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

1 Introduction

Mathematical models play an important role in analysing the spread of infectious diseases and get more and more attentions by many people, for example Hethcote [2], Yang and Mao [3], Artalejo and Lopez-Herrero [4], Bai and Mu [5].

Let S _t and I _t be the numbers of susceptible and infected individuals at time t, respectively. Hethcote and Yorke [6] introduced a classical susceptible-infectious-susceptible (SIS) epidemic model, described by the following ordinary differential equation (ODE) ( M 1 ) { dS t = [ μ N - β S t I t + γ I t - μ S t ] dt , dI t = [ β S t I t - ( μ + γ ) I t ] dt , where N is a constant and means the total size of the population among whom the disease is spreading at time t, μ is the natural-death rate of population per unit time, γ is the rate at which infected individuals become cured per unit time, β is the disease transmission coefficient per unit time and βS _t I _t is of bilinear incidence. For model (M1), it has powerful qualitative results such as globally asymptotically stable under some conditions. However, most of ODE epidemic models are inevitably affected by environment fluctuations. For better understanding of epidemic dynamics in reality, many scholars introduced white noise into deterministic models and established stochastic versions with the effect of noise. For example, Gray et al. [7] considered a stochastic differential equation (SDE) for SIS epidemic model as follows ( M 2 ) { dS t = [ μ N - β S t I t + γ I t - μ S t ] dt - σ S t I t dB t , dI t = [ β S t I t - ( μ + γ ) I t ] dt + σ S t I t dB t . Here, B _t is a Wiener process with B₀ = 0, defined in probability space (Ω, {ℱ _t } _t≥0, P) with a filtration {ℱ _t } _t≥0 satisfying the usual condition that contains all P-null sets. The intensity of white noise is σ _t (>0). Parameters N, μ, β and γ are defined as above. By the stochastic Lyapunov functional method, they obtained asymptotic behavior of the stochastic system around equilibria.

When no samples are available to estimate probability properties of the model (M2), we have to use a means of handling uncertainty rather than to randomness. One of the most important tools is fuzzy set theory, which the idea of fuzzy set was first proposed by Zaheh in 1965. A fuzzy set is a class of objects. Such a set is characterized by a set function (measure) which assigns to each object ranging between zero and one. In order to measure a fuzzy event, Liu [9] introduced a set function of fuzzy set named uncertain measure based on normality, duality, subadditivity and product axioms, and established uncertainty theory as a branch of mathematics for studying behavior of uncertainphenomena.

For a non-empty fuzzy set Γ, L is a σ-algebra on the set Γ and uncertain measure denoted by ℳ is a function from L to [0,1], satisfying four axioms. The triplet ( Γ , L , ℳ ) is called an uncertainty space. Uncertain process C _t is a given one-dimensional Liu process defined on uncertainty space. For the model (M1), there exist some uncertain phenomena influencing the cured rate γ. Suppose the rate γ is described by Liu process C _t , that is, γdt in model (M1) is replaced by γdt + σ _t dC _t . Li et al. [1] firstly established an uncertain differential equation (UDE) of SIS epidemic model with bilinear incidence, formed by ( M 3 ) { dS t = [ μ N - β S t I t + γ I t - μ S t ] dt + σ t I t dC t , dI t = [ β S t I t - ( μ + γ ) I t ] dt - σ t I t dC t , where σ _t (>0) is the intensity of Liu process C _t . Further, they obtained the solutions, convergence properties of α-paths of model (M3) and compared the results of ODE, SDE and UDE epidemic models, respectively. Comparing with the results of ODE model, those of UDE model reflected some practical problems better reasonably than those of SDE case without sample data. For model (M3), we have S _t + I _t = N - (N - N₀) e^−μt, where N₀ = S₀ + I₀. If t→ + ∞, then S _t + I _t → N. This means the model (M3) has constant population size.

However, for the long time the total population size may vary and the disease can cause deaths during the spread of epidemic. In this work, we applied the uncertainty theory to study a class of uncertain SIS epidemic model with standard incidence and demography. The rest is organized as follows. In Section 2, some basic concepts and notation of uncertainty theory are reviewed. Two modified SIS epidemic model are established in Section 3. In Section 4, α-paths of the models are obtained. Further, by α-paths, uncertainty distributions and expected values of uncertain SIS model are given. Some examples are provided to illustrate these results. A brief conclusion is in Section 5.

2 Preliminaries

This section will briefly review some basic concepts and notation of uncertainty theory, including uncertain measure, uncertain variable, uncertain process, uncertain differential equation and α-path. More details to see Liu [9–12], Yao and Chen [13], Yao [14].

A set function ℳ : L → [ 0 , 1 ] is called an uncertain measure if it is supposed to satisfy the following axioms:

Axiom 1 (Normality Axiom) ℳ {Γ} =1 for the universal set Γ.

Axiom 2 (Duality Axiom) ℳ {Λ} + ℳ {Λ ^c } =1 for any event Λ.

Axiom 3 (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂,..., ℳ { ⋃ i = 1 ∞ Λ i } ≤ ∑ i = 1 ∞ ℳ { Λ i } .

Axiom 4 (Product Axiom) Let ( Γ i , L i , ℳ i ) be uncertainty spaces and Λ _i be arbitrarily chosen events from L iitsc > for i = 1, 2,..., then the product uncertain measure ℳ is an uncertain measure satisfying ℳ { ∏ i = 1 ∞ Λ i } = ⋀ i = 1 ∞ ℳ i { Λ i } .

Let T be an index set and ( Γ , L , ℳ ) be an uncertainty space. An uncertain process X _t is a measurable function from T × ( Γ , L , ℳ ) to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set {X _t ∈ B} = {γ|X _t (γ) ∈ B} is an event. The uncertain process X _t is said to have an uncertainty distribution Φ _t (x) if at each time t, uncertain variable X _t has uncertainty distribution Φ _t (x) and Φ t ( x ) = ℳ { X t ≤ x } for any real number x ∈ ℝ .

Definition 1. An uncertain process C _t is said to be Liu process if

(i) C₀ = 0 and almost all sample paths are Lipschitz continuous.

(ii) C _t has stationary and independent increments.

(iii) Every increment C_t+s - C _s is a normal uncertain variable with an uncertainty distribution Φ t ( x ) = ( 1 + exp ( - π x 3 t ) ) - 1 , x ∈ ℝ .

Let X _t be an uncertain process and C _t be Liu process. For any partition of a closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as Δ = max 1 ≤ i ≤ k | t i + 1 - t i | . Then, Liu integral of X _t is defined by ∫ a b X t dC t = lim Δ → 0 ∑ i = 1 k X t i ( C t i + 1 - C t i ) provided that the limit exists almost surely and is finite.

Definition 2. defUDE Suppose C _t is Liu process, and f and g are two given functions. Then dX t = f ( t , X t ) dt + g ( t , X t ) dC t is called an uncertain differential equation (UDE). An uncertain process X _t is called a solution of the equation, if X _t satisfies dX _t ≡ f (t, X _t ) dt + g (t, X _t ) dC _t for any time t.

For 0 < α < 1, α-path of an uncertain differential equation dX _t = f (t, X _t ) dt + g (t, X _t ) dC _t with initial value X₀ is a deterministic function X t α with respect to t that solves the ordinary differential equation (ODE) dX t α = [ f ( t , X t α ) + | g ( t , X t α ) | Φ - 1 ( α ) ] dt , where Φ⁻¹ (α) is the inverse uncertainty distribution of standard normal uncertain variable, i.e., Φ - 1 ( α ) = 3 π ln α 1 - α , 0 < α < 1 .(1)

Let X _t be an uncertain process. If for each α ∈ (0, 1), there exists a real function X t α such that ℳ { X t ≤ X t α , ∀ t } = α , ℳ { X t > X t α , ∀ t } = 1 - α , then X _t is called a contour process. X t α is called α-path of the process X _t . Expected value of X _t is defined by E [ X t ] = ∫ 0 + ∞ ℳ { X t ≥ x } dx - ∫ - ∞ 0 ℳ { X t ≤ x } dx , provided that at least one of the two integrals is finite. If X _t has the regular uncertainty distribution Φ _t , then E [ X t ] = ∫ 0 1 Φ t - 1 ( α ) d α .

Let N _t be the total number of people at time t and N _t = S _t + I _t . The study of growth and change of populations is called demography. Thus, for some epidemic diseases without permanent immunity, it is appropriate to add the mortality and population variable N _t to SIS model. A new SIS model is established as follows { dS t = [ μ N t - β S t I t N t + γ I t - μ S t ] dt , dI t = [ β S t I t N t - ( μ + γ ) I t - θ I t ] dt(2) with the initial values S₀ and I₀, where θ (θ < β) is the mortality rate of the disease per unit time. μ, β and γ are defined in the model (M1). β S t N t I t is the average number of infection transmissions by all infectious individuals per unit time, called standard incidence. The bilinear and standard incidences agree when the total population size is a constant, but they differ if the total population size is variable. For the model (M1), N _t → N as t→ + ∞. However, for model (2), we have dN _t = - θI _t dt. Thus, the modified SIS model (2) is an SIS type with standard incidence rate and demography.

For simplicity, we nondimensionalize model (2) by defining X t = S t N t , Y t = I t N t , where X _t and Y _t are the proportions of susceptible and infected individuals, respectively. Thus, the model (2) is equivalent to the ODE model { dX t = [ ( μ + γ ) ( 1 - X t ) - ( β - θ ) ( 1 - X t ) X t ] dt , dY t = [ ( β - θ ) ( 1 - Y t ) Y t - ( μ + γ ) Y t ] dt(3) with the initial values X₀ = S₀/N₀, Y₀ = I₀/N₀. For any time t ≥ 0, X _t + Y _t = 1. By (3), we have dY t dt = ( β - θ - μ - γ ) Y t - ( β - θ ) Y t 2 . Denote a = β - θ - μ - γ and b = θ - β. Thus, the solution of the model (3) is { X t D = a + bY 0 - ( a + b ) Y 0 e at a + bY 0 ( 1 - e at ) , Y t D = aY 0 e at a + bY 0 ( 1 - e at ) .

For the ODE model (3), a basic reproductive number is defined as R 0 D = μ + γ β - θ .(4)

Through the fundamental theory of ordinary differential equation, it is easy to obtain the following results.

Theorem 1. (i) If R 0 D ≥ 1 , then the disease free-equilibrium ( X 0 , Y 0 ) = ( 1 , 0 ) of the model (3) is globally asymptotically stable.

(ii) If R 0 D < 1 , then there exists a unique endemic equilibrium ( X * , Y * ) = ( μ + γ β - θ , β - θ - μ - γ β - θ ) for model (3), which is globally asymptotic stability.

Let ( Γ , L , ℳ ) be an uncertainty space and C _t is Liu process defined on uncertainty space. Suppose the uncertain fluctuating environment in the form of Liu process effects the disease transmission coefficient β, that is, replace βdt with βdt + σ _t dC _t , then the model (3) becomes an uncertain differential equation (UDE) of SIS epidemic model in form of { dX t = [ ( μ + γ ) ( 1 - X t ) - ( β - θ ) ( 1 - X t ) X t ] dt - σ t ( 1 - X t ) X t dC t , dY t = [ ( β - θ ) ( 1 - Y t ) Y t - ( μ + γ ) Y t ] dt + σ t ( 1 - Y t ) Y t d C t ,(5) where σ _t (>0) is an integrable and differential function for any t, called the intensity of environmental disturbances.

In order to provide solutions of the uncertain SIS model (5), we discuss the solution of an uncertain differential equation and obtain the following result.

Lemma 1. Let u _it , v _it be two integrable and differential functions about time t for i = 1, 2 and v_1t ≠ 0. Then, the uncertain differential equation dY t = ( u 1 t Y t + u 2 t Y t 2 ) dt + ( v 1 t Y t + v 2 t Y t 2 ) dC t has a solution Y t = [ Z t - 1 W t - v 2 t v 1 t ] - 1 , where Z t = exp ( ∫ 0 t v 1 s dC s ) and dW t = Z t ( - u 1 t ( Z t - 1 W t - v 2 t v 1 t ) - u 2 t + ( v 2 t v 1 t ) ′ ) dt with W 0 = 1 Y 0 + v 20 v 10 .

Proof. By Theorem 14.2 in Liu [9], we have d ln Y t = ( u 1 t + u 2 t Y t ) dt + ( v 1 t + v 2 t Y t ) dC t . Denote Y ˜ t = - ln Y t , i.e. Y t = e - Y ˜ t . Thus, d Y ˜ t = - ( u 1 t + u 2 t e - Y ˜ t ) dt - ( v 1 t + v 2 t e - Y ˜ t ) dC t . Further, de Y ˜ t = - ( u 1 t e Y ˜ t + u 2 t ) dt - ( v 1 t e Y ˜ t + v 2 t ) dC t . Based on Theorem 1 of Liu [15], the above equation has a solution e Y ˜ t = Z t - 1 W t - v 2 t v 1 t . Therefore, Y t = e - Y ˜ t = [ Z t - 1 W t - v 2 t v 1 t ] - 1 . □

Since X _t + Y _t = 1, we only study the Y _t of the uncertain SIS model (5), that is, dY t = [ ( β - θ ) ( 1 - Y t ) Y t - ( μ + γ ) Y t ] dt(6) + σ t ( 1 - Y t ) Y t d C t .

Theorem 2. The uncertain SIS model (5) has a solution X t = W t 1 W t 1 + Z t 1 , Y t = Z t 1 W t 1 + Z t 1 ,(7) where Z t 1 = exp ( ∫ 0 t σ s dC s ) , and dW t 1 = [ ( μ + γ + θ - β ) ( W t 1 + Z t 1 ) + β - θ ] dt with W 01 = 1 - Y 0 Y 0 .

Proof. Denote u_1t = β - θ - μ - γ, u_2t = θ - β, v_1t = σ _t and v_2t = - σ _t . Then, for the equation (6), we obtain dY t = ( u 1 t Y t + u 2 t Y t 2 ) dt + ( v 1 t Y t + v 2 t Y t 2 ) dC t From Lemma 1 and Y _t = 1 - X _t , the solution of the model (5) can be obtained. □

An examples is provided to illustrate the above results.

Example 1. Take θ = 0.10, β = 0.50, μ = 0.03, γ = 0.27 and X₀ = 0.90, Y₀ = 0.10 for ODE SIS model (3) and UDE SIS model (5), respectively. Evidently, the solution of model (3) is { X t D = 0.96 + 0.03 e 0.1 t 0.1 - 0.04 ( 1 - e 0.1 t ) , Y t D = 0.01 e 0.1 t 0.1 - 0.04 ( 1 - e 0.1 t ) .

By the definition of R 0 D , we have R 0 D = 0.75 < 1 . From Theorem 1, there exists an endemic equilibrium (X*, Y*) = (0.75, 0.25), which is globally asymptotically stable.

Comparing with the model (3), by Theorem 2 the model (5) has the following solution X t = W t 1 W t 1 + e σ C t , Y t = e σ C t W t 1 + e σ C t , where dW_t1 = [-0.1 (W_t1 + e ^σC _t ) +0.4] dt with W₀₁ = 9.

Next we respectively provide two different intensities of environmental disturbances in model (5): (i) σ = 0.2; (ii) σ = 0.7. Fig. 1 shows curves of solutions for two models (3) and (5) with σ = 0.2 and σ = 0.7, respectively. Under uncertainty disturbance, trajectories of the model (5) is highly relevant to those of the model (3) in a timely manner to reflect changes, and will not deviate from the long-term value of model (3). On the other hand, for model (5), the larger the value of disturbance intensity is, the greater the fluctuation.

OPEN IN VIEWER

Fig. 1

Trajectories of solutions X _t ,Y _t in two models (3) and (5) with σ = 0.2 and σ = 0.7.

In order to obtain uncertainty distribution, we need to study α-path of the uncertain SIS model (5). Let the α-paths be denoted by X t α and Y t α , respectively. Yao [14] verified that a solution of the UDE is a contour process. Thus, solutions X _t and Y _t of the uncertain SIS model (5) are contour processes. An uncertain process Z _t is a contour process if and only if for each α ∈ (0, 1), there exists α-path Z t α suchthat ℳ { Z t < Z t α , ∀ t } = α , ℳ { Z t ≥ Z t α , ∀ t } = 1 - α . Then, for solution (X _t , Y _t ) of the uncertain model (5), we have ℳ { X t < X t α , ∀ t } = α , ℳ { X t ≥ X t α , ∀ t } = 1 - α and ℳ { Y t < Y t α , ∀ t } = α , ℳ { Y t ≥ Y t α , ∀ t } = 1 - α .

Based on equation (6), α-path Y t α is to solve the following ODE dY t α = { [ ( β - θ ) ( 1 - Y t α ) Y t α - ( μ + γ ) Y t α ](8) + σ t ( 1 - Y t α ) Y t α Φ - 1 ( α ) } dt with Y 0 α = Y 0 , where Φ⁻¹ (α) is defined by (φ).

Theorem 3. The α-paths of the model (5) are { X t α = 1 - exp ( - g ( t , 1 - α ) ) [ ∫ 0 t ( β - θ + σ s Φ - 1 ( 1 - α ) ) × exp ( - g ( s , 1 - α ) ) ds + 1 Y 0 ] - 1 , Y t α = exp ( - g ( t , α ) ) × [ ∫ 0 t ( β - θ + σ s Φ - 1 ( α ) ) × exp ( - g ( s , α ) ) ds + 1 Y 0 ] - 1 ,(9) where g ( t , α ) = ( μ + γ + θ - β ) t - Φ - 1 ( α ) ∫ 0 t σ _u du.

Proof. From X _t = 1 - Y _t , we have { Y t ≥ Y t 1 - α , ∀ t } = { X t ≤ 1 - Y t 1 - α , ∀ t } , { Y t < Y t 1 - α , ∀ t } = { X t > 1 - Y t 1 - α , ∀ t } . Since Y _t is a contour process, it is to get ℳ { X t ≤ 1 - Y t 1 - α , ∀ t } = ℳ { Y t ≥ Y t 1 - α , ∀ t } = α , ℳ { X t > 1 - Y t 1 - α , ∀ t } = ℳ { Y t < Y t 1 - α , ∀ t } = 1 - α .

Thus, X t α = 1 - Y t 1 - α , which is α-path of X _t in the uncertain SIS model (5). For the ODE (8), we denote u t = Y t α and get du t dt = [ ( β - θ ) ( 1 - u t ) u t - ( μ + γ ) u t ] + σ t ( 1 - u t ) u t Φ - 1 ( α ) = [ β - θ - μ - γ + σ t Φ - 1 ( α ) ] u t - [ β - θ + σ t Φ - 1 ( α ) ] u t 2 , which satisfies a Bernoulli differential equation. Due to the equation X t α = 1 - Y t 1 - α , the result follows. □

Let Φ Z t - 1 ( α ) ( 0 < α < 1 ) and Φ _{Z t} (x) be the inverse uncertainty distribution and uncertainty distribution of the contour process Z _t , where Z _t = X _t or Y _t . Based on Theorem 2 in Yao [14] and Theorem 4, the result below is obtained.

Theorem 4. For the model (5), the inverse uncertainty distributions of X _t and Y _t are Φ X t - 1 ( α ) = X t α , Φ Y t - 1 ( α ) = Y t α ,(10) and expected values are E [ X t ] = ∫ 0 1 X t α d α , E [ Y t ] = ∫ 0 1 Y t α d α ,(11) where X t α and Y t α are defined in (9).

By (10), we can obtain uncertainty distributions Φ _{X t} (x) and Φ _{Y t} (x) of X _t , Y _t , respectively. From Theorem 1, equilibria of the model (3) have globally asymptotically stable. The following results show asymptotic properties of α-paths in the model (5) and reveal the relationship of two models.

Theorem 5. Let σ _t = σ > 0 and R α U = R 0 D - σ Φ - 1 ( α ) β - θ with 0 < α < 1.

(i) If R α U ≥ 1 , then lim t → + ∞ X t α = X 0 , lim t → + ∞ Y t α = Y 0 , where (X⁰, Y⁰) = (1, 0) is the disease free equilibrium of the model (3).

(ii) If R α U < 1 , then lim t → + ∞ X t α = ( β - θ ) X * β - θ + σ Φ - 1 ( 1 - α ) , lim t → + ∞ Y t α = ( β - θ ) Y * + σ Φ - 1 ( α ) β - θ + σ Φ - 1 ( α ) . Especially, if α = 0.5, then lim t → + ∞ X t 0.5 = X * , lim t → + ∞ Y t 0.5 = Y * . where ( X * , Y * ) = ( μ + γ β - θ , β - θ - μ - γ β - θ ) is the endemic equilibrium of the model (3).

Proof. Since σ _t = σ, we have ∫ 0 t σ s ds = σ t and g (t, α) = (μ + γ + θ - β - σΦ⁻¹ (α)) t.

(i) By Theorem 3, if R α U = 1 , then lim t → + ∞ Y t α = lim t → + ∞ exp ( - g ( t , α ) ) × [ ∫ 0 t ( β - θ + σ s Φ - 1 ( α ) ) × exp ( - g ( s , α ) ) ds + 1 Y 0 ] - 1 = lim t → + ∞ [ ( μ + γ ) t + 1 Y 0 ] - 1 = 0 , lim t → + ∞ X t α = lim t → + ∞ ( 1 - Y t 1 - α ) = 1 .

If R α U > 1 , i.e. μ + γ + θ - β - σΦ⁻¹ (α) >0, we have lim t → + ∞ e - g ( t , α ) = 0 . Thus, lim t → + ∞ Y t α = lim t → + ∞ exp ( - g ( t , α ) ) × [ ∫ 0 t ( β - θ + σ s Φ - 1 ( α ) ) × exp ( - g ( s , α ) ) ds + 1 Y 0 ] - 1 = 0 , lim t → + ∞ X t α = lim t → + ∞ ( 1 - Y t 1 - α ) = 1 .

(ii) If R α U < 1 , i.e. μ + γ + θ - β - σΦ⁻¹ (α) <0, then lim t → + ∞ e g ( t , α ) = 0 . Further, lim t → + ∞ Y t α = lim t → + ∞ exp ( - g ( t , α ) ) [ ∫ 0 t ( β - θ + σ s Φ - 1 ( α ) ) × exp ( - g ( s , α ) ) ds + 1 Y 0 ] - 1 = lim t → + ∞ [ θ - β - σ Φ - 1 ( α ) μ + γ + θ - β - σ Φ - 1 ( α ) + e g ( t , α ) Y 0 ] - 1 = ( β - θ ) Y * + σ Φ - 1 ( α ) β - θ + σ Φ - 1 ( α ) , lim t → + ∞ X t α = lim t → + ∞ ( 1 - Y t 1 - α ) = ( β - θ ) X * β - θ + σ Φ - 1 ( 1 - α ) . For α = 0.5, we have Φ⁻¹ (α) = Φ⁻¹ (1 - α) =0. □

Theorem 5 reflects a fact: if α = 0.5, then α-paths of the uncertain model (5) are equivalent to model (3).

Example 2. (Example 1 Continue.) {In the model (5), take θ = 0.10, β = 0.50, μ = 0.03, γ = 0.27 and X₀ = 0.90, Y₀ = 0.10. From Theorem 5, we get α-paths as follows { X t α = 1 - e ( 0.1 + σ Φ - 1 ( 1 - α ) ) t [ 0.4 + σ Φ - 1 ( 1 - α ) 0.1 + σ Φ - 1 ( 1 - α ) × ( e ( 0.1 + σ Φ - 1 ( 1 - α ) ) t - 1 ) + 10 ] - 1 , Y t α = e ( 0.1 + σ Φ - 1 ( α ) ) t [ 0.4 + σ Φ - 1 ( α ) 0.1 + σ Φ - 1 ( α ) × ( e ( 0.1 + σ Φ - 1 ( α ) ) t - 1 ) + 10 ] - 1 . For α = 0.1, 0.2, …, 0.9, Figs. 2 and 3 respectively provide trajectories of α-paths with two cases σ = 0.2, 0.7. In addition, they reveal asymptotic behavior of α-paths X _t and Y _t with different value σ. OPEN IN VIEWER

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Fig. 3

Trajectories of α-paths Y t α with σ = 0.2, 0.7 for α = 0.1, 0.2, …, 0.9.

Table 1 provides the values of R α U with σ = 0.2, 0.7 when α = 0.1, 0.2, …, 0.9. By Theorem 4, if R α U ≥ 1 , then lim t → + ∞ X t α = 1 , lim t → + ∞ Y t α = 0 . Otherwise, lim t → + ∞ X t α = 0.3 0.4 + σ Φ - 1 ( 1 - α ) , lim t → + ∞ Y t α = 0.1 + σ Φ - 1 ( α ) 0.4 + σ Φ - 1 ( α ) . These results show uncertainty of extinction and permanence of the disease mainly depends on the value α, see Figs. 2 and Fig. 3. Especially, if α = 0.5, then lim t → + ∞ X t α = 0.75 , lim t → + ∞ Y t α = 0.25 , which is the same to the endemic equilibrium (X*, Y*) of model (3).

From (10) and α-paths X _t , Y _t , Fig. 4 provides uncertainty distributions of X _t and Y _t with σ = 0.2, 0.7. It is interesting to note that there exits a point of intersection for different uncertainty distributions when α = 0.5. Based on Theorem 5, we can obtain the expected value of X _t and Y _t . In Fig. 5, curves of E [X _t ] and E [Y _t ] are given under σ = 0.2, 0.7. OPEN IN VIEWER

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Fig. 5

The expected values of X _t and Y _t with (X₀, Y₀) = (0.90, 0.10).

This paper applies uncertain differential equation to study a class of SIS epidemic model with standard incidence and demography, which generalizes the main results of Li et al. [17]. For a classical SIS model, we introduce the mortality and population changes in it. By scale transformation, the ODE SIS model becomes an ODE proportion epidemic model with the mortality rate. Consider the disease transmission rate is uncertain, we obtain an UDE proportion epidemic model with the mortality rate. Further, we respectively provide the solutions of two models and obtain the corresponding asymptotic properties. According to the relationship of X _t and Y _t , α-paths of UDE model are acquired, respectively. Based on α-paths, we discuss threshold cases to reflect extinction and permanence of the disease. Comparing with the ODE and UDE models, we observe that they not only have some similarities, but also possess their respective characteristics.

Similar to uncertain differential equation, there exist other differential equations to investigate the spread dynamic of epidemic, for example, fuzzy differential equation and fuzzy fractional differential equations. More details about this topic to see He and Yi [16], Kanagarajan and Sambath [17], Agarwal et al. [18], Alikhani and Bahrami [19]. How to establish some epidemic models defined by these differential equations and reveal the corresponding dynamic properties of epidemic. This is an interesting, new and opening topic.