ 1 (α)) 1) + 1] 1.

8 934 J. Fang e al. / Uncerain SIS epidemic model wih sandard incidence and demography Fig. 4. Uncerainy disribuions of X and Y wih (X,Y ) = (.9,.1) a = 1. For α =.1,.2,...,.9, Figs. 2 and 3 respecively provide rajecories of α-pahs wih wo cases σ =.2,.7. In addiion, hey reveal asympoic behavior of α-pahs X and Y wih differen value σ. Table 1 provides he values of R U α wih σ =.2,.7 when α =.1,.2,...,.9. By Theorem 5, if R U α 1, hen Xα = 1, Y α =. Oherwise, Xα = σ 1 (1 α), Y α =.1 + σ 1 (α).4 + σ 1 (α). These resuls show uncerainy of exincion and permanence of he disease mainly depends on he value Fig. 5. The expeced values of X and Y wih (X,Y ) = (.9,.1). α, see Figs. 2 and 3. Especially, if α =.5, hen X α =.75, Y α =.25, which is he same o he endemic equilibrium (X,Y )of model (3). From (1) and α-pahs X, Y, Fig. 4 provides uncerainy disribuions of X and Y wih σ =.2,.7. I is ineresing o noe ha here exis a poin of inersecion for differen uncerainy disribuions when α =.5. Based on Theorem 5, we can obain he expeced value of X and Y. In Fig. 5, curves of E[X ] and E[Y ] are given under σ =.2, Conclusion This paper applies uncerain differenial equaion o sudy a class of SIS epidemic model wih sandard incidence and demography, which generalizes he main resuls of Li e al. [17]. For a classical SIS model, we inroduce he moraliy and populaion changes

9 J. Fang e al. / Uncerain SIS epidemic model wih sandard incidence and demography 935 in i. By scale ransformaion, he ODE SIS model becomes an ODE proporion epidemic model wih he moraliy rae. Consider he disease ransmission rae is uncerain, we obain an UDE proporion epidemic model wih he moraliy rae. Furher, we respecively provide he soluions of wo models and obain he corresponding asympoic properies. According o he relaionship of X and Y, α-pahs of UDE model are acquired, respecively. Based on α-pahs, we discuss hreshold cases o reflec exincion and permanence of he disease. Comparing wih he ODE and UDE models, we observe ha hey no only have some similariies, bu also possess heir respecive characerisics. Similar o uncerain differenial equaion, here exis oher differenial equaions o invesigae he spread dynamic of epidemic, for example, fuzzy differenial equaion and fuzzy fracional differenial equaions. More deails abou his opic o see He and Yi [16], Kanagarajan and Sambah [17], Agarwal e al. [18], Alikhani and Bahrami [19]. How o esablish some epidemic models defined by hese differenial equaions and reveal he corresponding dynamic properies of epidemic. This is an ineresing, new and opening opic. Acknowledgemens We hank he ediors and reviewers for heir insighful commens. This research was funded by he Naional Naural Science Foundaion of China (Gran No ), he Innovaive Training Program for College Sudens of Xinjiang Universiy (Gran No. XJU-SRT-1782), and he Naural Science Foundaion of Xinjiang (Gran No. 216D1C43). References [2] H.W. Hehcoe, The mahemaics of infecious disease, SIAM Rev 42 (2), [3] Q.S. Yang and X.R. Mao, Exincion and recurrence of muligroup SEIR epidemic models wih sochasic perurbaions, Nonlinear Anal Real World Appl 14 (213), [4] J.R. Aralejo and M.J. Lopez-Herrero, Sochasic epidemic models: New behavioral indicaors of he disease spreading, Appl Mah Model 38 (214), [5] Y.Z. Bai and X.Q. Mu, Global asympoic sabiliy of a generalized SIRS epidemic model wih ransfer from infecious o suscepible, J Applied Anal Compu 8 (218), [6] H.W. Hehcoe and J.A. Yorke, Gonorrhea Transmission Dynamics and Conrol, Lecure Noes in Biomahemaics 56. Springer-Verlag, [7] A. Gray, D. Greenhalgh, X. Mao and J. Pan, A sochasic differenial equaion SIS epidemic model, SIAM J Appl Mah 71 (211), [8] L.A. Zadeh, Fuzzy ses, Informa Conrol 33 (1965), [9] B. Liu, Uncerainy Theory, 2nd edn. Springer, Berlin, 27. [1] B. Liu, Fuzzy process, hybird process and uncerain process, J Uncerain Sys 2 (28), [11] B. Liu, Some research problems in uncerainy heory, J Uncerain Sys 3 (29), 3 1. [12] B. Liu, Uncerainy Theory. A Branch of Mahemaics for Modeling Human Uncerainy, Springer-Verlag, Berlin, 21. [13] K. Yao and X. Chen, A numberical mehod for solving uncerain differenial equaions, J Inell Fuzzy Sys 25 (213), [14] K. Yao, Uncerain conour process and i s applicaion in sock model wih floaing ineres rae, Fuzzy Opim Decis Ma 14 (215), [15] Y.H. Liu, An analyic mehod for solving uncerain differenial equaions, J Uncerain Sys 6 (212), [16] O. He and W. Yi, On fuzzy differenial equaions, Fuzzy Ses Sysems 24 (1989), [17] K. Kanagarajan and M. Sambah, Runge-kua nysrom mehod for solving fuzzy differenial equaions, J Compu Meh Appl Mah 1 (21), [18] R.P. Agarwal, S. Arshad, D. O Regan and V. Lupulescu, Fuzzy fracional inegral equaions under compacness ype condiion, Frac Calc Appl Anal 15 (212), [19] R. Alikhani and F. Bahrami, Global soluions for nonlinear fuzzy fracional inegral and inegrodifferenial equaions, Commun Nonlinear Sci Numer Simul 18 (213), [1] Z. Li, Y. Sheng, Z. Teng and H. Miao, An uncerain differenial equaion for SIS epidemic model, J Inell Fuzzy Sys 33 (217),