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In this paper, a modified SIS (susceptible-infected-susceptible) model with infective medium and feedback mechanism on scale-free networks is presented. The model is suited to describe some epidemic spreading which are not only transmitting by medium but also spreading between individuals by direct contacts. Considering biological relevance and people’s subjective consciousness, we introduce medium and feedback to describe the epidemic spreading. By mathematical analysis, we obtain the epidemic threshold and equilibriums. Simulation shows that the medium parameter can change the threshold, and the bigger it is, the easier epidemic breaks. Feedback parameter cannot change the basic reproductive number, but it can reduce the endemic level and weaken the epidemic spreading.

As is known to all that epidemic diseases (plague, kala-azar, H_{7}N_{9}, SARS, etc.) always bring great harm to the stability of the social and threat individual health. What’s more, the rapid development of social and frequent contacts among people contribute to the spread of disease. So, there is account for much attention to study the spreading mechanism and stability for formulating appropriate policies to prevent and control of epidemic diseases [

However, most of researcher works mentioned above, the medium influences on epidemic is not considerate when epidemic spreading. Nevertheless, in the transmission of diseases, such as malaria, dengue fever, those diseases spreading not only by connects between individuals, but contacts between individual and medium. So medium can change the transmission mechanism in epidemic spread. In Ref [

The rest of the paper is organized as follows: Section 2 derives a SIS model with infective medium and feedback mechanism on scale-free networks. Then, Section 3 obtains two equilibriums and basic reproductive number. Section 4 numerical simulations are performed. Finally, we conclude the paper in Section 5.

In order to investigate the efficiency of the medium and feedback mechanism policy, we present an SIS epidemic model with infective medium and feedback mechanism on scale-free networks. In this paper, the epidemic spread model is defined as follows: there are two types of nodes in a network, one is composed of A individuals and the other is composed of B medium. All nodes can only exist in one of the two discrete states, susceptible (i.e. healthy) or infected. We only consider that epidemic diseases can spread between individuals on scale-free networks, individuals and medium spreading by a homogeneous mixing method. In the process of disease transmission, λ is the infection rate, an infected individual is cured and become susceptible again with probability δ . The feedback parameter is α , it is determined by the fear degree of people to the epidemic disease, b is the natural birth rate, and μ is the death rate. Here, we assume that the birth rate equals the death rate, i.e. b = μ . A healthy person is infected by an infected medium with probability γ 1 . A susceptible medium receiving the disease from an infected individual with probability γ 2 , the infected medium dies with probability β . It is also assumed that there is no infection spreading between medium. Let S k ( t ) and I k ( t ) be the relative densities of susceptible and infected nodes of degree k at time t respectively. The density of infected medium is V ( t ) . I ¯ ( t ) = ∑ i = 1 n P ( i ) I i ( t ) is the density of infected individuals at time t.

Based on the above hypotheses, the dynamical mean-field reaction rate equations can be written as

{ d S k ( t ) d t = b − ( 1 − α Θ ( t ) ) λ k Θ ( t ) S k ( t ) + δ I k ( t ) − μ S k ( t ) − γ 1 S k ( t ) V ( t ) d I k ( t ) d t = ( 1 − α Θ ( t ) ) λ k Θ ( t ) ( 1 − I k ( t ) ) − δ I k ( t ) − μ I k ( t ) + γ 1 S k ( t ) V ( t ) d V ( t ) d t = − β V ( t ) + γ 2 ( 1 − V ( t ) ) I ¯ ( t ) (1)

where the probability Θ ( t ) can be described a link pointing to an infected

node, which written as Θ ( t ) = 1 〈 k 〉 ∑ i = 1 n φ ( i ) P ( i ) I i ( t ) . where 〈 k 〉 is the average degree of the network, i.e. 〈 k 〉 = ∑ k = 1 n k P ( k ) , and φ ( i ) is the infectivity of node with degree i. Furthermore S ( t ) = ∑ k = 1 n P ( k ) S k , I ( t ) = ∑ k = 1 n P ( k ) I k are

the global average densities of the two states. In view of the practical consideration, we noted that P ( k ) > 0 for k = 1 , 2 , ⋯ , n . Those variables obey the normalization condition, i.e. S k ( t ) + I k ( t ) = 1 , and the initial conditions for the system (1) satisfy:

S k ( 0 ) = 1 − I k ( 0 ) > 0 , I k ( 0 ) ≥ 0 , and Θ ( 0 ) > 0 .

In this section, we reveal some properties of the solutions and the equilibria of system (1).

Theory 1. Consider system (1), Define

R 0 = λ [ ( β ( δ + μ ) − γ 1 γ 2 ) 〈 k φ ( k ) 〉 + γ 1 γ 2 〈 φ ( k ) 〉 〈 k 〉 ] ( δ + μ ) ( β ( δ + μ ) − γ 1 γ 2 ) 〈 k 〉

The system always exits a disease-free equilibria E 0 = ( 1 , 0 ) and a unique epidemic equilibrium E ∗ = ( S k ∗ , I k ∗ ) if R 0 > 1 .

Proof. It is easily finding that E 0 is always an equilibrium of system (1). To obtain the equilibrium E ∗ , we need to impose the right side of system (1) to be equal to zero. In other words, the equilibrium E ∗ = ( S k ∗ , I k ∗ ) should satisfy

{ b + δ I k ∗ ( t ) − ( 1 − α Θ ∗ ( t ) ) λ k Θ ∗ ( t ) S k ∗ ( t ) − μ S k ∗ ( t ) − γ 1 S k ∗ ( t ) V ∗ ( t ) = 0 − δ I k ∗ ( t ) − μ I k ∗ ( t ) + ( 1 − α Θ ∗ ( t ) ) λ k Θ ∗ ( t ) ( 1 − I k ∗ ( t ) ) + γ 1 ( 1 − I k ∗ ( t ) ) V ∗ ( t ) = 0 − β V ∗ ( t ) + γ 2 ( 1 − V ∗ ( t ) ) I ∗ ¯ ( t ) = 0 (2)

where Θ ∗ ( t ) = 1 〈 k 〉 ∑ k = 1 n φ ( k ) P ( k ) I k ∗ ( t ) . We find that V ∗ ( t ) = γ 2 I ∗ ¯ ( t ) β + γ 2 I ∗ ¯ ( t ) , S k ∗ ( t ) = ( δ + μ ) ( β + γ 2 I ∗ ¯ ( t ) ) ( 1 − α Θ ∗ ) λ k Θ ∗ + γ 1 γ 2 I ∗ ¯ ( t ) I k ∗ ( t )

Inserting Equation into Equation, we have

I k ∗ ( t ) = ( β + γ 2 I ∗ ¯ ( t ) ) λ k [ 1 − α Θ ∗ ( t ) ] Θ ∗ ( t ) + γ 1 γ 2 I ∗ ¯ ( t ) ( δ + μ ) ( β + γ 2 I ∗ ¯ ( t ) ) + ( β + γ 2 I ∗ ¯ ( t ) ) λ k [ 1 − α Θ ∗ ( t ) ] Θ ∗ ( t ) + γ 1 γ 2 I ∗ ¯ ( t ) (3)

Let

F ( I ∗ ¯ , Θ ∗ ) = ( β + γ 2 I ∗ ¯ ( t ) ) λ k [ 1 − α Θ ∗ ( t ) ] Θ ∗ ( t ) + γ 1 γ 2 I ∗ ¯ ( t )

G ( I ∗ ¯ , Θ ∗ ) = ( δ + μ ) ( β + γ 2 I ∗ ¯ ( t ) ) + ( β + γ 2 I ∗ ¯ ( t ) ) λ k [ 1 − α Θ ∗ ( t ) ] Θ ∗ ( t ) + γ 1 γ 2 I ∗ ¯ ( t )

Then I k ( t ) = F ( I ∗ ¯ , Θ ∗ ) G ( I ∗ ¯ , Θ ∗ ) .

Therefore, a self-consistency equation can be obtained

Θ ∗ ( t ) = 1 〈 k 〉 ∑ i = 1 n φ ( k ) P ( k ) F ( I ∗ ¯ , Θ ∗ ) G ( I ∗ ¯ , Θ ∗ ) = W ( Θ ∗ ) (4)

Obviously, when I k = 0 , so I ¯ = 0 , Θ ( t ) ( t ) = 0 , it is a zero solution of (3). If there exist another positive solution Θ ( t ) > 0 , it follows that

d W ( Θ ∗ ) d Θ ∗ | Θ ∗ = 0 > 1 .

From the following form of I ∗ ¯

I ∗ ¯ ( t ) = ∑ k = 1 n P ( k ) I k ∗ ( t ) = ∑ k = 1 n P ( k ) ( β + γ 2 I ∗ ¯ ( t ) ) λ k [ 1 − α Θ ∗ ( t ) ] Θ ∗ ( t ) + γ 1 γ 2 I ∗ ¯ ( t ) ( δ + μ ) ( β + γ 2 I ∗ ¯ ( t ) ) + ( β + γ 2 I ∗ ¯ ( t ) ) λ k [ 1 − α Θ ∗ ( t ) ] Θ ∗ ( t ) + γ 1 γ 2 I ∗ ¯ ( t )

We get that d I ∗ ¯ d Θ ∗ | Θ ∗ = 0 = β λ 〈 k 〉 β ( δ + μ ) − γ 1 γ 2 .

Let

F = ( β + γ 2 I ∗ ¯ ( t ) ) λ k [ 1 − α Θ ∗ ( t ) ] Θ ∗ ( t ) + γ 1 γ 2 I ∗ ¯ ( t )

G = ( δ + μ ) ( β + γ 2 I ∗ ¯ ( t ) ) + ( β + γ 2 I ∗ ¯ ( t ) ) λ k [ 1 − α Θ ∗ ( t ) ] Θ ∗ ( t ) + γ 1 γ 2 I ∗ ¯ ( t )

Then, we have computed that

d W ( Θ ) d Θ | Θ = 0 = 1 〈 k 〉 ∑ i = 1 n φ ( i ) P ( i ) ( ∂ F ∂ I ¯ ⋅ ∂ I ¯ ∂ Θ + ∂ F ∂ Θ ) G − ( ∂ G ∂ I ¯ ⋅ ∂ I ¯ ∂ Θ + ∂ G ∂ Θ ) F G 2 | Θ = 0 = λ [ ( β ( δ + μ ) − γ 1 γ 2 ) 〈 k φ ( k ) 〉 + γ 1 γ 2 〈 φ ( k ) 〉 〈 k 〉 ] ( δ + μ ) ( β ( δ + μ ) − γ 1 γ 2 ) 〈 k 〉 > 1

We can get the basic reproduction number R 0 = λ [ ( β ( δ + μ ) − γ 1 γ 2 ) 〈 k φ ( k ) 〉 + γ 1 γ 2 〈 φ ( k ) 〉 〈 k 〉 ] ( δ + μ ) ( β ( δ + μ ) − γ 1 γ 2 ) 〈 k 〉 .

The unique nontrivial solution exist when R 0 > 1 . Therefore, the epidemic threshold can be defined by:

λ c = ( δ + μ ) ( β ( δ + μ ) − γ 1 γ 2 ) 〈 k 〉 ( β ( δ + μ ) − γ 1 γ 2 ) 〈 k φ ( k ) 〉 + γ 1 γ 2 〈 φ ( k ) 〉 〈 k 〉 (5)

Inserting the nontrivial solution of (4) into (3), we can get I k ∗ . Then we can obtain 0 < S k ∗ < 1 , 0 < I k ∗ < 1 for k = 1 , 2 , ⋯ , n . So, there is a unique epidemic equilibrium E ∗ = ( S k ∗ , I k ∗ ) if R 0 > 1 . The proof is completed.

Remark

(1) The epidemic threshold λ c is obtained by Equation (5). The epidemic threshold depends on the fluctuations of the degree distribution and some model parameters. Interestingly, the feedback mechanism parameter α cannot change the epidemic threshold λ c .

(2) If δ = 1 , μ = 0 and, α = 0 then the system (1) becomes the SIS model with the influence of the infective vector. Then the epidemic threshold

λ c = ( 1 − γ 1 γ 2 ) 〈 k 〉 ( 1 − γ 1 γ 2 ) 〈 k φ ( k ) 〉 + γ 1 γ 2 〈 φ ( k ) 〉 〈 k 〉

which consists with Ref [

(3) If φ ( k ) = k , δ = 1 , μ = 0 and α = 0 then the system (1) becomes the SIS model with the influence of the infective medium. Then the epidemic threshold

λ c = ( 1 − γ 1 γ 2 ) 〈 k 〉 γ 1 γ 2 ( 〈 k 〉 2 − 〈 k 2 〉 ) + 〈 k 2 〉

which consists with Ref [

(4) If φ ( k ) = k , δ = 1 , μ = 0 , α = 0 and γ 1 γ 2 = 0 , then the system (1) becomes the standard SIS model without the influence of the infective medium. Then the epidemic threshold λ c = 〈 k 〉 / 〈 k 2 〉 , which consists with Ref [

(5) If φ ( k ) = k and γ 1 γ 2 = 0 then the system (1) becomes the standard SIS model with feedback mechanism, which consists with Ref [

From above we discussed, the simulation can show our result in detail. The disease-free equilibrium E 0 is stable when R 0 < 1 ; and the disease is permanent on the network when R 0 > 1 . The medium can influence the epidemic outbreak, feedback parameter can’t change the threshold but can weaken the epidemic outbreak.

For epidemic spreading with infected medium and feedback mechanism on scale-free network, we know that the stability of the disease-free equilibrium and locally equilibrium depend on threshold. In this section, we study by Runge- Kutta numerical simulations to discuss the effects of some parameters on the epidemic threshold. We consider the system (1) on a scale-free network with the degree distribution P ( k ) = ξ k − r where r = 3 and ∑ i = 1 n P ( i ) = 1 , n = 1000 . We set the initial density of infected nodes to be 0.9, like studies [

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To study the influence of the spread of the disease and the topology of the network, we proposed a modified SIS model on scale-free network with infected medium and feedback mechanism. By using some mathematical analytic method, we found the basic reproduction number R 0 and equilibriums. Theoretical results indicate that the basic reproduction number is dependent on the topology of the underlying networks. The simulation has shown that the disease-free equilibrium E 0 is stable when R 0 < 1 ; and the disease is permanent on the network when R 0 > 1 . Here the rigorous proof that we could not complete will be discussed in the future. In addition, we added the medium into the normal SIS model, it shows medium contributes to epidemic spreading. As for feedback mechanism, we take note of that epidemic threshold bears no relation to the feedback parameter, but formulate shows that it can accelerate the extinction of the disease and weaken the endemic level, which complies with our realistic sense. The study may give us valuable guiding in effectively predicting and preventing epidemic spreading.

This work is supported by the National Natural Science Foundation of China under Grant 61672112 and Project in Hubei province department of education under Grant B2016036.

Liu, X.D., Li, T., Wang, Y.M., Wan, C. and Dong, J. (2017) An SIS Epidemic Model with Infective Medium and Feedback Mechanism on Scale-Free Networks. Open Access Library Journal, 4: e3598. https://doi.org/10.4236/oalib.1103598