Comparison of three SIS epidemic models: deterministic,stochastic and uncertain

Abstract

This paper applies uncertain theory to establish an uncertain differential equation (UDS) SIS epidemic model, comparing with deterministic and stochastic SIS models. Solution of the UDE model is obtained. Threshold conditions are derived for permanence and extinction of disease by the corresponding α-paths, which reveal relationships of three models. Numerical simulations are given to illustrate these results.

Keywords

Deterministic model stochastic model uncertain model Liu process

1 Introduction

An epidemic is the rapid spread of infectious disease to a large number of people in a given population within a short period of time. There is a wide range of deterministic and stochastic models to investigate spread of an epidemic, such as SIS, SIR, SEIR, SEIRS models and so on, see Kermack and Makendrick [1], Capasso and Serio [2], Liu et al. [3], Hethcote [4], Allen and Burgin [5], Li et al. [6], Yang and Mao [7], Artalejo and Lopez-Herrero [8].

Deterministic models are usually described by ordinary differential equations (ODEs) and provide local or global stability for persistence and extinction of disease by a basic reproduction number. For example, Korobeinikov and Wake [9] obtained global stability properties of SIS, SIR and SIRS epidemiological models by Lyapunov functions. Li and Jin [11] provided global stability of an SEIR epidemic model with infectious force. Bai and Mu [10] proved global asymptotic stability of a generalized SIRS epidemic model.

In order to explore the effect of environmental fluctuation on deterministic model, many scholars introduced different noises into deterministic model and established stochastic versions, which are generally formed by stochastic differential equations (SDEs). Stochastic model mainly deals with asymptotic behavior around equilibria of the deterministic case by a threshold value. Yang et al. [12] showed ergodicty and extinction of stochastic SIR and SEIR epidemic models with saturated incidence. Chen and Kang [13] proved asymptotic behavior of a stochastic SIR model.

Suppose the total population size at time t is given by N _t, which is divided into two sub-groups: the susceptible (S) and infected (I) classes. Let S _t and I _t be the numbers of susceptible and infectious individuals at time t, respectively. Consider a deterministic SIS epidemic model formed by the following ODEs

${\begin{matrix} {dS}_{t} & = & [μ N - β S_{t} I_{t} + γ I_{t} - μ S_{t}] dt, \\ {dI}_{t} & = & [β S_{t} I_{t} - (μ + γ) I_{t}] dt \end{matrix}$ (1)with initial values S ₀ and I ₀, called an ODE SIS model. Here, N is the total population size among whom the disease is spreading, μ is natural death rate of population per unit time, γ is recovery rate of the infectious per unit time, and β is infection coefficient per unit time. The model describes some diseases such as bacterial disease do not have permanent immunity, which individuals become susceptible again after infection.

Denote N _t = S _t + I _t and N ₀ = S ₀ + I ₀. For model (1), it is easy to get $\frac{{dN}_{t}}{dt} = μ N - N_{t} .$

Then, N _t = N - (N - N ₀) e ^-μt. If t→ + ∞, then N _t → N; that is, S _t + I _t → N. Without loss of generality, suppose S _t + I _t ≡ N. Define a basic reproduction number as

$R_{0}^{D} = \frac{β N}{μ + γ} .$ (2)

Thus, solution of the model (1) is obtained as follows $\begin{matrix} {\begin{matrix} S_{t} & = & N - [\frac{β}{β N - μ - γ} (1 - e^{- (β N - μ - γ) t}) \\ + \frac{1}{I_{0}} e^{- (β N - μ - γ) t}]^{- 1}, \\ I_{t} & = & [\frac{β}{β N - μ - γ} (1 - e^{- (β N - μ - γ) t}) \\ + \frac{1}{I_{0}} e^{- (β N - μ - γ) t}]^{- 1} \end{matrix} \end{matrix}$ for $R_{0}^{D} \neq 1$ . For $R_{0}^{D} = 1$ , one has $\begin{matrix} {\begin{matrix} S_{t} & = & N - [β t + \frac{1}{I_{0}}]^{- 1}, \\ I_{t} & = & [β t + \frac{1}{I_{0}}]^{- 1} . \end{matrix} \end{matrix}$

Based on the above solutions, the following results hold.

(A1) If $R_{0}^{D} \leq 1,$ then model (1) has a unique disease free equilibrium (S ⁰, I ⁰) = (N, 0), which is globally asymptotically stable. That is to say, the disease will die out.

(A2) If $R_{0}^{D} > 1,$ then there exists a unique endemic equilibrium $(S^{*}, I^{*}) = (\frac{μ + γ}{β}, N - \frac{μ + γ}{β}),$ which is globally asymptotically stable. That is to say, the disease will be persistent.

Consider random effects of environment make transmission coefficient β of disease, Gray et al. [14] established an Itô stochastic differential equation (SDE) SIS model by replacing βdt in model (1) by $\tilde{β} dt = β dt + σ d B_{t},$ where σ > 0 is white noise density. Here, B _t is Wiener process defined in probability space (Ω, {scripfontF _t} _t≥0, P) with a σ-algebra filtration {scripfontF _t} _t≥0 satisfying usual conditions that contain all P-null sets. Then, a stochastic SIS epidemic model is written by the following SDEs $\begin{matrix} {\begin{matrix} {dS}_{t} & = & [μ N - β S_{t} I_{t} + γ I_{t} - μ S_{t}] dt - σ S_{t} I_{t} {dB}_{t}, \\ {dI}_{t} & = & [β S_{t} I_{t} - (μ + γ) I_{t}] dt + σ S_{t} I_{t} {dB}_{t}, \end{matrix} \end{matrix}$ called an SDE SIS model. Similar to the model (1), assume S _t + I _t ≡ N. The above stochastic SIS model is equivalent to the following model

$\begin{matrix} {dI}_{t} & = & [β (N - I_{t}) I_{t} - (μ + γ) I_{t}] dt \\ + σ (N - I_{t}) I_{t} {dB}_{t} . \end{matrix}$ (3)

For model (3), define a threshold value as

$R_{0}^{S} = R_{0}^{D} - \frac{σ^{2} N^{2}}{2 (μ + γ)} .$ (4)

For a given initial value I ₀ and all t ≥ 0, Theorem 3.1 of Gray et al. [14] showed model (3) has a unique global positive solution I _t ∈ (0, N) with probability one. Moreover, the following results hold.

(B1) If $R_{0}^{S} < 1$ and σ ² ≤ β/N, then $lim_{t \to + \infty} sup \frac{1}{t} log I_{t} < 0,$ i.e. I _t tends to zero exponentially as t→ + ∞. That is to say, the disease dies out with probability one.

(B2) If $R_{0}^{S} > 1,$ then $lim_{t \to + \infty} sup I_{t} \geq ξ, lim_{t \to + \infty} inf I_{t} \leq ξ a . s .,$ where $ξ = \frac{1}{σ^{2}} (\sqrt{β^{2} - 2 σ^{2} (μ + γ)} - (β - σ^{2} N)) .$ Namely, I _t will rise to or above the level ξ infinitely often with probability one. It reflects the disease is persistent.

In practices, the number of infectious individuals perhaps continues to increase or decrease at the beginning or the end of an epidemic. Thus, the total number of compartments may not behave like Ito’s SDE. On the other hand, in order to investigate transmission mechanism of disease, it need to collect a set of the available samples and provide numerical simulations. However, for such kinds of epidemics with few data in short time or missing data in long time, we have to invite some domain experts to evaluate belief degree that each event will happen. Liu [15] first introduced the concept of belief degree to solve the above questions, which represents strength with which we believe the event will happen. Based on belief degree, an uncertain theory is established to reflect uncertain phenomena according to normality, duality, subadditivity and product axioms, different from probability theory. In order to indicate belief degree, an uncertain measure is defined as a set function on a σ-algebra, which is interpreted as personal belief degree. Thus, an uncertain variable is used to represent quantities with uncertainty and uncertainty distribution is used to describe an uncertain variable.

In 2008, Liu introduced uncertain process and proposed a type of differential equations driven by Liu process, called uncertain differential equation (UDE). So far, uncertain differential equation has become a main tool to deal with dynamic uncertain system such as finance and control fields. Liu [18] supposed stock price followed an uncertain differential equation and produced a new topic of uncertain finance. Chen and Gao [19] analyzed an uncertain interest rate model with uncertain short rate. Yao [20] studied a stock model with uncertain interest rate. Liu [21] proposed an uncertain insurance risk process, and Yao and Qin [22] modified the uncertain risk model.

Recently, Li et al. [23] firstly applied uncertain differential equation to the domain of infectious diseases and established an uncertain SIS epidemic model with uncertain recovered rate γ. Replace γdt in the model (1) by $\tilde{γ} dt = γ dt + σ_{t} {dC}_{t}$ , where C _t is a Liu process defined in an uncertain space, and σ _t > 0 is intensity of Liu process. Then, an SIS model is gotten by uncertain differential equations (UDEs) as follows $\begin{matrix} {\begin{matrix} {dS}_{t} = [μ N - β S_{t} I_{t} + γ I_{t} - μ S_{t}] dt + σ_{t} I_{t} {dC}_{t}, \\ {dI}_{t} = [β S_{t} I_{t} - (μ + γ) I_{t}] dt - σ_{t} I_{t} {dC}_{t}, \end{matrix} \end{matrix}$ called an UDE SIS model. Further, they obtained solution, convergence properties of the UDE SIS model.

Consider the infection coefficient rate β of model (1) is a main way that people infected with disease. Similar to the above UDE model of Li et al. [23], this paper tries to establish an UDE SIS model with uncertain transmission rate and reveals relationships of deterministic, stochastic and uncertain models. The rest is organized as follows. In Section 2, we first review some basic concepts of uncertain theory. Then, an UDE SIS epidemic model with uncertain transmission rate is established. We provide analytic solution of the model and obtain threshold conditions by α-paths. Further, uncertain distributions of solution can be obtained by α-paths, too. In Section 3, we provide comparison of ODE, SDE and UDE SIS models and reveal relationship of them. In order to illustrate these results, numerical simulations are shown in Section 4 and a brief conclusion is in Section 5.

2 SIS epidemic model with uncertain infection coefficient

2.1 Uncertain theory

Some definitions about uncertain measure, uncertain variable, uncertain process, uncertain calculus and uncertain differential equation refer to Liu [15], Liu [24] and Yao [25].

Definition 1. [15] Let $L$ be a σ-algebra on a nonempty set Γ. A set function $M : L \to [0, 1]$ is called an uncertain measure if it is supposed to satisfy:

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

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

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ ₁, Λ ₂,…, $M {⋃_{i = 1}^{+ \infty} Λ_{i}} \leq \sum_{i = 1}^{+ \infty} M {Λ_{i}} .$

In this case, the triple $(Γ, L, M)$ is called an uncertainty space. The product uncertain measure on the product σ-algebra $L$ was defined by Liu [18].

Axiom 4. (Product Axiom) Let $(Γ_{i}, L_{i}, M_{i})$ be uncertainty spaces for i = 1, 2,…, then the product uncertain measure scripfontM is an uncertain measure satisfying $M {\prod_{i = 1}^{+ \infty} Λ_{i}} = ⋀_{i = 1}^{+ \infty} M_{i} {Λ_{i}},$ where Λ _i are arbitrarily chosen events from $L_{i}$ .

An uncertain variable ξ is a measurable function from uncertainty space $(Γ, L, M)$ to the real number set $ℝ$ , i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ|ξ (γ) ∈ B} is an event. The uncertainty distribution Φ of uncertain variable ξ is defined by Φ (x) = scripfontM {ξ ≤ x} for any real number x. If the inverse function Φ ^-1 exists and is unique for each α ∈ (0, 1), then it is called the inverse uncertainty distribution of ξ. In this case, the uncertainty distribution Φ is said to be regular. An uncertain variable ξ is called normal if it has a normal uncertainty distribution $Φ (x) = (1 + exp (\frac{π (e - x)}{\sqrt{3} σ}))^{- 1}, x \in ℝ,$ where e and σ are real numbers with σ > 0, denoted by ξ ∼ scripfontN (e, σ). If e = 0 and σ = 1, ξ is called standard normal, denoted by ξ ∼ scripfontN (0, 1).

Let T be an index set and $(Γ, L, M)$ be an uncertainty space. An uncertain process X _t is a measurable function from $T \times (Γ, L, M)$ 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, the uncertain variable X _t has the uncertainty distribution Φ _t (x).

Definition 2. [18] An uncertain process C _t is said to be a Liu process if

C ₀ = 0 and almost all sample paths are Lipschitz continuous.

C _t has stationary and independent increments.

Every increment C _t+s - C _s is a normal uncertain variable with an uncertainty distribution $Φ_{t} (x) = (1 + exp (- \frac{π x}{\sqrt{3} t}))^{- 1}, x \in ℝ .$

Let X _t be an uncertain process and C _t be a Liu process. For any partition of a closed interval [a, b] with a = t ₁ < t ₂ < ⋯ < t _k+1 = b, the mesh is written as $Δ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$ Then, the time integral of X _t with respect to t is $\int_{a}^{b} X_{t} dt = lim_{▵ \to 0} \sum_{i = 1}^{k} X_{t_{i}} \cdot (t_{i + 1} - t_{i}),$ provided that the limit exists almost surely and is finite. Liu integral of X _t is defined by $\int_{a}^{b} X_{t} {dC}_{t} = lim_{Δ \to 0} \sum_{i = 1}^{k} X_{t_{i}} (C_{t_{i + 1}} - C_{t_{i}})$ provided that the limit exists almost surely and is finite.

Definition 3. [16] Suppose C _t is a Liu process, f and g are two given functions. Then, ${dX}_{t} = f (t, X_{t}) dt + g (t, X_{t}) {dC}_{t}$ (5)is called an uncertain differential equation (UDE). An uncertain process X _t is called a solution of the equation (5), if X _t satisfies the equation for anytime t.

Definition 4. [17] for 0 < α < 1, α-path of an UDE (5) with initial value X ₀ is a deterministic function $X_{t}^{α}$ with respect to t that solves an 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., $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α} .$ (6)

Let X _t be an uncertain process. For each α ∈ (0, 1), there exists a real function $X_{t}^{α}$ such that $M {X_{t} \leq X_{t}^{α}, \forall t} = α, M {X_{t} > X_{t}^{α}, \forall t} = 1 - α .$ then, X _t is called a contour process and $X_{t}^{α}$ is called the α-path of the process X _t.

2.2 An UDE SIS epidemic model

For the model (1), suppose uncertain fluctuations in the environment will manifest themselves mainly as fluctuations in the infection coefficient β, so that βdt is replaced by βdt + σ _t dC _t, where C _t is Liu process with intensity σ _t > 0, instead of Wiener process B _t. Thus, an UDE SIS epidemic model with uncertain infection coefficient is established as follows

${\begin{matrix} {dS}_{t} & = & [μ N - β S_{t} I_{t} + γ I_{t} - μ S_{t}] dt \\ - σ_{t} S_{t} I_{t} {dC}_{t}, \\ {dI}_{t} & = & [β S_{t} I_{t} - (μ + γ) I_{t}] dt + σ_{t} S_{t} I_{t} {dC}_{t}, \end{matrix}$ (7)

In order to get solution of model (7), we will solve a special type of nonlinear uncertain differential equation and obtain the corresponding analytic solution.

Lemma 1. Let u _it, v _it be two continuous and differential functions at time t for i = 1, 2, and v _1t ≠ 0. Then, the UDE ${dX}_{t} = (u_{1 t} X_{t} + u_{2 t} X_{t}^{2}) dt + (v_{1 t} X_{t} + v_{2 t} X_{t}^{2}) {dC}_{t}$ has a solution $X_{t} = [Y_{t}^{- 1} Z_{t} - \frac{v_{2 t}}{v_{1 t}}]^{- 1},$ where $Y_{t} = exp (\int_{0}^{t} v_{1 s} {dC}_{s})$ and Z _t satisfies $\begin{matrix} {dZ}_{t} & = & Y_{t} (- u_{1 t} (Y_{t}^{- 1} Z_{t} - \frac{v_{2 t}}{v_{1 t}}) \\ - u_{2 t} + (\frac{v_{2 t}}{v_{1 t}})^{'}) dt \end{matrix}$ with $Z_{0} = \frac{1}{X_{0}} + \frac{v_{20}}{v_{10}}$ .

Proof. Based on Theorem 14.2 in Liu [15], we have $\begin{matrix} d ln X_{t} = (u_{1 t} + u_{2 t} X_{t}) dt + (v_{1 t} + v_{2 t} X_{t}) {dC}_{t} . \end{matrix}$

Denote ${\tilde{X}}_{t} = - ln X_{t}$ , i.e. $X_{t} = e^{- {\tilde{X}}_{t}}$ . Thus, $\begin{matrix} d {\tilde{X}}_{t} = - (u_{1 t} + u_{2 t} e^{- {\tilde{X}}_{t}}) dt - (v_{1 t} + v_{2 t} e^{- {\tilde{X}}_{t}}) {dC}_{t} . \end{matrix}$

Further, $\begin{matrix} {de}^{{\tilde{X}}_{t}} = - (u_{1 t} e^{{\tilde{X}}_{t}} + u_{2 t}) dt - (v_{1 t} e^{{\tilde{X}}_{t}} + v_{2 t}) {dC}_{t} . \end{matrix}$

By Theorem 1 of Liu (2012), a solution of the above equation is $e^{{\tilde{X}}_{t}} = Y_{t}^{- 1} Z_{t} - v_{2 t} / v_{1 t} .$ Therefore, $X_{t} = e^{- {\tilde{X}}_{t}} = [Y_{t}^{- 1} Z_{t} - \frac{v_{2 t}}{v_{1 t}}]^{- 1} .$ □

Based on Lemma 1, we obtain the solution of the uncertain model (7).

Theorem 1. The UDE SIS epidemic model (7) has a solution $\begin{matrix} S_{t} = \frac{N^{2} Z_{t 1}}{{NZ}_{t 1} + Y_{t 1}}, I_{t} = \frac{{NY}_{t 1}}{{NZ}_{t 1} + Y_{t 1}}, \end{matrix}$ where $Y_{t 1} = exp (N \int_{0}^{t} σ_{s} {dC}_{s})$ and Z _t satisfies ${dZ}_{t 1} = ((μ + γ - β N) Z_{t} + \frac{μ + γ}{N} Y_{t 1}) dt$ with $Z_{01} = \frac{1}{I_{0}} - \frac{1}{N} .$

Proof. For model (7), it is easy to get d (S _t + I _t) = μ (N - (S _t + I _t)) dt. Obviously, S _t + I _t → N as t→ + ∞. Without loss of generality, suppose S _t + I _t ≡ N. Thus, it is sufficient to study model (7) by the variable I _t and

$\begin{matrix} {dI}_{t} & = & [β (N - I_{t}) I_{t} - (μ + γ) I_{t}] dt \\ + σ_{t} (N - I_{t}) I_{t} {dC}_{t} . \end{matrix}$ (8)

Denote u _1t = βN - μ - γ, u _2t = - β, v _1t = σ _t N and v _2t = - σ _t. Then, for the UDE (8) we obtain $\begin{matrix} {dI}_{t} = (u_{1 t} I_{t} + u_{2 t} I_{t}^{2}) dt + (v_{1 t} I_{t} + v_{2 t} I_{t}^{2}) {dC}_{t} . \end{matrix}$

Based on Lemma 1 and S _t = N - I _t, solution of the model (7) follows. □

Yao [25] proved that solution of an UDE is a contour process. Thus, solutions S _t and I _t of UDE SIS model (7) are two contour processes. In order to obtain an uncertainty distribution of an UDE, we need to discuss α-path. For model (7), α-paths are denoted by $S_{t}^{α}$ and $I_{t}^{α}$ , respectively. Based on equation (8), α-path $I_{t}^{α}$ is to solve the ODE

$\begin{matrix} {dI}_{t}^{α} & = & {[β (N - I_{t}^{α}) I_{t}^{α} - (μ + γ) I_{t}^{α}] \\ + σ_{t} (N - I_{t}^{α}) I_{t}^{α} Φ^{- 1} (α)} dt, I_{0}^{α} = I_{0}, \end{matrix}$ (9)where Φ ^-1 (α) is defined in (6).

Remark 1. In Yao [25], an uncertain process X _t is a contour process if and only if for each α ∈ (0, 1), there exists α-path $X_{t}^{α}$ such that $M {X_{t} < X_{t}^{α}, \forall t} = α, M {X_{t} \geq X_{t}^{α}, \forall t} = 1 - α .$ Then, for solution (S _t, I _t) of UDE SIS model (7), we have $M {S_{t} < S_{t}^{α}, \forall t} = α, M {S_{t} \geq S_{t}^{α}, \forall t} = 1 - α$ and $M {I_{t} < I_{t}^{α}, \forall t} = α, M {I_{t} \geq I_{t}^{α}, \forall t} = 1 - α .$

Theorem 2. For the model (7), $S_{t}^{α} = N - I_{t}^{1 - α}$ or $S_{t}^{1 - α} = N - I_{t}^{α}$ , where 0 < α < 1.

Proof. From S _t = N - I _t, one has $\begin{matrix} {I_{t} \geq I_{t}^{1 - α}, \forall t} = {S_{t} \leq N - I_{t}^{1 - α}, \forall t}, \\ {I_{t} < I_{t}^{1 - α}, \forall t} = {S_{t} > N - I_{t}^{1 - α}, \forall t} . \end{matrix}$

Since I _t is a contour process, it is to get $\begin{matrix} M {S_{t} \leq N - I_{t}^{1 - α}, \forall t} & = & M {I_{t} \geq I_{t}^{1 - α}, \forall t} = α, \\ M {S_{t} > N - I_{t}^{1 - α}, \forall t} & = & M {I_{t} < I_{t}^{1 - α}, \forall t} \\ = & 1 - α . \end{matrix}$

Thus, $S_{t}^{α} = N - I_{t}^{1 - α}$ . □

Theorem 3. Let $R_{t}^{U} = R_{0}^{D} + \frac{σ_{t} N Φ^{- 1} (α)}{μ + γ}$ for any t.

(1) If $R_{t}^{U} \equiv 1$ , then α-paths of UDE model (7) are $\begin{matrix} {\begin{matrix} S_{t}^{α} = N - [β t + Φ^{- 1} (1 - α) \int_{0}^{t} σ_{s} ds + \frac{1}{I_{0}}]^{- 1}, \\ I_{t}^{α} = [β t + Φ^{- 1} (α) \int_{0}^{t} σ_{s} ds + \frac{1}{I_{0}}]^{- 1} . \end{matrix} \end{matrix}$

(2) If $R_{t}^{U} \neq 1$ , then α-paths of UDE model (7) are $\begin{matrix} {\begin{matrix} S_{t}^{α} = N - e^{- g (t, 1 - α)} \\ [\int_{0}^{t} (β + σ_{s} Φ^{- 1} (1 - α)) e^{- g (s, 1 - α)} ds + \frac{1}{I_{0}}]^{- 1}, \\ I_{t}^{α} = e^{- g (t, α)} [\int_{0}^{t} (β + σ_{s} Φ^{- 1} (α)) e^{- g (s, α)} ds + \frac{1}{I_{0}}]^{- 1}, \end{matrix} \end{matrix}$

where $g (t, α) = (μ + γ - β N) t - N Φ^{- 1} (α) \int_{0}^{t} σ_{u} du .$

Proof. (1) For convenience, for ODE (9), let $u_{t} = I_{t}^{α}$ . If $R_{t}^{U} \equiv 1$ , then $\begin{matrix} {du}_{t} = - (β + σ_{t} Φ^{- 1} (α)) u_{t}^{2} dt, u_{0} = I_{0} . \end{matrix}$

Evidently, $1 / u_{t} = \int_{0}^{t} (β + σ_{s} Φ^{- 1} (α)) ds + \frac{1}{I_{0}} .$ Thus, $I_{t}^{α} = [β t + Φ^{- 1} (α) \int_{0}^{t} σ_{s} ds + \frac{1}{I_{0}}]^{- 1} .$

Moreover, by Theorem 2, $\begin{matrix} S_{t}^{α} & = & N - I^{1 - α} \\ = & N - [β t + Φ^{- 1} (1 - α) \int_{0}^{t} σ_{s} ds + \frac{1}{I_{0}}]^{- 1} . \end{matrix}$

(2) If $R_{t}^{U} \neq 1$ , for ODE (9), $\begin{matrix} \frac{{du}_{t}}{dt} & = & [β (N - u_{t}) u_{t} - (μ + γ) u_{t}] \\ + σ_{t} (N - u_{t}) u_{t} Φ^{- 1} (α) \\ = & [β N - μ - γ + σ_{t} N Φ^{- 1} (α)] u_{t} \\ - [β + σ_{t} Φ^{- 1} (α)] u_{t}^{2}, \end{matrix}$ which corresponds to a Bernoulli differential equation. Together with the equation $S_{t}^{α} = N - I_{t}^{1 - α}$ , the results follow. □

Let Φ _{X
_t} (x) and $Φ_{X_{t}}^{- 1} (α)$ (0 < α < 1) be uncertainty distribution and inverse uncertainty distribution of contour process X _t, where X _t = S _t or I _t. By Theorem 2 in Yao [25], we have $Φ_{S_{t}}^{- 1} (α) = S_{t}^{α}, Φ_{I_{t}}^{- 1} (α) = I_{t}^{α} .$

Theorem 4. Let σ _t = σ > 0 and

$R_{0}^{U} = R_{0}^{D} + \frac{σ N Φ^{- 1} (α)}{μ + γ}$ (10)with 0 < α < 1. For UDE model (7), the following conclusions hold.

(1) If $R_{0}^{U} \leq 1$ , then $lim_{t \to + \infty} S_{t}^{α} = S^{0}, lim_{t \to + \infty} I_{t}^{α} = I^{0},$ where (S ⁰, I ⁰) = (N, 0) is the disease free equilibrium of model (1).

(2) If $R_{0}^{U} > 1$ , then $\begin{matrix} lim_{t \to + \infty} S_{t}^{α} = \frac{β S^{*}}{β + σ Φ^{- 1} (1 - α)}, \\ lim_{t \to + \infty} I_{t}^{α} = \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}, \end{matrix}$ where $(S^{*}, I^{*}) = (\frac{μ + γ}{β}, N - \frac{μ + γ}{β})$ is the endemic equilibrium of model (1).

Proof. Since σ _t = σ, we have $\int_{0}^{t} σ_{s} ds = σ t$ and $g (t, α) = (μ + γ - β N - σ N Φ^{- 1} (α)) t .$

(1) By Theorem 3, if $R_{0}^{U} = 1$ , then $\begin{matrix} lim_{t \to + \infty} S_{t}^{α} \\ = lim_{t \to + \infty} (N - [(β + σ Φ^{- 1} (1 - α)) t + \frac{1}{I_{0}}]^{- 1}) \\ = N, \\ lim_{t \to + \infty} I_{t}^{α} \\ = lim_{t \to + \infty} [(β + σ Φ^{- 1} (1 - α)) t + \frac{1}{I_{0}}]^{- 1} = 0 . \end{matrix}$

If $R_{0}^{U} < 1$ , i.e. μ + γ > βN + σNΦ ^-1 (α), we have $lim_{t \to + \infty} e^{g (t, α)} = + \infty$ . Thus, $\begin{matrix} lim_{t \to + \infty} I_{t}^{α} & = & lim_{t \to + \infty} [\frac{β + σ Φ^{- 1} (α)}{β N + σ N Φ^{- 1} (α) - μ - γ} \\ (1 - e^{g (t, α)}) + \frac{1}{I_{0}} e^{g (t, α)})]^{- 1} = 0, \\ lim_{t \to + \infty} S_{t}^{α} & = & lim_{t \to + \infty} (N - I_{t}^{1 - α}) = N . \end{matrix}$

(2) If $R_{0}^{U} > 1$ , i.e. μ + γ < βN + σNΦ ^-1 (α), then $lim_{t \to + \infty} e^{g (t, α)} = 0$ . Further, $\begin{matrix} lim_{t \to + \infty} I_{t}^{α} & = & \frac{β N + σ N Φ^{- 1} (α) - μ - γ}{β + σ Φ^{- 1} (α)} \\ = & \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}, \\ lim_{t \to + \infty} S_{t}^{α} & = & N - lim_{t \to + \infty} I_{t}^{1 - α} \\ = & \frac{β S^{*}}{β + σ Φ^{- 1} (1 - α)} . \end{matrix}$

3 Comparison of deterministic, stochastic and uncertain SIS models

In this section, we compare dynamic properties of deterministic, stochastic and uncertain SIS epidemic models and reveal relationship among three models. In ODE SIS model, the basic reproductive number $R_{0}^{D}$ determines persistence or extinction of disease. In SDE SIS model, probability of disease elimination ultimately approaches 1, dependent of the threshold value $R_{0}^{S}$ . For UDE SIS model, asymptotic properties of the α-paths are closely related to the number $R_{0}^{U}$ . Next we study the relationship between ODE and UDE SIS models, SDE and UDE SIS models.

3.1 The relationship of ODE and UDE SIS models

By the definition of Φ ^-1 (α) in (6), we have $[Φ^{- 1} (α)]_{α}^{'} = \frac{\sqrt{3}}{π} \frac{1}{α (1 - α)} > 0$ for 0 < α < 1, and $lim_{α \to 0 +} Φ^{- 1} (α) = - \infty, lim_{α \to 1 -} Φ^{- 1} (α) = + \infty .$

Thus, there only exists a unique value α ^* such that Φ ^-1 (α ^*) =0. Obviously, α ^* = 0.5. Further, if 0 < α < 0.5, then Φ ^-1 (α) <0; if 0.5 < α < 1, then Φ ^-1 (α) >0.

Since solution I _t of UDE model (7) is a contour process, it is easy to get $M {I_{t} \leq I_{t}^{α}, \forall t} = α, M {I_{t} > I_{t}^{α}, \forall t} = 1 - α .$

Next consider three cases of uncertain measure α:

(i) If α = 0.5, then $R_{0}^{U} = R_{0}^{D}$ because of Φ ^-1 (α) = Φ ^-1 (1 - α) =0. In this case, α-paths $S_{t}^{0.5}$ and $I_{t}^{0.5}$ of model (7) are equivalent to ODE SIS model (1). Thus, for model (7), $M {I_{t} \leq I_{t}^{0.5}, \forall t} = M {I_{t} > I_{t}^{0.5}, \forall t} = 0.5,$ which means $I_{t} \leq I_{t}^{0.5}$ or $I_{t} > I_{t}^{0.5}$ with uncertain measure value 0.5. When $R_{0}^{U} = R_{0}^{D} \leq 1,$ for model (1) the disease dies out and $M {I_{t} \leq 0} = M {I_{t} > 0} = 0.5,$ for model (7). Otherwise, the disease will be persistent in model (1) and $M {I_{t} \leq I^{*}} = M {I_{t} > I^{*}} = 0.5$ in model (7) as t→ + ∞.

(ii) If 0 < α < 0.5, then $R_{0}^{U} < R_{0}^{D}$ because of Φ ^-1 (α) <0. When $R_{0}^{D} \leq 1$ , it is to get $R_{0}^{U} < 1$ . Thus, $lim_{t \to + \infty} I_{t}^{α} = 0,$ that is, for model (7), $M {I_{t} \leq 0} = α < 0.5, M {I_{t} > 0} = 1 - α > 0.5$ as t→ + ∞. When $R_{0}^{U} > 1$ , one has $R_{0}^{D} > 1$ , which reflects $\begin{matrix} M {I_{t} \leq \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} = α < 0.5, \\ M {I_{t} > \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} = 1 - α > 0.5 \end{matrix}$ in model (7) as t→ + ∞.

(iii) If 0.5 < α < 1, then $R_{0}^{U} > R_{0}^{D}$ because of Φ ^-1 (α) >0. When $R_{0}^{D} > 1$ , it is to get $R_{0}^{U} > 1$ . Thus, $lim_{t \to + \infty} I_{t}^{α} = \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)},$ which shows $\begin{matrix} M {I_{t} \leq \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} = α > 0.5, \\ M {I_{t} > \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} = 1 - α < 0.5 \end{matrix}$ as t→ + ∞. When $R_{0}^{U} < 1$ , one has $R_{0}^{D} < 1$ , which reflects $M {I_{t} \leq 0} = α > 0.5, M {I_{t} > 0} = 1 - α < 0.5$ as t→ + ∞.

Remark 2. Based on the results (i), (ii) and (iii), we observe ODE SIS model directly has globally asymptotic stability according to $R_{0}^{D} \leq$ or >1. However, UDE SIS model has asymptotic properties with uncertain measure α or 1 - α, dependent of threshold value $R_{0}^{U} \leq 1$ or >1.

3.2 The relationship of SDE and UDE SIS models

By the definitions of $R_{0}^{U}$ and $R_{0}^{S}$ , let $f (α) = R_{0}^{U} - R_{0}^{S}$ for 0 < α < 1. Then, $f (α) = \frac{σ N}{μ + γ} [Φ^{- 1} (α) + \frac{σ N}{2}] .$

Since $f^{'} (α) = \frac{\sqrt{3} σ N}{π (μ + γ) α (1 - α)} > 0$ and $lim_{α \to 0 +} f (α) = - \infty, lim_{α \to 1 -} f (α) = + \infty,$ there exists a unique value, denoted by $α^{*} = 1 / (1 + e^{\frac{σ N π}{2 \sqrt{3}}})$ such that f (α ^*) =0. Obviously, α ^* < 0.5. For 0 < α < 1, we have three cases as follows.

(I) If α = α ^*, one has $R_{0}^{U} = R_{0}^{S}$ . Further, if $R_{0}^{U} \leq 1$ and σ ² ≤ β/N, the disease will die out with probability one for model (3). For model (7), $M {I_{t} \leq 0} = α^{*} < 0.5, M {I_{t} > 0} = 1 - α^{*} > 0.5$ as t→ + ∞. If $R_{0}^{U} > 1$ , the disease will be persistent for model (3) with probability one and $\begin{matrix} M {I_{t} \leq \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} = α^{*} < 0.5, \\ M {I_{t} > \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} = 1 - α^{*} > 0.5 \end{matrix}$ for model (7) as t→ + ∞.

(II) If 0 < α < α ^*, one has $R_{0}^{U} < R_{0}^{S}$ . When $R_{0}^{S} \leq 1$ and σ ² ≤ β/N, it is to get $R_{0}^{U} < 1$ . Thus, for model (3) the disease will die out with probability one, and $\begin{matrix} M {I_{t} \leq 0} = α < α^{*} < 0.5, \\ M {I_{t} > 0} = 1 - α > 1 - α^{*} > 0.5 \end{matrix}$ for model (7) as t→ + ∞. If $R_{0}^{U} > 1$ , one has $R_{0}^{S} > 1$ , which shows the disease is persistent with probability one and $\begin{matrix} M {I_{t} \leq \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} = α < α^{*} < 0.5, \\ M {I_{t} > \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} \\ = 1 - α > 1 - α^{*} > 0.5 \end{matrix}$ for model (7) as t→ + ∞.

(III) If α ^* < α < 1, one has $R_{0}^{U} > R_{0}^{S}$ . When $R_{0}^{S} > 1$ , it is to get $R_{0}^{U} > 1$ . Thus, $\begin{matrix} M {I_{t} \leq \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} = α > α^{*} > 0.5, \\ M {I_{t} > \frac{β I^{*} - σ N Φ^{- 1} (α)}{β + σ Φ^{- 1} (α)}} \\ = 1 - α < 1 - α^{*} < 0.5 \end{matrix}$ as t→ + ∞. If $R_{0}^{U} \leq 1$ , one has R ₀ < 1. Then, in model (3) the disease will die out when σ ² ≤ β/N, and $\begin{matrix} M {I_{t} \leq 0} = α > α^{*} > 0.5, \\ M {I_{t} > 0} = 1 - α < 1 - α^{*} < 0.5 \end{matrix}$ in model (7) as t→ + ∞.

Remark 3. Based on the results (I), (II) and (III), SDE SIS model (3) reflects the disease dies out or is persistent with probability one according to $R_{0}^{S} \leq 1$ or >1. This point is different from those of UDE model (7).

4 Numerical simulation

In order to reveal the same and different properties of ODE model (1), SDE model (3) and UDE model (7), we provide three cases according to threshold values $R_{0}^{D}$ , $R_{0}^{S}$ and $R_{0}^{U}$ . The discrete equations of the model (1) is described by

$\begin{matrix} {\begin{matrix} S_{n + 1} = S_{n} + [μ N - β S_{n} I_{n} + γ I_{n} - μ S_{n}] Δ t, \\ I_{n + 1} = I_{n} + [β S_{n} I_{n} + (μ + γ) I_{n}] Δ t . \end{matrix} \end{matrix}$ (11)

The discrete equations of the model (3) is shown by

$\begin{matrix} {\begin{matrix} S_{n + 1} = & S_{n} + [μ N - β S_{n} I_{n} + γ I_{n} - μ S_{n}] Δ t \\ - σ S_{n} I_{n} ε_{1 n} (Δ t)^{1 / 2} - σ^{2} S_{n} I_{n} \frac{(ε_{1 n}^{2} - 1)}{2} Δ t, \\ I_{n + 1} = & I_{n} + [β S_{n} I_{n} + (μ + γ) I_{n}] Δ t \\ + σ S_{n} I_{n} ε_{1 n} (Δ t)^{1 / 2} + σ^{2} S_{n} I_{n} \frac{(ε_{1 n}^{2} - 1)}{2} Δ t, \end{matrix} \end{matrix}$ (12)and that of model (7) is formed by

${\begin{matrix} S_{n + 1} = & S_{n} + [μ N - β S_{n} I_{n} + γ I_{n} - μ S_{n} \\ - σ S_{n} I_{n} ε_{2 n}] Δ t, \\ I_{n + 1} = & I_{n} + [β S_{n} I_{n} + (μ + γ) I_{n} \\ + σ S_{n} I_{n} ε_{2 n}] Δ t, \end{matrix}$ (13)where Δt = 0.002, ε _1n satisfies the normal distribution N (0, Δt), ε _2n satisfies the uncertain normal distribution scripfontN (0, Δt), and n = 1, 2, …, 3000.

Case 1: $R_{0}^{D} \leq 1$ and $R_{0}^{S} < 1$ .

Take N = 100, β = 0.00005, μ = 0.02, γ = 0.2, σ = 10^-3 and S ₀ = 80, I ₀=20. The trajectories of S _t and I _t in three models are shown in Fig. 1 by the discrete equations (11), (12) and (13).

Fig.1

Trajectories of the solutions S _t and I _t with β = 0.00005 and σ = 10^-3.

Taking the above parameters into the equation (2), it has $R_{0}^{D} = 0.0227 < 1 .$ Based on (A1), the disease free equilibrium (S ⁰, I ⁰) = (100, 0) is globally asymptotically stable for model (1). Similarly, by the equation (4), $R_{0}^{S} = 0 < 1$ with σ ² - β/N = 5 *10^-7 ≈ 0. Thus, for model (3) the disease will die out with probability one because of (B1).

On the other hand, by (10), values of $R_{0}^{U}$ is obtained for α = 0.1, 0.2, …, 0.9 in Table 1. Clearly, $R_{0}^{U} < 1$ . Thus, by Theorem 2.2 we have $lim_{t \to + \infty} S_{t}^{α} = 100$ and $lim_{t \to + \infty} I_{t}^{α} = 0$ for α = 0.1, 0.2, …, 0.9. Thus, the curve of model (7) is coincidence with those of models (1) and (3).

Table 1

The values of $R_{0}^{U}$ in Case 1

α	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$R_{0}^{U}$	-0.5279	-0.3247	-0.1896	-0.0789	0.0227	0.1243	0.2351	0.3701	0.5734

Case 2: $R_{0}^{D} > 1$ and $R_{0}^{S} < 1$ .

Take N = 100, β = 0.005, μ = 0.02, γ = 0.2, σ = 10^-1.8 and S ₀ = 80, I ₀=20. Based on the discrete Equations (11), (12), (13) and all given parameters, Fig. 2 reflects trajectories of S _t and I _t in these models.

Fig.2

Trajectories of the solutions S _t and I _t with β = 0.005 and σ = 10^-1.8.

Similar to Case 1, we have $R_{0}^{D} = 2.27 > 1$ . By (A2), the endemic equilibrium (44, 56) of model (1) is asymptotically stable, that is $lim_{t \to + \infty} S_{t} = 44, lim_{t \to + \infty} I_{t} = 56 .$

It reflects the disease will be persistent.

For model (3), $R_{0}^{S} = - 3.4361 < 1$ and σ ² - β/N = 2.0119 * 10^-4 ≈ 0. Thus, by (B1), I _t tends to zero exponentially as t→ ∞. Different from the result of model (1), the disease dies out with probability one.

Table 2 provides the values of $R_{0}^{U}$ when α = 0.1, 0.2, …, 0.9. If $R_{0}^{U} < 1$ , then $lim_{t \to + \infty} S_{t}^{α} = 100, lim_{t \to + \infty} I_{t}^{α} = 0 .$

Table 2

The values of $R_{0}^{U}$ in Case 2

α	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$R_{0}^{U}$	-6.4542	-3.2334	-1.0926	0.6623	2.2727	3.8832	5.6380	7.7788	10.9997

If $R_{0}^{U} > 1$ , then $\begin{matrix} lim_{t \to + \infty} S_{t}^{α} = \frac{0.22}{0.005 + 10^{- 1.8} Φ^{- 1} (1 - α)}, \\ lim_{t \to + \infty} I_{t}^{α} = \frac{0.28 - 0.1 Φ^{- 1} (α)}{0.005 + 10^{- 1.8} Φ^{- 1} (1 - α)} . \end{matrix}$ for α = 0.1, 0.2, …, 0.9. Thus, the curve of model (7) show irregular fluctuations, different from the deterministic and stochastic models.

Case 3: $R_{0}^{D} > 1$ and $R_{0}^{S} > 1$ .

Take N = 100, β = 0.006, μ = 0.02, γ = 0.2, σ = 10^-3 and S ₀ = 80, I ₀=20. Similar to Case 1 or Case 2, Fig. 3 reflects the trajectories fluctuation of models (1), (3) and (7).

Fig.3

Trajectories of the solutions S _t and I _t with β = 0.005 and σ = 10^-3.

For model (1), $R_{0}^{D} = 2.7273 > 1$ , which means the endemic equilibrium (37, 64) is globally asymptotically stable. For model (3), $R_{0}^{S} = 2.7045 > 1$ . By (B2), I _t will rise to or above the level ξ = 63.2206; that is, $lim_{t \to + \infty} sup I_{t} \geq 63.2206, lim_{t \to + \infty} inf I_{t} \leq 63.2206 .$

It reflects the disease is persistent. Comparing with model (1), trajectories fluctuation of model (3) is greater than that of (7), see Fig. 3.

For α = 0.1, 0.2, …, 0.9, Table 3 provides the values of $R_{0}^{U}$ for model (7). Since $R_{0}^{U} > 1$ , one has $\begin{matrix} lim_{t \to + \infty} S_{t}^{α} = \frac{0.22}{0.006 + 10^{- 3} Φ^{- 1} (1 - α)}, \\ lim_{t \to + \infty} I_{t}^{α} = \frac{0.38 - 0.1 Φ^{- 1} (α)}{0.006 + 10^{- 3} Φ^{- 1} (1 - α)} . \end{matrix}$ for α = 0.1, 0.2, …, 0.9. Thus, the curve of model (7) is coincided with that of model (1).

Table 3

The values of $R_{0}^{U}$ in Case 3

α	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$R_{0}^{U}$	2.1766	2.3799	2.5149	2.6257	2.7273	2.8289	2.9396	3.0747	3.2779

5 Conclusion

This paper applies uncertain differential equation to study SIS epidemic model. By relationship of S _t and I _t, solution of UDE SIS model and the corresponding α-paths are obtained, respectively. Based on α-paths, we obtain threshold conditions to reflect permanence and extinction of disease.

In order to reveal the relationships of deterministic, stochastic and uncertain model, we study the dynamic properties of these models by their threshold values $R_{0}^{D}$ , $R_{0}^{S}$ and $R_{0}^{U}$ when σ is a constant. If uncertain measure α = 0.5, α-pathes of model (7) are equivalent to model (1), which reveal the deterministic model is α-paths of uncertain model (7). If $R_{0}^{D} \leq$ or >1, then the deterministic model has globally asymptotic stability for disease free or endemic equilibria. However, the uncertain model has asymptotic properties with uncertain measure α or 1 - α according to $R_{0}^{U} \leq 1$ or >1. Denote $α^{*} = 1 / (1 + e^{\frac{σ N π}{2 \sqrt{3}}})$ . If α = α ^*, then $R_{0}^{U} = R_{0}^{S}$ . If $R_{0}^{S} \leq 1$ or >1, the disease will die out or be persistent with probability one in stochastic model. However, for the uncertain model the disease will die out or be persistent with uncertain measure α ^* or 1 - α ^* according to $R_{0}^{U} \leq 1$ or >1. Under the discrete equations and the fixed parameters of three models, some numerical simulations are provided to illustrate their same and different properties.

Besides the deterministic, stochastic and uncertain differential equations, there exist other differential equations such as fuzzy differential equations and fractional differential equations, see Nieto et al. [26], Guo (2009), Khastan et al. [27, 28], Cecconello [29] and Zhang et al. [30 –32]. How to apply these differential equations to investigate spread of an epidemic and reveal the relationships of epidemic models defined by these equations. This is an interesting, new and opening topic.

Footnotes

Acknowledgments

We thank the reviewers for their insightful comments which led to a significant improvement of the paper. This research was funded by the National Natural Science Foundation of China (Grant Nos. 11661076, 61563050) and the Science and Technology Project Foundation of Xinjiang Uygur Autonomous Region (Grant No. 2016D01C043).

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