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The article investigates a SIQR epidemic model with specific nonlinear incidence rate and stochastic model based on the former, respectively. For deterministic model, we study the existence and stability of the equilibrium points by controlling threshold parameter
*R*
_{0} which determines whether the disease disappears or prevails. Then by using Routh-Hurwitz criteria and constructing suitable Lyapunov function, we get that the disease-free equilibrium is globally asymptotically stable if
*R*
_{0}
<1 or unstable if
*R*
_{0}>1. In addition, the endemic equilibrium point is globally asymptotically stable in certain region when
*R*
_{0}>1. For the corresponding stochastic model, the existence and uniqueness of the global positive solution are discussed and some sufficient conditions for the extinction of the disease and the persistence in the mean are established by defining its related stochastic threshold
*R*
_{0}^{s}. Moreover, our analytical results show that the introduction of random fluctuations can suppress disease outbreak. And numerical simulations are used to confirm the theoretical results.

Infectious diseases have always been a thorny issue that endangers human health, triggers social unrest and even affects national stability. Therefore, it is of great significance to take effective prevention measures to control the epidemic by establishing mathematical models with typical characteristics, discovering the transmission and development trends of infectious diseases. In the last decades, many authors have made a great headway on SIR (Susceptible-Infected-Removed) epidemic models [

In order to realistically reflect the process of human-to-human disease transmission, it is very important to determine the specific form of the incidence function which describes the increased number of infected people per unit time and plays a vital role in epidemiological dynamics research. Due to the complexity of disease transmission in real life, many scholars admit that the nonlinear incidence function is more reasonable than the bilinear incidence and standard incidence. The specific nonlinear incidence β S ( t ) I ( t ) f ( S ( t ) , I ( t ) ) was proposed in 2013 [

In this paper, to improve and generalize the model of Joshi et al. [

{ d S ( t ) d t = A − μ S ( t ) − β S ( t ) I ( t ) f ( S ( t ) , I ( t ) ) , d I ( t ) d t = β S ( t ) I ( t ) f ( S ( t ) , I ( t ) ) − ( δ + γ + μ + μ 1 ) I ( t ) , d Q ( t ) d t = δ I ( t ) − ( ρ + μ + μ 2 ) Q ( t ) , d R ( t ) d t = γ I ( t ) + ρ Q ( t ) − μ R ( t ) . (1)

The total population N ( t ) is divided into four compartments and N ( t ) = S ( t ) + I ( t ) + Q ( t ) + R ( t ) , where S ( t ) , I ( t ) , Q ( t ) and R ( t ) are the number of the susceptible, infected, quarantined and recovered individuals at time t, respectively. The parameter constants have the following biological meanings: A is the recruitment rate of the susceptible through birth and immigration; μ is the natural death rate of the population; μ 1 is the disease-caused mortality of infective individuals; μ 2 is the disease-caused mortality of quarantined individuals; β represents contact rate of an infected person with other compartment members per unit time; δ is the isolation rate of the compartment I quarantined directly to enter Q; γ is the recovery rate of infected individuals; ρ is the recovery rate of quarantined individuals. In addition, all parameters of model (1) are supposed to be nonnegative constants. Especially, A and μ are positive constants.

In fact, any system is more or less affected by environmental factors. Stochastic models can predict the future dynamics of the system accurately compared to their corresponding deterministic models. Therefore, when establishing population model, many stochastic biological systems and stochastic epidemic models have been presented and studied [

{ d S ( t ) = ( A − μ S ( t ) − β S ( t ) I ( t ) f ( S ( t ) , I ( t ) ) ) d t − σ 3 S ( t ) I ( t ) f ( S ( t ) , I ( t ) ) d B 3 ( t ) , d I ( t ) = ( β S ( t ) I ( t ) f ( S ( t ) , I ( t ) ) − ( δ + γ + μ + μ 1 ) I ( t ) ) d t − σ 1 I ( t ) d B 1 ( t ) + σ 3 S ( t ) I ( t ) f ( S ( t ) , I ( t ) ) d B 3 ( t ) , d Q ( t ) = ( δ I ( t ) − ( ρ + μ + μ 2 ) Q ( t ) ) d t − σ 2 Q ( t ) d B 2 ( t ) , d R ( t ) = ( γ I ( t ) + ρ Q ( t ) − μ R ( t ) ) d t . (2)

This paper is organized as follows. In Section 2, we present some preliminaries which will be used in our following analysis. In Section 3, the existence and stability of the equilibrium points of deterministic system is analyzed. In Section 4, we study dynamics of the stochastic model. Firstly, the existence and uniqueness of the global positive solution is proved. Then, the extinction and persistence of the disease under certain conditions is discussed. Finally, numerical simulations are presented to illustrate our main results. Section 5 just provides a brief discussion and the summary.

In this section, some notations, definitions and lemmas are provided to prove our main results. Let ( Ω , F , ℙ ) be a complete probability space with a filtration { F t } t ≥ 0 satisfying the usual conditions (i.e. it is increasing and right continuous while F 0 contains all ℙ -null sets). And B i ( t ) ( i = 1 , 2 , 3 ) are defined on this complete probability space.

Consider the 4-dimensional stochastic differential equation

d x ( t ) = f ( x ( t ) , t ) d t + g ( x ( t ) , t ) d B ( t ) for t ≥ t 0 , (3)

with initial value x 0 ∈ ℝ + 4 . We define the differential operator L of Equation (3) as follows:

L = ∂ ∂ t + ∑ i = 1 4 f i ( x , t ) ∂ ∂ x i + 1 2 ∑ i , j = 1 4 [ g T ( x , t ) g ( x , t ) ] i j ∂ 2 ∂ x i ∂ x j .

Let L act on a nonnegative function V ( x , t ) ∈ C 2,1 ( ℝ + 4 × [ t 0 , ∞ ) ; ℝ + ) . Then,

L V ( x , t ) = V t ( x , t ) + V x ( x , t ) f ( x , t ) + 1 2 t r a c e [ g T ( x , t ) V x x ( x , t ) g ( x , t ) ] ,

where ℝ + 4 = { x i > 0 , i = 1 , 2 , 3 , 4 } . By Itô’s formula, d V ( x , t ) = L V ( x , t ) d t + V x ( x , t ) g ( x , t ) d B ( t ) . For an integrable function χ on [ 0, + ∞ ) , we define

〈 χ ( t ) 〉 = 1 t ∫ 0 t χ ( s ) d s .

Definition 2.1 System (2) is said to be persistent in the mean if

lim inf t → ∞ 〈 I ( t ) 〉 > 0 a . s ..

Moreover, we need the following lemma (see Lemma 5.1 in [

Lemma 2.2. Let g ∈ C ( ℝ + × Ω , ℝ ) and G ∈ C ( ℝ + × Ω , ℝ ) . If there exist two real numbers λ 0 ≥ 0 and λ > 0 for all t ≥ 0 , such that

ln g ( t ) ≥ λ 0 t − λ ∫ 0 t g ( s ) d s + G ( t ) and lim t → ∞ G ( t ) t = 0 a . s . ,

then

lim inf t → ∞ 〈 g ( t ) 〉 ≥ λ 0 λ a . s ..

Lemma 2.3. Consider the following two systems

d x d t = f ( t , x ) , d y d t = g ( y ) ,

where x , y ∈ ℝ n , f and g are continuous, satisfy local Lipschitz conditions in any compact set X ⊂ ℝ n , and f ( t , x ) → g ( x ) as t → + ∞ , so that the second system is the limit system for the first system. Let Φ ( t , t 0 , x 0 ) and φ ( t , t 0 , y 0 ) be solutions of these systems, respectively. Suppose that e ∈ X is a locally asymptotically stable equilibrium of the limit system and its attractive region is

W ( e ) = { y ∈ X | φ ( t , t 0 , y ) → e , t → + ∞ } .

Let W Φ be the omega limit set of Φ ( t , t 0 , x 0 ) . If W Φ ∩ W ( e ) ≠ ∅ , then lim t → + ∞ Φ ( t , t 0 , x 0 ) = e .

For a population dynamics system, studying its equilibrium points is the precondition for predicting the development trend of populations within the system.

Theorem 3.1 System (1) has two equilibrium points, E 0 = ( A μ , 0 , 0 , 0 ) for all parameter values and E * = ( S * , I * , Q * , R * ) for R 0 > 1 , here S * ∈ ( 0, A μ ) , I * , Q * and R * > 0 .

Proof. Summing up all the equations of model (1), we find the following differential equation: d N d t = A − μ N − μ 1 I − μ 2 Q . By comparison theorem, we obtain that the solutions of model (1) exist in the region defined by Γ = { ( S , I , Q , R ) ∈ ℝ + 4 : S + I + Q + R ≤ A μ , S ≥ 0 , I ≥ 0 , Q ≥ 0 , R ≥ 0 } . To get the equilibrium points, we set the right-side of equations to be 0,

{ A − μ S − β S I f ( S , I ) = 0 , β S I f ( S , I ) − ( δ + γ + μ + μ 1 ) I = 0 , δ I − ( ρ + μ + μ 2 ) Q = 0 , γ I + ρ Q − μ R = 0 , (4)

which yields

I = A − μ S δ + γ + μ + μ 1 , Q = δ ρ + μ + μ 2 I , R = ( γ μ + ρ δ μ ( ρ + μ + μ 2 ) ) I ,

β S f ( S , A − μ S δ + γ + μ + μ 1 ) = δ + γ + μ + μ 1 .

If I = 0 , the model (1) has a disease-free equilibrium E 0 = ( A μ , 0 , 0 , 0 ) for all parameter values. And we can get the basic reproduction number R 0 = β A ( μ + α 1 A ) ( δ + γ + μ + μ 1 ) by using next generation method. The value R 0 represents the average number of secondary infections when an infected person enters fully susceptible population. If I ≠ 0 , I = A − μ S δ + γ + μ + μ 1 > 0 implies S < A μ . Hence, there is no positive equilibrium point if S > A μ . Now, we consider the function g ( S ) defined on the interval [ 0, A μ ] , where

g ( S ) = β S f ( S , A − μ S δ + γ + μ + μ 1 ) − ( δ + γ + μ + μ 1 ) ≜ h ( S , I ) − ( δ + γ + μ + μ 1 ) .

Obviously, g ( 0 ) = − ( δ + γ + μ + μ 1 ) < 0 and g ( A μ ) = β A μ + α 1 A − ( δ + γ + μ + μ 1 ) = ( δ + γ + μ + μ 1 ) ( R 0 − 1 ) > 0 when R 0 > 1 . Simultaneously, differentiating the function g, we gain g ′ ( S ) = ∂ h ∂ S − μ δ + γ + μ + μ 1 ∂ h ∂ I > 0 . Because g ( S ) is monotonically increasing in the interval [ 0, A μ ] , g ( 0 ) < 0 and g ( A μ ) > 0 , the equation g ( S ) = 0 has only one positive root by the zero theorem. That is, there exists a unique endemic equilibrium E * = ( S * , I * , Q * , R * ) with S * ∈ ( 0, A μ ) .

In the biological sense, we analyze the stability of the disease-free equilibrium point and the endemic equilibrium point.

Theorem 3.2. The disease-free equilibrium E 0 of system (1) is globally asymptotically stable if R 0 < 1 and unstable if R 0 > 1 .

Proof. Consider the Jacobian matrix of system (1) at E 0

J ( E 0 ) = ( − μ − β A μ + α 1 A 0 0 0 β A μ + α 1 A − ( δ + γ + μ + μ 1 ) 0 0 0 δ − ( ρ + μ + μ 2 ) 0 0 γ ρ − μ ) .

The characteristic equation of system (1) at E 0 is

( λ + μ ) 2 [ λ + ( ρ + μ + μ 2 ) ] [ λ − β A μ + α 1 A + ( δ + γ + μ + μ 1 ) ] = 0.

Clearly, λ 1 , 2 = − μ < 0 , λ 3 = − ( ρ + μ + μ 2 ) < 0 and the positive and negative of the fourth eigenvalue depends on R 0 . That is, λ 4 = β A μ + α 1 A − ( δ + γ + μ + μ 1 ) = ( δ + γ + μ + μ 1 ) ( R 0 − 1 ) < 0 when R 0 < 1 , λ 4 > 0 when R 0 > 1 . Hence the disease-free equilibrium E 0 is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1 .

Then we prove the global stability of the system (1) at the equilibrium E 0 when R 0 < 1 . Taking the Lyapunov function W 1 ( t ) = I ( t ) into consideration, we get

W ˙ 1 = ( β S f ( S , I ) − ( δ + γ + μ + μ 1 ) ) I ≤ ( β A μ + α 1 A − ( δ + γ + μ + μ 1 ) ) I = ( δ + γ + μ + μ 1 ) ( R 0 − 1 ) I ≤ 0.

Thus if R 0 < 1 , W ˙ 1 ≤ 0 . And W ˙ 1 = 0 if and only if I = 0 . In this case, d S d t = A − μ S indicates S → A μ as t → ∞ . Similarly, Q → 0 and R → 0 as t → ∞ . So the largest positive invariant set in { ( S , I , Q , R ) ∈ Γ : W ˙ 1 = 0 } is the singleton E 0 . By Liapunov-Lasalle theorem, E 0 = ( A μ , 0 , 0 , 0 ) is globally asymptotically stable in Γ .

Theorem 3.3. If R 0 > 1 , the endemic equilibrium point E * of the system (1) is globally asymptotically stable in the region Ω = Γ − { ( S , I , Q , R ) ∈ Γ : I = 0 } .

Proof. Consider the Jacobian matrix of system (1) at E *

J ( E * ) = ( − μ − β I * + α 2 β I * 2 f 2 ( S * , I * ) − β S * + α 1 β S * 2 f 2 ( S * , I * ) 0 0 β I * + α 2 β I * 2 f 2 ( S * , I * ) β S * + α 1 β S * 2 f 2 ( S * , I * ) − ( δ + γ + μ + μ 1 ) 0 0 0 δ − ( ρ + μ + μ 2 ) 0 0 γ ρ − μ ) .

Let C 1 = β I * + α 2 β I * 2 f 2 ( S * , I * ) and C 2 = β S * + α 1 β S * 2 f 2 ( S * , I * ) , then

J ( E * ) = ( − μ − C 1 − C 2 0 0 C 1 C 2 − ( δ + γ + μ + μ 1 ) 0 0 0 δ − ( ρ + μ + μ 2 ) 0 0 γ ρ − μ ) .

Therefore the characteristic equation of system (1) at E * is

( λ + μ ) [ λ + ( ρ + μ + μ 2 ) ] { ( λ + μ + C 1 ) [ λ + ( δ + γ + μ + μ 1 ) − C 2 ] + C 1 C 2 } = 0.

Obviously, λ 1 = − μ < 0 , λ 2 = − ( ρ + μ + μ 2 ) < 0 and the other two eigenvalues are determined by the following quadratic equation

λ 2 + [ C 1 + μ + ( δ + γ + μ + μ 1 ) − C 2 ] λ + ( C 1 + μ ) [ ( δ + γ + μ + μ 1 ) − C 2 ] + C 1 C 2 = 0

⇒ λ 2 + a 1 λ + a 2 = 0 ,

where

a 1 = C 1 + μ + [ ( δ + γ + μ + μ 1 ) − C 2 ] , a 2 = C 1 ( δ + γ + μ + μ 1 ) + μ [ ( δ + γ + μ + μ 1 ) − C 2 ] .

By utilizing Routh-Hurwitz criteria, we know that the system is stable if a 1 , a 2 > 0 and unstable if a 1 , a 2 < 0 . From the second equation of (4), we obtain ( δ + γ + μ + μ 1 ) > C 2 , thus all eigenvalues have negative real parts. The endemic equilibrium E * is locally asymptotically stable.

Now we confirm the global stability at the equilibrium E * when R 0 > 1 . The first two equations of system (1) do not contain Q and R, so we consider the following Lyapunov function in the positive quadrant of the two-dimensional plane SI.

W 2 ( t ) = S − S * − ∫ S * S l ( S * , I * ) l ( x , I * ) d x + I * Ψ ( I I * ) ,

where l ( S , I ) = β S f ( S , I ) , Ψ ( x ) = x − 1 − ln x , x > 0 . Clearly,

Due to

we have

In order to study the dynamics of stochastic models, the primary question to be considered is whether the solution is global and nonnegative existence. Although the coefficients of the model (2) satisfy the local Lipschitz condition, it’s not enough to prove that the solution does not explode within a finite time for any given initial value. Hence in this section, we will show that the solution of model (2) is positive and global.

Theorem 4.1. For any given initial value

Proof. Since the system (2) has locally Lipschitz continuous coefficients, then for any initial value

Set

Define a

Applying Itô’s formula, for all

where

Since

Hence,

From the definition of

Letting

which is a contradiction and we confirmed

Remark 4.2. The region

One of the most concerning issues in epidemiology is how to establish the threshold condition for the extinction and persistence of the disease. The target of this section is to study the extinction and persistence of the disease. First of all, we define corresponding random threshold as follows:

Theorem 4.3. Let

1) If

2) If

which means that

Proof. Define Lyapunov function

where

Suppose 1) holds. Noting that

Then,

Integrating both sides of the above inequality from 0 to t and dividing by t, we obtain

where

We obtain the desired assertion (6).

If 2) holds, from Equation (9), we get

Since

We obtain the desired assertion (7). And so

From the first two equations of system (2), there is

Integrating both sides of (13) from 0 to t and dividing by t, we have

Therefore,

where

Therefore the assertion (8) holds. The conclusion is proven.

Next, the conditions for the persistence of the disease are presented.

Theorem 4.4. Suppose that

where

Proof. Since

Then

Integrating both sides of (19) from 0 to t, there is

From (14), we have

where

This is the required inequality (15), and from (14), we have

Therefore,

the inequality (16) is valid. From the third equation of system (2), we have

Then,

where

The last equation of system (2) gives

Then,

So we have

This is the required inequality (18).

In this section, we numerically simulate solutions of the models by using the Milstein’s method [

where

Example 4.3.1 For the deterministic system (1), we choose the initial value

Example 4.3.2 For the stochastic system (2), we choose the initial value

Example 4.3.3 Now, we reselect the parameters

In this work, we investigate the deterministic and stochastic SIQR epidemic models with the specific nonlinear incidence. This incidence rate can become multiple types, and is more abundant than saturation incidence. We obtain the

dynamics properties of the SIQR model based on two threshold parameters

In future work, we will further consider the delayed SIQR model with this incidence and the SIQS model without permanent immunity.

The authors declare no conflicts of interest regarding the publication of this paper.

Xu, J. and Zhang, T.S. (2019) Dynamic Analysis for a SIQR Epidemic Model with Specific Nonlinear Incidence Rate. Journal of Applied Mathematics and Physics, 7, 1840-1860. https://doi.org/10.4236/jamp.2019.78126