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How the individuals’ risk aversion affect the epidemic spreading
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How the individuals’ risk aversion affect the epidemic spreading Dun Han a,c,∗, Qi Shao a, Dandan Li b, Mei Sun a a
Institute of Applied Systems Analysis, Jiangsu University, Zhenjiang, Jiangsu 212013, People’s Republic of China School of Management, Jiangsu University Zhenjiang, Jiangsu 212013, People’s Republic of China c The Adaptive Networks and Control Laboratory, Department of Electronic Engineering, and the Research Center of Smart Networks and Systems, School of Information Science and Engineering, Fudan University, Shanghai 200433, People’s Republic of China b
a r t i c l e
i n f o
Article history: Received 10 August 2019 Revised 15 October 2019 Accepted 27 October 2019 Available online xxx Keywords: Epidemic spreading Individuals’ sensitivity Asymptotically stable Network
a b s t r a c t Some diseases can live in people for many years without making them sick, during this time, the bacteria can spread to others who come in contact with the infected person. However, explosive individuals in infected people will exhibit certain dominant states associated with infectious diseases, as a results, the uninfected persons will avoid contact with such individuals with dominant infectious diseases. Considering the individual’s ability to avoid risks in the epidemic season, we propose the epidemic spreading model with individuals’ sensitivity. The epidemic spreading threshold is calculated by means of the mean-field theory and the next-generation matrix method. In addition, the locally, globally and exponential asymptotically stable conditions in the disease-free equilibrium state are given. Finally, we simulate the proposed epidemic spreading modeling in the ER random network and the BA scale-free network. The numerical simulations results show that the probability of the latent individuals transforming into explosive individuals has a greater impact on the spread of infectious diseases. Meanwhile, self-protection is an effective measure to reduce the outbreak of infectious diseases. © 2019 Elsevier Inc. All rights reserved.
1. Introduction There has been a long struggle between human society and infectious diseases, research on the occurrence and development of infectious diseases, the spreading mechanism of infectious diseases and the corresponding prevention and control measures has always been the focus of scientific research. Various infectious diseases, such as SARS, Ebola, plague, influenza, plague, AIDS, cholera and avian influenza, have affected many parts of the world and claimed the lives of thousands of people [1–7]. Infectious diseases are mainly transmitted through contact between individuals, so the contact network can more realistically depict the transmission process of infectious diseases [8,9]. Scholars have adopted a variety of theoretical methods to study the spread of infectious diseases, such as percolation theory, mean field theory, game theory, stochastic processes and cellular automata [10–13]. With the discovery and establishment of small world networks and scale-free networks in real systems, the research on the transmission dynamics of infectious diseases on complex networks has developed rapidly [14–17]. Many models of infectious disease transmission based on complex networks have been proposed by scholars, among which the most three classical models are: (1) the mean field model proposed by Pastor-Satorras and Vespignani ∗
Corresponding author at: Institute of Applied Systems Analysis, Jiangsu University, Zhenjiang, Jiangsu 212013, People’s Republic of China. E-mail addresses: [email protected] (D. Han), [email protected] (M. Sun).
https://doi.org/10.1016/j.amc.2019.124894 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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[18]; (2) the percolation model proposed by Newman and Watts [19]; (3) the discrete probability model proposed by Wang et al. [20]. Although immunization is a very effective measure to control the spread of infectious diseases [21,22], human awareness plays an important role in the spread of infectious diseases and the control of propagation patterns. Faced with the complicated social environment, it shows that, human behavior is extremely complex. With the awareness of the epidemic spreading in their surroundings, healthy individuals may tend to take certain actions (such as wearing protective masks, washing hands frequently, staying home, etc.), which generally help reduce their risks of getting infected [23,24]. However, those who do not pay attention to protecting themselves will suffer a higher risk of infection than those who are diligent in protecting themselves. A conclusion is that risk awareness can suppress epidemic spreading [25]. Clara et al. studied the dynamical interplay between awareness and epidemic spreading in multiplex networks, they claimed that the critical point for the onset of the epidemics has a critical value defined by the awareness dynamics and the topology of the virtual network [26]. Sun and co-workers explored the epidemic spread in bipartite network by considering risk awareness. They results showed that, the final infection density is negative linear with the value of individuals!‘ risk awareness [27]. Wu and Xiao analyzed the effects of awareness on epidemic spreading, they obtained that only epidemic information, being available in network nodes’ neighborhood overlapping in two layers, helps change the epidemic threshold [28]. Juang and Liang studied the impact of vaccine success and awareness on epidemic dynamics. They results highlighted the important and crucial roles of both vaccine success and contact infection awareness on epidemic dynamics [29]. Although there are many studies on the spread of infectious diseases [30,31], with the extreme complexity of individual behavior [32,33], it is still a problem worth to study how to integrate individual risk-avoidance behaviors into the spread of infectious diseases. Antibiotic-resistant bacteria like the Methicillin-resistant Staphylococcus aureus (MRSA) can live in people for many years without making them sick. During this time, the bacteria can spread to others who come in contact with the MRSA-infected person. It should be noted that these latent-infected people have a strong ability to spread, the spread ability of individuals with some outbreaks or those who are labeled infected is reduced. Whereas, previous studies on epidemic spreading rarely focused on the those effects. Therefore, in this paper we assumed that only the latent individuals have the ability to spread, while explosive individuals have no ability to transmit. Meanwhile, we showed the epidemic spreading modeling with individuals’ awareness, and we presented the threshold of infectious disease transmission and the conditions of the locally, globally and exponential asymptotically stable. In addition, we have performed our proposed model in the ER random network and the BA scale-free network. The results indicated that the probability of the latent individuals transforming into explosive individuals has a greater impact on the spread of infectious diseases, and individuals strengthen self-protection. It is an effective measure to reduce the outbreak of infectious diseases. The rest of the paper is organized as follows. The model and theoretical analysis are presented in Section 2. Numerical simulations are presented to illustrate the behaviors of cooperative spreading processes in Section 3. Finally, conclusions are given in Section 4. 2. Infectious disease transmission model 2.1. Preliminaries and model formulations Antibiotic-resistant bacteria like the Methicillin-resistant Staphylococcus aureus (MRSA) can live in people for many years without making them sick. During this time, the bacteria can spread to others who come in contact with the MRSA-infected person. During the epidemic spreading season, some individuals prefer to strengthen self-protection, while others have less self-protection awareness, which leads to the different of being infected. We divide the total individuals into four groups: (i) the individuals without self-protection (S); (ii) the individuals with self-protection awareness (Sa ); (iii) the latent individuals (E); (iv) and the explosive (outbreak) individuals (I). S-state individuals and Sa -state individuals are collectively referred to as uninfected individuals, and E-state individuals and I-state individuals are collectively referred to as infected individuals. Since outbreak individuals in infected people will exhibit certain dominant states associated with infectious diseases, such as sneezing and fever, uninfected persons will avoid contact with such individuals with dominant infectious diseases. Therefore, we assume that only latent individuals can infect uninfected individuals. In this paper, d is used to indicate mortality; we suppose that a new individual is either an S-state individual or an Sa -state individual, α 1 and α 2 represent the birth rate of S-state individuals and Sa -state individuals, respectively. The total birth rate is equal to the mortality rate, i.e., α1 + α2 = d. μ1 (μ2 ) indicates the probability that an individual who has (does not has) protection consciousness is converted into an individual who does not have (has) self-protection consciousness. β 1 (β 2 ) indicates the probability that an individual who does not have (has) protection consciousness is infected by the latent individuals, and in general β 1 ≥ β 2 . λ indicates the probability that a latent individual transformed into an explosive individual; γ 1 (γ 2 ) indicates the probability that an explosive individual converted into an individual without (with) self-protection awareness. The transmission diagram of infectious diseases is shown in Fig. 1. 2.2. Analysis of infectious disease transmission model in the homogeneous mixed population We let ρ S , ρSa , ρ R and ρ I represent the density of the individuals without self-protection consciousness, the individuals with self-protection consciousness, the latent individuals, and the explosive individuals, respectively. It is easy to know that Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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3
Fig. 1. Schematic diagram of the epidemic spreading. We divide the total individuals into four groups: (i) the individuals without self-protection (S); (ii) the individuals with self-protection awareness (Sa ); (iii) the latent individuals (E); (iv) and the explosive (outbreak) individuals (I).
at any time t, we have ρS (t ) + ρSa (t ) + ρE (t ) + ρI (t ) = 1. In an uniformly mixed population, we can obtain the change of different types of individuals density as follows:
⎧ dρ S ⎪ = α1 − μ1 ρS + μ2 ρSa − β1 ρS ρE + γ1 ρI − dρS , ⎪ ⎪ dt ⎪ ⎪ ⎪ a ⎪ dρ ⎪ ⎨ S = α2 + μ1 ρS − μ2 ρSa − β2 ρSa ρE + γ2 ρI − dρSa , dt
(1)
⎪ d ρE ⎪ = β1 ρS ρE + β2 ρSa ρE − λρE − dρE , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dρI = λρ − γ ρ − γ ρ − dρ , E 1 I 2 I I dt
According to the system (1), we can calculate the disease-free equilibrium point G0 = ρS0 , ρS0a , ρE0 , ρI0 =
α1 +μ2 μ1 + μ2 + d ,
α2 +μ1 μ1 +μ2 +d , 0, 0 . By means of the next-generation matrix algorithm [34], the basic reproduction number could be calculated β ( α + μ )+ β ( a + μ ) as R0 = 1 (λ1+d )(2μ +2μ +2 d ) 1 . Next, we study the locally or globally stability of system (1) at the disease-free equilibrium 1 2
point.
Theorem 1. If the basic reproduction number R0 =
0
point G0 = ρS0 , ρS0a , ρE0 , ρI
=
β1 (α1 +μ2 )+β2 (a2 +μ1 ) < 1, then the system (1) at the disease-free equilibrium (λ +d )(μ1 +μ2 +d )
α1 +μ2 α2 +μ1 μ1 + μ2 + d , μ1 + μ2 + d , 0 , 0
is locally asymptotically stable.
Proof. The Jacobian matrix of system (1) at the disease-free equilibrium point G0 = ρS0 , ρS0a , ρE0 , ρI0 = α2 +μ1 μ1 + μ2 + d , 0 , 0
−μ1 − d
⎜ ⎜ ⎝
α1 +μ2 μ1 + μ2 + d ,
is:
⎛
J=⎜
μ1
μ2 −μ2 − d
0
0
0
0
−β1 ρS0 −β2 ρS0a
β1 ρS0 + β2 ρS0a − λ − d λ
γ1 γ1 0
⎞ ⎟ ⎟ ⎟ ⎠
(2)
−γ1 − γ2 − d
Therefore, we can calculate the characteristic function of the Jacobian matrix as:
f (g) = −(g − λ − d + β1 ρS0 + β2 ρS0a )(g + γ1 + γ2 + d )(g + μ2 + d )(g + μ1 + d )
(3)
Further, the eigenvalues of the Jacobian matrix can be obtained as follows: g1 = −(γ1 + γ2 + d ), g2 =−(μ2 + d ), g3 = −(μ1 + d ), g4 = −λ − d + β1 ρS0 + β2 ρS0a . Because g1 < 0, g2 < 0, g3 < 0, g4 = (λ + d ) d)
β1 ρS0 +β2 ρS0a −1 λ+ d
= (λ +
β1 (α1 +μ2 )+β2 (a2 +μ1 ) β ( α + μ )+ β ( a + μ ) − 1 . So when R0 = 1 (λ1+d )(2μ +2μ +2 d ) 1 < 1, all the eigenvalues of the Jacobian matrix at the (λ+d )(μ1 +μ2 +d ) 1 2
disease-free equilibrium point less than 0, that is, the system (1) is locally asymptotically stable at the disease-free equilibrium. Remark 1. If the birth rate and mortality are not considered, α1 = α2 = d = 0, then the basic reproduction number can be β μ +β μ obtained as R0 = λ1(μ2 +μ2 )1 . In particular, if the individuals who strengthen self-protection and the individuals who do not 1 2 strengthen self-protection are infected with the same probability, i.e., β1 = β2 = β , then the basic reproduction number can be written as R0 = βλ . Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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β1 (α1 +μ2 )+β2 (a2 +μ1 ) < 1, γ1 = γ2 , then the system (1) at the disease-free (λ+d )(μ1 +μ2 + d ) α1 +μ2 α2 +μ1 μ1 +μ2 +d , μ1 +μ2 +d , 0, 0 is globally asymptotically stable.
Theorem 2. If the basic reproduction number R0 =
0
equilibrium point G0 = ρS0 , ρS0a , ρE0 , ρI
=
Proof. Construct the Lyapunov function as follows:
V (t ) =
1 (ρS − ρS0 )2 + φ1 ρE 2
(4)
Where φ 1 ≥ 0, then we could get V(t) ≥ 0. Combining Eqs. (1) and (4), one can get:
V˙ (t ) = ρ˙ S (ρS − ρS0 ) + φ1 ρ˙ E
α1 ρS − μ1 ρS2 + μ2 ρS − μ2 ρS2 − μ2 ρS ρE − μ2 ρS ρI − β1 ρS2 ρE + γ1 ρS ρI − dρS2 − (α1 ρS0 − μ1 ρS ρS0 + μ2 ρS0 − μ2 ρS ρS0 − μ2 ρS0 ρE − μ2 ρS0 ρI − dρS0 ρS + φ1 (β1 ρS ρE + β2 ρE − β2 ρS ρE − β2 ρE2 − β2 ρI ρE − β3 ρE − dρE ) = (α1 + μ2 )ρS − (α1 + μ2 )ρS0 − (μ1 + μ2 + d )ρS2 + (ρ1 + μ2 + d )ρS0 ρS + μ2 (ρS0 − ρS )ρE + (μ2 − γ1 )ρS0 ρI − (μ2 − γ1 )ρS ρI − β1 ρS2 ρE + β1 ρS0 ρS ρE + (φ1 β1 − φ1 β2 )ρS ρE + φ1 β2 ρE − φ1 β2 ρE2 − φ1 β2 ρI ρE − (β3 + d )φ1 ρE = − (μ1 + μ2 + d )(ρS0 − ρS )2 − φ1 (λ + d )(1 − R0 ) + (φ1 β1 − φ1 β2 − μ2 − β1 ρS )(ρS − ρS0 )ρE − φ1 β2 ρE2 − φ1 β2 IρE =
(5)
Because ρ S ≥ 0, ρSa ≥ 0, ρ E ≥ 0, ρ I ≥ 0, β 1 > β 2 , so there is at least one φ 1 > 0 such that φ1 β1 − φ1 β2 − μ2 − β1 ρS = 0. Then, when R0 < 1 we can get V˙ (t ) ≤ 0. Then the system (1) at the disease-free equilibrium point G0 = (ρS0 , ρS0a , ρE0 , ρI0 ) = ( μα+1 +μμ+2 d , μα+2 +μμ+1 d , 0, 0 ) is globally asymptotically stable. 1
2
1
2
Remark 2. The condition of Theorem 2 requires γ1 = γ2 , this is a very strong constraint, but Theorem 2 is a sufficient condition for the globally asymptotic stable of system (1) at the disease-free equilibrium. Although there are many sufficient conditions for the global asymptotic stability of system (1) at the disease-free equilibrium, the calculation is too cumbersome, we have not done that. 2.3. Analysis of infectious disease transmission model in heterogeneous mixed population Let A = (Ai j )N×N denote the connection matrix between individuals. If there is a connection between an individual i and an individual j, Ai j = 1, otherwise Ai j = 0. We let ρSi (t ), ρSa (t ), ρEi (t ), ρIi (t ) denote the probability that an individual i is i
an non-self-protection consciousness, a self-protection consciousness, a latent-state or an explosive state, respectively. It is easy to conclude that i (ρSi + ρSa + ρEi + ρIi ) = 1. Then, we could write down the differential equation of infectious disease i
propagation dynamics as follows:
⎧ dρ ⎪ ⎪ Si = α1 − μ1 ρSi + μ2 ρSia − ρSi [1 − j (1 − β1 Ai j ρE j )] + γ1 ρIi − dρSi , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dρ a ⎪ ⎨ Si = α2 + μ1 ρSi − μ2 ρSa ρSa [1 − j (1 − β2 Ai j ρE j )] + γ2 ρIi − dρSa , i i i dt
(6)
⎪ d ρE i ⎪ ⎪ = ρSi [1 − j (1 − β1 Ai j ρE j )] + ρSia [1 − j (1 − β2 Ai j ρE j )] − λρEi − dρEi , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dρIi = λρE − γ1 ρI − γ2 ρI − dρI , dt
i
i
i
i
It is easy to conclude that the disease-free equilibrium point of the system (6) is G0 = α +μ
α +μ
α +μ
α +α
1 N
α1 +μ2 α2 +μ1 μ1 + μ2 + d , μ1 + μ2 + d ,
1 2 1 2 0, 0, . . . , μ + , 2 1 , 0, 0, . . . , μ + , 2 1 , 0, 0 . 1 μ2 + d μ1 + μ2 + d 1 μ2 + d μ1 + μ2 + d According to Eq. (6), we can get:
⎧ d ρS i ⎪ ⎪ = α1 − μ1 ρSi + μ2 ρSia − β1 ρSi j Ai j ρE j + γ1 ρIi − dρSi , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dρ a ⎪ ⎨ Si = α2 + μ1 ρSi − μ2 ρSa − β2 ρSa j Ai j ρE j + γ2 ρIi − dρSa , i i i dt
⎪ d ρE i ⎪ ⎪ = β1 ρSi j Ai j ρE j + β2 ρSia j Ai j ρE j − λρEi − dρEi , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dρIi = λρE − γ1 ρI − γ2 ρI − dρI , dt
i
i
i
(7)
i
In the following, we divide the dynamic behavior of the system (7). Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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(i) Individuals who are not self-protected and those who strengthen self-protection have the same probability of infection, i.e., β1 = β2 = β . Theorem 3. If β1 = β2 = β , and λmax (β A + (1 − λ − d )I ) < 1, then the disease-free equilibrium of the system (7) is asymptotically stable. Proof. According to the formula (7), one can obtain:
⎧ dρ ⎪ ⎨ Ei = (1 − ρEi − ρIi )β j Ai j ρE j − λρEi − dρEi , dt
(8)
⎪ ⎩ dρIi = λρE − γ1 ρI − γ2 ρ Ii − dρI , i i i dt
According to the formula (8) we could get:
ρEi (t + 1 ) = (1 − ρEi (t ) − ρIi (t ))β
Ai j ρE j (t ) + (1 − λ − d )ρEi (t )
(9)
j
Furthermore, from the formula (7) we can get:
ρEi (t + 1 ) ≤ β
Ai j ρE j (t ) + (1 − λ − d )ρEi (t )
(10)
j
For two real-valued column vectors u = (u1 , u2 , . . . , uN )T , v = (v1 , v2 , . . . , vN )T ∈ RN , we say uv if ui ≤ vi for all i ∈ {1, 2, . . . , N}. Hence, we could rewrite the formula (10) as follows:
ρE (t + 1 ) (β A + (1 − λ − d )I )ρEi (t )
(11)
Therefore, if λmax (β A + (1 − λ − d )UN ) < 1 (UN is the N-order identity matrix), formula (7) is asymptotically stable at disease-free equilibrium point. Theorem 4. If λmax (A ) < λβ+d , then the disease-free equilibrium point of the system (7) is exponential asymptotically stable. Proof. Consider the third formula of the system (7), we can get:
d ρE i = dt =
β1 ρSi
Ai j ρE j + β2 ρS0
i
j
β (ρSi + ρSi0 )
Ai j ρE j − λρEi − dρEi
j
A i j ρE j − ( λ + d ) ρE i
(12)
j
Because ρSi + ρS0 ≤ 1, then we can obtain i
d ρE i ≤β dt
A i j ρE j − ( λ + d ) ρE i
(13)
j
Let ρE = (ρE1 , ρE2 , . . . , ρEN )T , then we can get
d ρE ≤ [ β A − ( λ + d ) ] ρE dt
(14)
Therefore, we can get ρE (t ) ≤ e[β A−(λ+d )]t ρE (0 ). As 0ρ E 1, so if λmax (A ) < λβ+d , the disease-free equilibrium point of
the system (7) is exponential asymptotically stable.
(ii) Individuals who are the non-self-protected are more likely to be infected than the individuals who strengthen self-protection, i.e., β 1 > β 2 . Let ρ = (ρS1 , ρS2 , . . . , ρSN , ρSa , ρSa , . . . , ρSa , ρE1 , ρE2 , . . . , ρEN , ρI1 , ρI2 , . . . , ρIN ), then, according to (7) one can get 1
2
N
dρ = α + Bρ + F ( ρ ) dt
(15)
Where
α = (α1 , . . . , α1 , α2 , . . . , α2 , 0, . . . , 0 )T , ⎛ ⎞ ⎛ B11 B12 0 B14 −μ1 − d ⎜B ⎟ ⎜ B 0 B 21 22 24 ⎜ ⎟ .. B=⎜ ⎟, B11 = ⎜ . ⎝ 0 B33 0 ⎠ ⎝0 0
0
B43
B44
0
···
0
..
.. .
.
···
−μ1 − d
⎞
⎛
−μ2
⎟ ⎜ ⎟, B12 = ⎜ .. ⎠ ⎝ .
0
···
0
..
.. .
.
···
⎞ ⎟ ⎟, ⎠
−μ2
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⎛
−γ1
⎜ . ⎜ . ⎝ .
B14 =
⎛
..
.. .
.
⎞
⎛
−μ1
⎟ ⎜ ⎟, B21 = ⎜ .. ⎠ ⎝ .
−γ1
···
0
···
0
.. .
..
.. .
0
···
⎛
B43
0
−μ2 − d
⎜ ⎜ ⎝
B22 =
···
−λ ⎜ . = ⎜ .. ⎝
··· ..
0 F = (−β1 ρS1
0
⎞
..
.. .
.
⎛
−μ2 − d
Ai j ρE j , . . . , −β2 ρS1a
···
0
..
.. .
.
⎞
−γ2
···
0
.. .
..
.. .
⎛
−λ − d
···
0
.. .
..
.. .
0
···
⎟ ⎜ ⎟, B33 = ⎜ ⎠ ⎝
−γ1 − γ2 − d
⎟ ⎜ ⎟, B44 = ⎜ ⎠ ⎝ −λ
j
⎟ ⎟, ⎠
···
0
⎛
⎞
−μ1
−γ2
⎟ ⎜ ⎟, B24 = ⎜ .. ⎠ ⎝ .
.. .
.
···
.
0
···
0
⎞
···
.
⎞
.
⎞ ⎟ ⎟, ⎠
−λ − d
⎟ ⎟, ⎠
−γ1 − γ2 − d Ai j ρE j , . . . , β1 ρS1 Ai j ρE j + β2 ρS1a A i j ρE j , . . . , 0 , . . . ) . ···
0
j
j
j
(μ +μ +d )(λ+d )
Theorem 5. The trivial equilibrium G0 of system (7) is asymptotically stable if λmax (A ) < β (α 1+μ 2)+β (α +μ ) , where λmax (A ) 1 1 2 2 2 1 is the greatest eigenvalue of the adjacency matrix A of the propagating network. Proof. At the point of disease-free equilibrium, according to Eqs. (7) and (15), we can get the Jacobian matrix of Eq. (7) at equilibrium point G0
⎛
−(μ1 + d )UN
⎜ ⎜ ⎝
J=⎜
μ1UN
μ2UN −(μ2 + d )UN
T1 A
0
0
T2 A − (λ + d )UN
0
0
λUN
β ( α +μ )
β ( α +μ )
Where T0 = μ1 +1μ +2d , T1 = μ2 +2μ +1d , T2 = 1 2 1 2 the above Jacobian matrix is
T0 A
⎞ γ1UN γ2UN ⎟ ⎟ ⎟ 0 ⎠
(16)
T3UN
β1 (α1 +mu2 )+β2 (α2 +μ1 ) , T3 = −(γ1 + γ2 + d ). Then, the characteristic equation of μ1 +mu2 +d
f (g) = (g + μ1 + d )N (g + μ2 + d )N (g + γ1 + γ2 + d )N × det (T2 A − (λ + d )UN )
(17)
We can get the eigenvalues of the Jacobian matrix: g = − (μ1 + d ) < 0 (N double roots), g = −(μ2 + d ) < 0 (N double β ( α + μ )+ β ( α + μ ) roots), g = −(γ1 + γ2 + d ) < 0 (N double roots). If g = det 1 1 μ 2+μ 2+d 2 1 A − (λ + d )UN ≤ 0, i.e., when λmax (A ) < 1
2
(μ1 +μ2 +d )(λ+d ) β1 (α1 +μ2 )+β2 (α2 +μ1 ) , all eigenvalues are less than 0, and the system (7) is asymptotically stable at the disease-free equi-
librium point.
Theorem 6. If λmax (A ) < λβ+d , β = max{β1 , β2 }, then the disease-free equilibrium point of the system (7) is exponential asymptotically stable. Proof. The proof process is similarity with Theorem 4.
3. Numerical simulation Supposing that the initial density of individuals without self-protection consciousness is ρS (0 ) = 0.5, the initial density of the self-protected individuals is ρSa (0 ) = 0.49, and the density of latent individuals is ρE (0 ) = 0.01, the individual density of the explosive state ρI (0 ) = 0. In addition, we let ρ E (∞) and ρ I (∞) denote the density of the latent individual and the explosive individual when the system reaches equilibrium, respectively. Let ρEI (∞ ) = ρE (∞ ) + ρI (∞ ) indicate the total infected density when the system approach to the stable equilibrium state. Fig. 2 depicts the variation of ρ EI (∞) and R0 as a function of λ in the infectious disease transmission system when the individuals are uniformly mixed. As can be seen from Fig. 2, as the parameter λ increases, the basic reproduction number R0 gradually reduces, and the total density of the latent and explosive individuals ρEI (∞ ) = ρE (∞ ) + ρI (∞ ) gradually decrease when the system reaches equilibrium. When λ < 0.1, R0 > 1, the final latent individual and the explosive individual in the system will always exist. However, once the values of λ > 0.1, R0 < 1, the infected individuals could not exist in the final system. This result is consistent with our theoretical analysis. In addition, by observing Fig. 2, it can be found that in the presence of endemic diseases (the left half of the red dotted line), with the increase of R0 , the value of ρ EI (∞) gradually increases. Fig. 3 shows how the total density of ρEI (∞ ) = ρE (∞ ) + ρI (∞ ) changes with β 1 and β 2 , we find that the value of ρ EI (∞) increase with the values of β 1 and β 2 . Individuals’ ability to prevent infectious diseases can effectively reduce the Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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Fig. 2. The basic reproducible number R0 and the total density of the latent individual and the explosive individual in the final system ρEI (∞ ) = ρE (∞ ) + ρI (∞ ) change with the variation of the parameter λ. Other parameters are set to be: α1 = α2 = γ1 = γ2 = 0.2, β1 = 0.6, β2 = 0.4, μ1 = μ2 = 0.3, d = 0.4. According to R0 = β1 ((αλ1++dμ)(2μ)++β2μ(a+2 +d )μ1 ) , we can calculate R0 = 1 when λ = 0.1. The left half of the red dotted line is endemic infectious disease, the right half 1
2
is disease free. It can be found that in the presence of endemic diseases, the value of ρ EI (∞) gradually increases with the R0 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. The value of ρ EI (∞) in the final system changes with β 1 and β 2 . We only describe the case of β 1 > β 2 . Other parameters are set to be: α1 = α2 = γ1 = γ2 = 0.2, μ1 = μ2 = 0.3, d = 0.4. (a ) λ = 0.1. (b) λ = 0.2. It can be found that with the increase of parameter λ, the density of infected persons in the final system can be effectively reduced.
probability of infection. By comparing Fig. 3(a) and (b), we observe that with the same parameter value (β 1 , λ), the value of ρ EI (∞) declines with the β 2 . For instant, in Fig. 3(a), even the value of β 1 > 0.8, when β 2 ≈ 0, we could get ρ EI (∞) ≈ 0. At the same time, it can be found that with the increase of parameter λ, the density of infected persons in the final system can be effectively reduced. Fig. 4 depicts the total density of ρ EI (∞) changes with the values of μ1 and μ2 . From Fig. 4(a) and (b), it can be found that with the increase of the parameter μ1 , the density of infected persons ρ EI (∞) in the final system gradually decreases. But with the increase of the parameter μ2 , the density of infected persons in the final system ρ EI (∞) gradually increases. The magnitudes of the parameters μ1 and μ2 reflect the possibility of individuals taking self-protection measures. With the increase of the parameter μ1 value, the individuals S who have not taken self-protection measures have a higher probability of taking the initiative to take self-protection; on the other hand, as the value of the parameter μ2 decreases, the probability that the individual Sa take the self-protection measure continues to maintain good protection measures becomes larger. Comparing Fig. 4(a) and (b), it can be found that when the value of μ1 is larger and the value of μ2 is smaller, there will be no infected person in the whole system in the last. We study the propagation dynamics of infectious diseases under different network topologies. For convenience, we assume that birth and death do not occur in the short term, i.e. α1 = α2 = d = 0. Then we can get the basic reproduction β μ +β μ number R0 = λ1(μ2 +μ2 )1 . It should be noted that the basic reproduction number R0 is obtained in a homogeneous mixed 1
2
population, However, for the general inter-individual connection structure, we may not get such a result. Next, we mainly perform the proposed model on the following two network topologies. (1) The ER random network with N = 10 0 0 players and the average degree < k >= 6, or (2) the BA scare-free network with N = 10 0 0, the degree obeys the distribution Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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Fig. 4. The total density of ρ EI (∞) changes with μ1 and μ2 . Others parameters are set to be: α1 = α2 = γ1 = γ2 = 0.2, β1 = 0.6, β2 = 0.4, d = 0.4. (a ) λ = 0.05. (b) λ = 0.1. It can be found that when the value of μ1 is larger and the value of μ2 is smaller, there will be no infected person in the whole system in the last.
Fig. 5. Evolution of the total density of E and I in different network topologies over time. The maximum eigenvalues of the ER random network and the BA BA scale-free network connection topology matrix are gER max = 7.2159 and gmax = 15, respectively. The parameters are set to be: β1 = 0.06, β2 = 0.05, μ1 = μ2 = 0.3, γ1 = γ2 = λ = 0.2. (a) The change of ρ EI (∞) in ER random network; (b) The change of ρ EI (∞) in BA scale-free network.
P (k ) ∼ k−γ1 with γ1 = 2.1 and the average degree < k >= 6. Based on the Monte Carlo method, we simulate proposed model. Since the results differ for each Monte Carlo trial, we present the results averaged over 100 independent runs. Fig. 5 describe the evolution of the total density of E and I in different network topologies over time. According to calculation, the maximum eigenvalues of the ER random network and the BA scale-free network connection topology matrix β 1 μ2 + β 2 μ1 BA 0 are gER max = 7.2159 and gmax = 15. The ER random network is a homogeneous network, so in Fig. 5(a), when R = λ(μ +μ ) = 1
2
0.3 < 1 the infection density in the whole system ρEI (∞ ) = 0. However, since we consider the connection topology between β μ +β μ individuals and the degree of heterogeneity of the BA scale-free network is large, even in Fig. 5(b), R0 = λ1(μ2 +μ2 )1 = 0.3 < 1, 1
2
and final density of the infected person in the whole system ρ EI (∞) > 0. Fig. 6 depicts the change in the final infected density ρ EI (∞) in the ER random network and the BA scale-free network with changes in infection rate β 2 and parameter λ. It can be found from Fig. 6(a) and (b) that under different network structures, as the parameter λ increases, the final infected density is getting smaller. Therefore, the parameter λ is an important parameter for controlling the spread of infectious diseases. When it is greater than a certain threshold, there will be no infected person in the final infectious disease transmission system. In addition, as the infection rate β 2 increases, the density of the final infected person is also becoming smaller. It is worth noting that, in comparison with Fig. 6(a) and (b), it can be found that, the value of λ, when infected person disappears in the BA free-scale network, is 1.3 times that of the ER random network. Finally, we analyze how the value of β 1 and β 2 affect the final infection density ρ EI (∞) in ER random networks and BA scale-free networks. By comparing the analysis the Fig. 7(a) and (b), one can find that with the infection rates β 1 and β 2 increase, the final infected person density ρ EI (∞) gradually increases. Interestingly, even if β 1 ≈ 1 and β 2 ≈ 1, ρ EI (∞) ≈ 0.8 < 1, however, once β 1 → 0 and β 2 → 0, there will be no infected people in the network. Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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Fig. 6. Under different β 2 values, the change of ρ EI (∞) in ER random network and BA scale-free network with λ. Other parameters are set to be: μ1 = μ2 = 0.3, γ1 = γ2 = λ = 0.2. (a) The change of the ρ EI (∞) in the ER random network; (b) The change of the ρ EI (∞) in the BA scale-free network. It can be found that under different network structures, as the parameter λ increases, the final infected density is getting smaller.
Fig. 7. The change of final infection density ρ EI (∞) in ER random network and BA scale-free network with the infection rates β 1 and β 2 . Parameters are set to be: μ1 = μ2 = 0.3, γ1 = γ2 = λ = 0.2. (a) The change of ρ EI (∞) value in ER random network; (b) The change of ρ EI (∞) value in BA scale-free network. One can find that with the infection rates β 1 and β 2 increase, the final infected person density ρ EI (∞) gradually increases.
4. Conclusion Considering the individuals’ ability to avoid risks in the epidemic season, this paper proposed a new SSa EI model for the spread of disease. Based on the mean field theory and the next-generation matrix method, we calculated the threshold of infectious disease transmission. At the same time, we presented the conditions of the locally, globally and exponential asymptotically stable at the disease-free equilibrium state. In addition, we have performed our models in the ER random network and the BA scale-free network, respectively. The results showed that the probability of the latent individuals transforming into explosive individuals has a greater impact on the spread of infectious diseases. In short, we provided a theoretical framework to study the epidemic spread, our results suggested that improving individuals’ risk awareness is beneficial to suppressing the epidemic spreading. In the future, we will do more efforts on the epidemic spreading with the eyes of empirical data and experimental supports.
Acknowledgments This research was supported by the National Nature Science Foundation of China (No. 61803184, 71774070, 61973143, 71974080); National Nature Science Foundation of Jiangsu Province (Nos. BK20180851, BK20190832); China Postdoctoral Science Foundation (2018M640326); Social Science Fund of Jiangsu Province (No. 18TQD002); the Natural Science Research Projects of Jiangsu Higher Education Institutions (No. 19KJB120 0 01); the Jiangsu Postdoctoral Research Funding Project (No. 2019K172). Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
How the individuals’ risk aversion affect the epidemic spreading Dun Han a,c,∗, Qi Shao a, Dandan Li b, Mei Sun a a
Institute of Applied Systems Analysis, Jiangsu University, Zhenjiang, Jiangsu 212013, People’s Republic of China School of Management, Jiangsu University Zhenjiang, Jiangsu 212013, People’s Republic of China c The Adaptive Networks and Control Laboratory, Department of Electronic Engineering, and the Research Center of Smart Networks and Systems, School of Information Science and Engineering, Fudan University, Shanghai 200433, People’s Republic of China b
a r t i c l e
i n f o
Article history: Received 10 August 2019 Revised 15 October 2019 Accepted 27 October 2019 Available online xxx Keywords: Epidemic spreading Individuals’ sensitivity Asymptotically stable Network
a b s t r a c t Some diseases can live in people for many years without making them sick, during this time, the bacteria can spread to others who come in contact with the infected person. However, explosive individuals in infected people will exhibit certain dominant states associated with infectious diseases, as a results, the uninfected persons will avoid contact with such individuals with dominant infectious diseases. Considering the individual’s ability to avoid risks in the epidemic season, we propose the epidemic spreading model with individuals’ sensitivity. The epidemic spreading threshold is calculated by means of the mean-field theory and the next-generation matrix method. In addition, the locally, globally and exponential asymptotically stable conditions in the disease-free equilibrium state are given. Finally, we simulate the proposed epidemic spreading modeling in the ER random network and the BA scale-free network. The numerical simulations results show that the probability of the latent individuals transforming into explosive individuals has a greater impact on the spread of infectious diseases. Meanwhile, self-protection is an effective measure to reduce the outbreak of infectious diseases. © 2019 Elsevier Inc. All rights reserved.
1. Introduction There has been a long struggle between human society and infectious diseases, research on the occurrence and development of infectious diseases, the spreading mechanism of infectious diseases and the corresponding prevention and control measures has always been the focus of scientific research. Various infectious diseases, such as SARS, Ebola, plague, influenza, plague, AIDS, cholera and avian influenza, have affected many parts of the world and claimed the lives of thousands of people [1–7]. Infectious diseases are mainly transmitted through contact between individuals, so the contact network can more realistically depict the transmission process of infectious diseases [8,9]. Scholars have adopted a variety of theoretical methods to study the spread of infectious diseases, such as percolation theory, mean field theory, game theory, stochastic processes and cellular automata [10–13]. With the discovery and establishment of small world networks and scale-free networks in real systems, the research on the transmission dynamics of infectious diseases on complex networks has developed rapidly [14–17]. Many models of infectious disease transmission based on complex networks have been proposed by scholars, among which the most three classical models are: (1) the mean field model proposed by Pastor-Satorras and Vespignani ∗
Corresponding author at: Institute of Applied Systems Analysis, Jiangsu University, Zhenjiang, Jiangsu 212013, People’s Republic of China. E-mail addresses: [email protected] (D. Han), [email protected] (M. Sun).
https://doi.org/10.1016/j.amc.2019.124894 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
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[18]; (2) the percolation model proposed by Newman and Watts [19]; (3) the discrete probability model proposed by Wang et al. [20]. Although immunization is a very effective measure to control the spread of infectious diseases [21,22], human awareness plays an important role in the spread of infectious diseases and the control of propagation patterns. Faced with the complicated social environment, it shows that, human behavior is extremely complex. With the awareness of the epidemic spreading in their surroundings, healthy individuals may tend to take certain actions (such as wearing protective masks, washing hands frequently, staying home, etc.), which generally help reduce their risks of getting infected [23,24]. However, those who do not pay attention to protecting themselves will suffer a higher risk of infection than those who are diligent in protecting themselves. A conclusion is that risk awareness can suppress epidemic spreading [25]. Clara et al. studied the dynamical interplay between awareness and epidemic spreading in multiplex networks, they claimed that the critical point for the onset of the epidemics has a critical value defined by the awareness dynamics and the topology of the virtual network [26]. Sun and co-workers explored the epidemic spread in bipartite network by considering risk awareness. They results showed that, the final infection density is negative linear with the value of individuals!‘ risk awareness [27]. Wu and Xiao analyzed the effects of awareness on epidemic spreading, they obtained that only epidemic information, being available in network nodes’ neighborhood overlapping in two layers, helps change the epidemic threshold [28]. Juang and Liang studied the impact of vaccine success and awareness on epidemic dynamics. They results highlighted the important and crucial roles of both vaccine success and contact infection awareness on epidemic dynamics [29]. Although there are many studies on the spread of infectious diseases [30,31], with the extreme complexity of individual behavior [32,33], it is still a problem worth to study how to integrate individual risk-avoidance behaviors into the spread of infectious diseases. Antibiotic-resistant bacteria like the Methicillin-resistant Staphylococcus aureus (MRSA) can live in people for many years without making them sick. During this time, the bacteria can spread to others who come in contact with the MRSA-infected person. It should be noted that these latent-infected people have a strong ability to spread, the spread ability of individuals with some outbreaks or those who are labeled infected is reduced. Whereas, previous studies on epidemic spreading rarely focused on the those effects. Therefore, in this paper we assumed that only the latent individuals have the ability to spread, while explosive individuals have no ability to transmit. Meanwhile, we showed the epidemic spreading modeling with individuals’ awareness, and we presented the threshold of infectious disease transmission and the conditions of the locally, globally and exponential asymptotically stable. In addition, we have performed our proposed model in the ER random network and the BA scale-free network. The results indicated that the probability of the latent individuals transforming into explosive individuals has a greater impact on the spread of infectious diseases, and individuals strengthen self-protection. It is an effective measure to reduce the outbreak of infectious diseases. The rest of the paper is organized as follows. The model and theoretical analysis are presented in Section 2. Numerical simulations are presented to illustrate the behaviors of cooperative spreading processes in Section 3. Finally, conclusions are given in Section 4. 2. Infectious disease transmission model 2.1. Preliminaries and model formulations Antibiotic-resistant bacteria like the Methicillin-resistant Staphylococcus aureus (MRSA) can live in people for many years without making them sick. During this time, the bacteria can spread to others who come in contact with the MRSA-infected person. During the epidemic spreading season, some individuals prefer to strengthen self-protection, while others have less self-protection awareness, which leads to the different of being infected. We divide the total individuals into four groups: (i) the individuals without self-protection (S); (ii) the individuals with self-protection awareness (Sa ); (iii) the latent individuals (E); (iv) and the explosive (outbreak) individuals (I). S-state individuals and Sa -state individuals are collectively referred to as uninfected individuals, and E-state individuals and I-state individuals are collectively referred to as infected individuals. Since outbreak individuals in infected people will exhibit certain dominant states associated with infectious diseases, such as sneezing and fever, uninfected persons will avoid contact with such individuals with dominant infectious diseases. Therefore, we assume that only latent individuals can infect uninfected individuals. In this paper, d is used to indicate mortality; we suppose that a new individual is either an S-state individual or an Sa -state individual, α 1 and α 2 represent the birth rate of S-state individuals and Sa -state individuals, respectively. The total birth rate is equal to the mortality rate, i.e., α1 + α2 = d. μ1 (μ2 ) indicates the probability that an individual who has (does not has) protection consciousness is converted into an individual who does not have (has) self-protection consciousness. β 1 (β 2 ) indicates the probability that an individual who does not have (has) protection consciousness is infected by the latent individuals, and in general β 1 ≥ β 2 . λ indicates the probability that a latent individual transformed into an explosive individual; γ 1 (γ 2 ) indicates the probability that an explosive individual converted into an individual without (with) self-protection awareness. The transmission diagram of infectious diseases is shown in Fig. 1. 2.2. Analysis of infectious disease transmission model in the homogeneous mixed population We let ρ S , ρSa , ρ R and ρ I represent the density of the individuals without self-protection consciousness, the individuals with self-protection consciousness, the latent individuals, and the explosive individuals, respectively. It is easy to know that Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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3
Fig. 1. Schematic diagram of the epidemic spreading. We divide the total individuals into four groups: (i) the individuals without self-protection (S); (ii) the individuals with self-protection awareness (Sa ); (iii) the latent individuals (E); (iv) and the explosive (outbreak) individuals (I).
at any time t, we have ρS (t ) + ρSa (t ) + ρE (t ) + ρI (t ) = 1. In an uniformly mixed population, we can obtain the change of different types of individuals density as follows:
⎧ dρ S ⎪ = α1 − μ1 ρS + μ2 ρSa − β1 ρS ρE + γ1 ρI − dρS , ⎪ ⎪ dt ⎪ ⎪ ⎪ a ⎪ dρ ⎪ ⎨ S = α2 + μ1 ρS − μ2 ρSa − β2 ρSa ρE + γ2 ρI − dρSa , dt
(1)
⎪ d ρE ⎪ = β1 ρS ρE + β2 ρSa ρE − λρE − dρE , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dρI = λρ − γ ρ − γ ρ − dρ , E 1 I 2 I I dt
According to the system (1), we can calculate the disease-free equilibrium point G0 = ρS0 , ρS0a , ρE0 , ρI0 =
α1 +μ2 μ1 + μ2 + d ,
α2 +μ1 μ1 +μ2 +d , 0, 0 . By means of the next-generation matrix algorithm [34], the basic reproduction number could be calculated β ( α + μ )+ β ( a + μ ) as R0 = 1 (λ1+d )(2μ +2μ +2 d ) 1 . Next, we study the locally or globally stability of system (1) at the disease-free equilibrium 1 2
point.
Theorem 1. If the basic reproduction number R0 =
0
point G0 = ρS0 , ρS0a , ρE0 , ρI
=
β1 (α1 +μ2 )+β2 (a2 +μ1 ) < 1, then the system (1) at the disease-free equilibrium (λ +d )(μ1 +μ2 +d )
α1 +μ2 α2 +μ1 μ1 + μ2 + d , μ1 + μ2 + d , 0 , 0
is locally asymptotically stable.
Proof. The Jacobian matrix of system (1) at the disease-free equilibrium point G0 = ρS0 , ρS0a , ρE0 , ρI0 = α2 +μ1 μ1 + μ2 + d , 0 , 0
−μ1 − d
⎜ ⎜ ⎝
α1 +μ2 μ1 + μ2 + d ,
is:
⎛
J=⎜
μ1
μ2 −μ2 − d
0
0
0
0
−β1 ρS0 −β2 ρS0a
β1 ρS0 + β2 ρS0a − λ − d λ
γ1 γ1 0
⎞ ⎟ ⎟ ⎟ ⎠
(2)
−γ1 − γ2 − d
Therefore, we can calculate the characteristic function of the Jacobian matrix as:
f (g) = −(g − λ − d + β1 ρS0 + β2 ρS0a )(g + γ1 + γ2 + d )(g + μ2 + d )(g + μ1 + d )
(3)
Further, the eigenvalues of the Jacobian matrix can be obtained as follows: g1 = −(γ1 + γ2 + d ), g2 =−(μ2 + d ), g3 = −(μ1 + d ), g4 = −λ − d + β1 ρS0 + β2 ρS0a . Because g1 < 0, g2 < 0, g3 < 0, g4 = (λ + d ) d)
β1 ρS0 +β2 ρS0a −1 λ+ d
= (λ +
β1 (α1 +μ2 )+β2 (a2 +μ1 ) β ( α + μ )+ β ( a + μ ) − 1 . So when R0 = 1 (λ1+d )(2μ +2μ +2 d ) 1 < 1, all the eigenvalues of the Jacobian matrix at the (λ+d )(μ1 +μ2 +d ) 1 2
disease-free equilibrium point less than 0, that is, the system (1) is locally asymptotically stable at the disease-free equilibrium. Remark 1. If the birth rate and mortality are not considered, α1 = α2 = d = 0, then the basic reproduction number can be β μ +β μ obtained as R0 = λ1(μ2 +μ2 )1 . In particular, if the individuals who strengthen self-protection and the individuals who do not 1 2 strengthen self-protection are infected with the same probability, i.e., β1 = β2 = β , then the basic reproduction number can be written as R0 = βλ . Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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β1 (α1 +μ2 )+β2 (a2 +μ1 ) < 1, γ1 = γ2 , then the system (1) at the disease-free (λ+d )(μ1 +μ2 + d ) α1 +μ2 α2 +μ1 μ1 +μ2 +d , μ1 +μ2 +d , 0, 0 is globally asymptotically stable.
Theorem 2. If the basic reproduction number R0 =
0
equilibrium point G0 = ρS0 , ρS0a , ρE0 , ρI
=
Proof. Construct the Lyapunov function as follows:
V (t ) =
1 (ρS − ρS0 )2 + φ1 ρE 2
(4)
Where φ 1 ≥ 0, then we could get V(t) ≥ 0. Combining Eqs. (1) and (4), one can get:
V˙ (t ) = ρ˙ S (ρS − ρS0 ) + φ1 ρ˙ E
α1 ρS − μ1 ρS2 + μ2 ρS − μ2 ρS2 − μ2 ρS ρE − μ2 ρS ρI − β1 ρS2 ρE + γ1 ρS ρI − dρS2 − (α1 ρS0 − μ1 ρS ρS0 + μ2 ρS0 − μ2 ρS ρS0 − μ2 ρS0 ρE − μ2 ρS0 ρI − dρS0 ρS + φ1 (β1 ρS ρE + β2 ρE − β2 ρS ρE − β2 ρE2 − β2 ρI ρE − β3 ρE − dρE ) = (α1 + μ2 )ρS − (α1 + μ2 )ρS0 − (μ1 + μ2 + d )ρS2 + (ρ1 + μ2 + d )ρS0 ρS + μ2 (ρS0 − ρS )ρE + (μ2 − γ1 )ρS0 ρI − (μ2 − γ1 )ρS ρI − β1 ρS2 ρE + β1 ρS0 ρS ρE + (φ1 β1 − φ1 β2 )ρS ρE + φ1 β2 ρE − φ1 β2 ρE2 − φ1 β2 ρI ρE − (β3 + d )φ1 ρE = − (μ1 + μ2 + d )(ρS0 − ρS )2 − φ1 (λ + d )(1 − R0 ) + (φ1 β1 − φ1 β2 − μ2 − β1 ρS )(ρS − ρS0 )ρE − φ1 β2 ρE2 − φ1 β2 IρE =
(5)
Because ρ S ≥ 0, ρSa ≥ 0, ρ E ≥ 0, ρ I ≥ 0, β 1 > β 2 , so there is at least one φ 1 > 0 such that φ1 β1 − φ1 β2 − μ2 − β1 ρS = 0. Then, when R0 < 1 we can get V˙ (t ) ≤ 0. Then the system (1) at the disease-free equilibrium point G0 = (ρS0 , ρS0a , ρE0 , ρI0 ) = ( μα+1 +μμ+2 d , μα+2 +μμ+1 d , 0, 0 ) is globally asymptotically stable. 1
2
1
2
Remark 2. The condition of Theorem 2 requires γ1 = γ2 , this is a very strong constraint, but Theorem 2 is a sufficient condition for the globally asymptotic stable of system (1) at the disease-free equilibrium. Although there are many sufficient conditions for the global asymptotic stability of system (1) at the disease-free equilibrium, the calculation is too cumbersome, we have not done that. 2.3. Analysis of infectious disease transmission model in heterogeneous mixed population Let A = (Ai j )N×N denote the connection matrix between individuals. If there is a connection between an individual i and an individual j, Ai j = 1, otherwise Ai j = 0. We let ρSi (t ), ρSa (t ), ρEi (t ), ρIi (t ) denote the probability that an individual i is i
an non-self-protection consciousness, a self-protection consciousness, a latent-state or an explosive state, respectively. It is easy to conclude that i (ρSi + ρSa + ρEi + ρIi ) = 1. Then, we could write down the differential equation of infectious disease i
propagation dynamics as follows:
⎧ dρ ⎪ ⎪ Si = α1 − μ1 ρSi + μ2 ρSia − ρSi [1 − j (1 − β1 Ai j ρE j )] + γ1 ρIi − dρSi , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dρ a ⎪ ⎨ Si = α2 + μ1 ρSi − μ2 ρSa ρSa [1 − j (1 − β2 Ai j ρE j )] + γ2 ρIi − dρSa , i i i dt
(6)
⎪ d ρE i ⎪ ⎪ = ρSi [1 − j (1 − β1 Ai j ρE j )] + ρSia [1 − j (1 − β2 Ai j ρE j )] − λρEi − dρEi , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dρIi = λρE − γ1 ρI − γ2 ρI − dρI , dt
i
i
i
i
It is easy to conclude that the disease-free equilibrium point of the system (6) is G0 = α +μ
α +μ
α +μ
α +α
1 N
α1 +μ2 α2 +μ1 μ1 + μ2 + d , μ1 + μ2 + d ,
1 2 1 2 0, 0, . . . , μ + , 2 1 , 0, 0, . . . , μ + , 2 1 , 0, 0 . 1 μ2 + d μ1 + μ2 + d 1 μ2 + d μ1 + μ2 + d According to Eq. (6), we can get:
⎧ d ρS i ⎪ ⎪ = α1 − μ1 ρSi + μ2 ρSia − β1 ρSi j Ai j ρE j + γ1 ρIi − dρSi , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dρ a ⎪ ⎨ Si = α2 + μ1 ρSi − μ2 ρSa − β2 ρSa j Ai j ρE j + γ2 ρIi − dρSa , i i i dt
⎪ d ρE i ⎪ ⎪ = β1 ρSi j Ai j ρE j + β2 ρSia j Ai j ρE j − λρEi − dρEi , ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dρIi = λρE − γ1 ρI − γ2 ρI − dρI , dt
i
i
i
(7)
i
In the following, we divide the dynamic behavior of the system (7). Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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5
(i) Individuals who are not self-protected and those who strengthen self-protection have the same probability of infection, i.e., β1 = β2 = β . Theorem 3. If β1 = β2 = β , and λmax (β A + (1 − λ − d )I ) < 1, then the disease-free equilibrium of the system (7) is asymptotically stable. Proof. According to the formula (7), one can obtain:
⎧ dρ ⎪ ⎨ Ei = (1 − ρEi − ρIi )β j Ai j ρE j − λρEi − dρEi , dt
(8)
⎪ ⎩ dρIi = λρE − γ1 ρI − γ2 ρ Ii − dρI , i i i dt
According to the formula (8) we could get:
ρEi (t + 1 ) = (1 − ρEi (t ) − ρIi (t ))β
Ai j ρE j (t ) + (1 − λ − d )ρEi (t )
(9)
j
Furthermore, from the formula (7) we can get:
ρEi (t + 1 ) ≤ β
Ai j ρE j (t ) + (1 − λ − d )ρEi (t )
(10)
j
For two real-valued column vectors u = (u1 , u2 , . . . , uN )T , v = (v1 , v2 , . . . , vN )T ∈ RN , we say uv if ui ≤ vi for all i ∈ {1, 2, . . . , N}. Hence, we could rewrite the formula (10) as follows:
ρE (t + 1 ) (β A + (1 − λ − d )I )ρEi (t )
(11)
Therefore, if λmax (β A + (1 − λ − d )UN ) < 1 (UN is the N-order identity matrix), formula (7) is asymptotically stable at disease-free equilibrium point. Theorem 4. If λmax (A ) < λβ+d , then the disease-free equilibrium point of the system (7) is exponential asymptotically stable. Proof. Consider the third formula of the system (7), we can get:
d ρE i = dt =
β1 ρSi
Ai j ρE j + β2 ρS0
i
j
β (ρSi + ρSi0 )
Ai j ρE j − λρEi − dρEi
j
A i j ρE j − ( λ + d ) ρE i
(12)
j
Because ρSi + ρS0 ≤ 1, then we can obtain i
d ρE i ≤β dt
A i j ρE j − ( λ + d ) ρE i
(13)
j
Let ρE = (ρE1 , ρE2 , . . . , ρEN )T , then we can get
d ρE ≤ [ β A − ( λ + d ) ] ρE dt
(14)
Therefore, we can get ρE (t ) ≤ e[β A−(λ+d )]t ρE (0 ). As 0ρ E 1, so if λmax (A ) < λβ+d , the disease-free equilibrium point of
the system (7) is exponential asymptotically stable.
(ii) Individuals who are the non-self-protected are more likely to be infected than the individuals who strengthen self-protection, i.e., β 1 > β 2 . Let ρ = (ρS1 , ρS2 , . . . , ρSN , ρSa , ρSa , . . . , ρSa , ρE1 , ρE2 , . . . , ρEN , ρI1 , ρI2 , . . . , ρIN ), then, according to (7) one can get 1
2
N
dρ = α + Bρ + F ( ρ ) dt
(15)
Where
α = (α1 , . . . , α1 , α2 , . . . , α2 , 0, . . . , 0 )T , ⎛ ⎞ ⎛ B11 B12 0 B14 −μ1 − d ⎜B ⎟ ⎜ B 0 B 21 22 24 ⎜ ⎟ .. B=⎜ ⎟, B11 = ⎜ . ⎝ 0 B33 0 ⎠ ⎝0 0
0
B43
B44
0
···
0
..
.. .
.
···
−μ1 − d
⎞
⎛
−μ2
⎟ ⎜ ⎟, B12 = ⎜ .. ⎠ ⎝ .
0
···
0
..
.. .
.
···
⎞ ⎟ ⎟, ⎠
−μ2
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⎛
−γ1
⎜ . ⎜ . ⎝ .
B14 =
⎛
..
.. .
.
⎞
⎛
−μ1
⎟ ⎜ ⎟, B21 = ⎜ .. ⎠ ⎝ .
−γ1
···
0
···
0
.. .
..
.. .
0
···
⎛
B43
0
−μ2 − d
⎜ ⎜ ⎝
B22 =
···
−λ ⎜ . = ⎜ .. ⎝
··· ..
0 F = (−β1 ρS1
0
⎞
..
.. .
.
⎛
−μ2 − d
Ai j ρE j , . . . , −β2 ρS1a
···
0
..
.. .
.
⎞
−γ2
···
0
.. .
..
.. .
⎛
−λ − d
···
0
.. .
..
.. .
0
···
⎟ ⎜ ⎟, B33 = ⎜ ⎠ ⎝
−γ1 − γ2 − d
⎟ ⎜ ⎟, B44 = ⎜ ⎠ ⎝ −λ
j
⎟ ⎟, ⎠
···
0
⎛
⎞
−μ1
−γ2
⎟ ⎜ ⎟, B24 = ⎜ .. ⎠ ⎝ .
.. .
.
···
.
0
···
0
⎞
···
.
⎞
.
⎞ ⎟ ⎟, ⎠
−λ − d
⎟ ⎟, ⎠
−γ1 − γ2 − d Ai j ρE j , . . . , β1 ρS1 Ai j ρE j + β2 ρS1a A i j ρE j , . . . , 0 , . . . ) . ···
0
j
j
j
(μ +μ +d )(λ+d )
Theorem 5. The trivial equilibrium G0 of system (7) is asymptotically stable if λmax (A ) < β (α 1+μ 2)+β (α +μ ) , where λmax (A ) 1 1 2 2 2 1 is the greatest eigenvalue of the adjacency matrix A of the propagating network. Proof. At the point of disease-free equilibrium, according to Eqs. (7) and (15), we can get the Jacobian matrix of Eq. (7) at equilibrium point G0
⎛
−(μ1 + d )UN
⎜ ⎜ ⎝
J=⎜
μ1UN
μ2UN −(μ2 + d )UN
T1 A
0
0
T2 A − (λ + d )UN
0
0
λUN
β ( α +μ )
β ( α +μ )
Where T0 = μ1 +1μ +2d , T1 = μ2 +2μ +1d , T2 = 1 2 1 2 the above Jacobian matrix is
T0 A
⎞ γ1UN γ2UN ⎟ ⎟ ⎟ 0 ⎠
(16)
T3UN
β1 (α1 +mu2 )+β2 (α2 +μ1 ) , T3 = −(γ1 + γ2 + d ). Then, the characteristic equation of μ1 +mu2 +d
f (g) = (g + μ1 + d )N (g + μ2 + d )N (g + γ1 + γ2 + d )N × det (T2 A − (λ + d )UN )
(17)
We can get the eigenvalues of the Jacobian matrix: g = − (μ1 + d ) < 0 (N double roots), g = −(μ2 + d ) < 0 (N double β ( α + μ )+ β ( α + μ ) roots), g = −(γ1 + γ2 + d ) < 0 (N double roots). If g = det 1 1 μ 2+μ 2+d 2 1 A − (λ + d )UN ≤ 0, i.e., when λmax (A ) < 1
2
(μ1 +μ2 +d )(λ+d ) β1 (α1 +μ2 )+β2 (α2 +μ1 ) , all eigenvalues are less than 0, and the system (7) is asymptotically stable at the disease-free equi-
librium point.
Theorem 6. If λmax (A ) < λβ+d , β = max{β1 , β2 }, then the disease-free equilibrium point of the system (7) is exponential asymptotically stable. Proof. The proof process is similarity with Theorem 4.
3. Numerical simulation Supposing that the initial density of individuals without self-protection consciousness is ρS (0 ) = 0.5, the initial density of the self-protected individuals is ρSa (0 ) = 0.49, and the density of latent individuals is ρE (0 ) = 0.01, the individual density of the explosive state ρI (0 ) = 0. In addition, we let ρ E (∞) and ρ I (∞) denote the density of the latent individual and the explosive individual when the system reaches equilibrium, respectively. Let ρEI (∞ ) = ρE (∞ ) + ρI (∞ ) indicate the total infected density when the system approach to the stable equilibrium state. Fig. 2 depicts the variation of ρ EI (∞) and R0 as a function of λ in the infectious disease transmission system when the individuals are uniformly mixed. As can be seen from Fig. 2, as the parameter λ increases, the basic reproduction number R0 gradually reduces, and the total density of the latent and explosive individuals ρEI (∞ ) = ρE (∞ ) + ρI (∞ ) gradually decrease when the system reaches equilibrium. When λ < 0.1, R0 > 1, the final latent individual and the explosive individual in the system will always exist. However, once the values of λ > 0.1, R0 < 1, the infected individuals could not exist in the final system. This result is consistent with our theoretical analysis. In addition, by observing Fig. 2, it can be found that in the presence of endemic diseases (the left half of the red dotted line), with the increase of R0 , the value of ρ EI (∞) gradually increases. Fig. 3 shows how the total density of ρEI (∞ ) = ρE (∞ ) + ρI (∞ ) changes with β 1 and β 2 , we find that the value of ρ EI (∞) increase with the values of β 1 and β 2 . Individuals’ ability to prevent infectious diseases can effectively reduce the Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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Fig. 2. The basic reproducible number R0 and the total density of the latent individual and the explosive individual in the final system ρEI (∞ ) = ρE (∞ ) + ρI (∞ ) change with the variation of the parameter λ. Other parameters are set to be: α1 = α2 = γ1 = γ2 = 0.2, β1 = 0.6, β2 = 0.4, μ1 = μ2 = 0.3, d = 0.4. According to R0 = β1 ((αλ1++dμ)(2μ)++β2μ(a+2 +d )μ1 ) , we can calculate R0 = 1 when λ = 0.1. The left half of the red dotted line is endemic infectious disease, the right half 1
2
is disease free. It can be found that in the presence of endemic diseases, the value of ρ EI (∞) gradually increases with the R0 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. The value of ρ EI (∞) in the final system changes with β 1 and β 2 . We only describe the case of β 1 > β 2 . Other parameters are set to be: α1 = α2 = γ1 = γ2 = 0.2, μ1 = μ2 = 0.3, d = 0.4. (a ) λ = 0.1. (b) λ = 0.2. It can be found that with the increase of parameter λ, the density of infected persons in the final system can be effectively reduced.
probability of infection. By comparing Fig. 3(a) and (b), we observe that with the same parameter value (β 1 , λ), the value of ρ EI (∞) declines with the β 2 . For instant, in Fig. 3(a), even the value of β 1 > 0.8, when β 2 ≈ 0, we could get ρ EI (∞) ≈ 0. At the same time, it can be found that with the increase of parameter λ, the density of infected persons in the final system can be effectively reduced. Fig. 4 depicts the total density of ρ EI (∞) changes with the values of μ1 and μ2 . From Fig. 4(a) and (b), it can be found that with the increase of the parameter μ1 , the density of infected persons ρ EI (∞) in the final system gradually decreases. But with the increase of the parameter μ2 , the density of infected persons in the final system ρ EI (∞) gradually increases. The magnitudes of the parameters μ1 and μ2 reflect the possibility of individuals taking self-protection measures. With the increase of the parameter μ1 value, the individuals S who have not taken self-protection measures have a higher probability of taking the initiative to take self-protection; on the other hand, as the value of the parameter μ2 decreases, the probability that the individual Sa take the self-protection measure continues to maintain good protection measures becomes larger. Comparing Fig. 4(a) and (b), it can be found that when the value of μ1 is larger and the value of μ2 is smaller, there will be no infected person in the whole system in the last. We study the propagation dynamics of infectious diseases under different network topologies. For convenience, we assume that birth and death do not occur in the short term, i.e. α1 = α2 = d = 0. Then we can get the basic reproduction β μ +β μ number R0 = λ1(μ2 +μ2 )1 . It should be noted that the basic reproduction number R0 is obtained in a homogeneous mixed 1
2
population, However, for the general inter-individual connection structure, we may not get such a result. Next, we mainly perform the proposed model on the following two network topologies. (1) The ER random network with N = 10 0 0 players and the average degree < k >= 6, or (2) the BA scare-free network with N = 10 0 0, the degree obeys the distribution Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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Fig. 4. The total density of ρ EI (∞) changes with μ1 and μ2 . Others parameters are set to be: α1 = α2 = γ1 = γ2 = 0.2, β1 = 0.6, β2 = 0.4, d = 0.4. (a ) λ = 0.05. (b) λ = 0.1. It can be found that when the value of μ1 is larger and the value of μ2 is smaller, there will be no infected person in the whole system in the last.
Fig. 5. Evolution of the total density of E and I in different network topologies over time. The maximum eigenvalues of the ER random network and the BA BA scale-free network connection topology matrix are gER max = 7.2159 and gmax = 15, respectively. The parameters are set to be: β1 = 0.06, β2 = 0.05, μ1 = μ2 = 0.3, γ1 = γ2 = λ = 0.2. (a) The change of ρ EI (∞) in ER random network; (b) The change of ρ EI (∞) in BA scale-free network.
P (k ) ∼ k−γ1 with γ1 = 2.1 and the average degree < k >= 6. Based on the Monte Carlo method, we simulate proposed model. Since the results differ for each Monte Carlo trial, we present the results averaged over 100 independent runs. Fig. 5 describe the evolution of the total density of E and I in different network topologies over time. According to calculation, the maximum eigenvalues of the ER random network and the BA scale-free network connection topology matrix β 1 μ2 + β 2 μ1 BA 0 are gER max = 7.2159 and gmax = 15. The ER random network is a homogeneous network, so in Fig. 5(a), when R = λ(μ +μ ) = 1
2
0.3 < 1 the infection density in the whole system ρEI (∞ ) = 0. However, since we consider the connection topology between β μ +β μ individuals and the degree of heterogeneity of the BA scale-free network is large, even in Fig. 5(b), R0 = λ1(μ2 +μ2 )1 = 0.3 < 1, 1
2
and final density of the infected person in the whole system ρ EI (∞) > 0. Fig. 6 depicts the change in the final infected density ρ EI (∞) in the ER random network and the BA scale-free network with changes in infection rate β 2 and parameter λ. It can be found from Fig. 6(a) and (b) that under different network structures, as the parameter λ increases, the final infected density is getting smaller. Therefore, the parameter λ is an important parameter for controlling the spread of infectious diseases. When it is greater than a certain threshold, there will be no infected person in the final infectious disease transmission system. In addition, as the infection rate β 2 increases, the density of the final infected person is also becoming smaller. It is worth noting that, in comparison with Fig. 6(a) and (b), it can be found that, the value of λ, when infected person disappears in the BA free-scale network, is 1.3 times that of the ER random network. Finally, we analyze how the value of β 1 and β 2 affect the final infection density ρ EI (∞) in ER random networks and BA scale-free networks. By comparing the analysis the Fig. 7(a) and (b), one can find that with the infection rates β 1 and β 2 increase, the final infected person density ρ EI (∞) gradually increases. Interestingly, even if β 1 ≈ 1 and β 2 ≈ 1, ρ EI (∞) ≈ 0.8 < 1, however, once β 1 → 0 and β 2 → 0, there will be no infected people in the network. Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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Fig. 6. Under different β 2 values, the change of ρ EI (∞) in ER random network and BA scale-free network with λ. Other parameters are set to be: μ1 = μ2 = 0.3, γ1 = γ2 = λ = 0.2. (a) The change of the ρ EI (∞) in the ER random network; (b) The change of the ρ EI (∞) in the BA scale-free network. It can be found that under different network structures, as the parameter λ increases, the final infected density is getting smaller.
Fig. 7. The change of final infection density ρ EI (∞) in ER random network and BA scale-free network with the infection rates β 1 and β 2 . Parameters are set to be: μ1 = μ2 = 0.3, γ1 = γ2 = λ = 0.2. (a) The change of ρ EI (∞) value in ER random network; (b) The change of ρ EI (∞) value in BA scale-free network. One can find that with the infection rates β 1 and β 2 increase, the final infected person density ρ EI (∞) gradually increases.
4. Conclusion Considering the individuals’ ability to avoid risks in the epidemic season, this paper proposed a new SSa EI model for the spread of disease. Based on the mean field theory and the next-generation matrix method, we calculated the threshold of infectious disease transmission. At the same time, we presented the conditions of the locally, globally and exponential asymptotically stable at the disease-free equilibrium state. In addition, we have performed our models in the ER random network and the BA scale-free network, respectively. The results showed that the probability of the latent individuals transforming into explosive individuals has a greater impact on the spread of infectious diseases. In short, we provided a theoretical framework to study the epidemic spread, our results suggested that improving individuals’ risk awareness is beneficial to suppressing the epidemic spreading. In the future, we will do more efforts on the epidemic spreading with the eyes of empirical data and experimental supports.
Acknowledgments This research was supported by the National Nature Science Foundation of China (No. 61803184, 71774070, 61973143, 71974080); National Nature Science Foundation of Jiangsu Province (Nos. BK20180851, BK20190832); China Postdoctoral Science Foundation (2018M640326); Social Science Fund of Jiangsu Province (No. 18TQD002); the Natural Science Research Projects of Jiangsu Higher Education Institutions (No. 19KJB120 0 01); the Jiangsu Postdoctoral Research Funding Project (No. 2019K172). Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894
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Please cite this article as: D. Han, Q. Shao and D. Li et al., How the individuals’ risk aversion affect the epidemic spreading, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124894