Dynamics of discrete epidemic models on heterogeneous networks

Journal Pre-proof Dynamics of discrete epidemic models on heterogeneous networks Xinhe Wang, Junwei Lu, Zhen Wang, Yuxia Li

PII: DOI: Reference:

S0378-4371(19)31690-5 https://doi.org/10.1016/j.physa.2019.122991 PHYSA 122991

To appear in:

Physica A

Received date : 26 June 2019 Revised date : 26 August 2019 Please cite this article as: X. Wang, J. Lu, Z. Wang et al., Dynamics of discrete epidemic models on heterogeneous networks, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.122991. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Journal Pre-proof

Dynamics of discrete epidemic models on heterogeneous networks

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Xinhe Wang1 , Junwei Lu2 , ∗Zhen Wang1 , Yuxia Li3 1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China 2 School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210023, China 3 College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China

Abstract

Keywords:

Discrete epidemic model, Heterogeneous networks, Basic reproduction number, Stability

and bifurcation

1

Pr e-

In this paper, one considers the discrete epidemic models on heterogeneous networks, stability and bifurcation of the endemic equilibrium of SIS model is studied, the dynamics analysis of SIRS model and the immunization are discussed. Compared with the epidemic model established by differential equations, the difference systems have the same stability conditions at the disease free equilibrium, and the dynamical behaviours of endemic equilibrium is more complex for discrete epidemic model. These obtained results can be used to explore the complex epidemic propagation dynamics on the networks.

Introduction

Jo

urn

al

There are a large number of complex systems in nature and human society, they can be described by networks consist of many different individuals, called vertices or nodes, and the intricacy connections between the vertices, called links [1, 2]. As can be seen, complex networks played an important role in many different fields, such as the internet, the social networks, the neural networks and the biology networks [3, 4, 5, 6]. In mathematics, the network has been studied for a long period in the form of graph theory, and the random graph theory has pioneered the systematic study of the complex networks [7, 8, 9]. Since the introduction of small world networks [10] and scale-free networks [11], complex networks have received widespread attention and entered a new era [12]. Many researchers were committed to studying the topological properties of networks [13, 14, 16] and the dynamical propagation on networks [15, 17, 18, 19, 20]. In [21], the authors proposed a epidemic model on scale-free networks with N nodes. The numbers of the nodes with degree k is Nk , k = 1, 2 . . . , n. The model is defined as dρk (t) = −ρk (t) + λk(1 − ρk (t))Θ(t), dt

where ρk (t) representsPthe relative density of infected nodes with degree k, and λ is the spreading rate of n kp(k)ρ(t) the disease. Θ(t) = k=1 hki is the probability that a link points to a infected node, in which, hki = Pn hki Nk k=1 kp(k) and p(k) = N . According the study of Ref. [21], the threshold is calculated as λc = hk2 i , P n where hk 2 i = k=1 k 2 p(k). Under certain conditions, hk 2 i is infinite and the authors discovered the absence of the threshold, which overturns the threshold theory of the traditional epidemic dynamics. Since then, people begun to recognize the important impact of network topologies on the dynamics of epidemic model [22, 23]. ∗ Corresponding

author: wangzhen [email protected]

1

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In [24, 25], the stability of the endemic equilibrium was discussed, it indicates that the endemic equilibrium is asymptotically stable if λ > λc . Some immunization strategies were studied and compared in [26, 27, 28], it shows that the immune strategy against network structure is more effective. Up to now, based on differential equations, there have been a lot of achievements in the modelling and dynamics of epidemic models on the networks [29, 30, 31]. Difference equations have a long history of being used to establish biological models, Fibonacci sequence is a good example [32]. As a recursive relation, the difference system is easier to understand and accept, because the data collection has a certain time interval, which fits with the characteristics of the difference equation. More importantly, dynamics of difference equations are more complex and challenging, even a simple form like logistic model has complex dynamics [33, 34, 35, 36]. On the other hand, difference system is the discretization of the continuous equation [37], it is more convenient in numerical calculation. Thus, the investigation of the discrete epidemic model on networks is valuable. Recently, a discrete-time SIS epidemic model on heterogeneous networks is established in [38], the model is formulated as    Sk (t + 1) = Sk (t)(1 − hβkΘ(I(t))) + hrIk (t), Ik (t + 1) = Ik (t)(1 − rh) + βhkSk (t)Θ(I(t)), t = 0, 1, 2, · · · ,   0 ≤Sk (0) ≤ Nk , 0 ≤ Ik (0) ≤ Nk ,

2

urn

al

Pr e-

where Sk (t) and Ik (t) denotes susceptible and infected individuals with degree k. h is the time step-size. β and r represent the transmission rate of the disease and recover rate. The properties of the solutions and the basic reproduction number are studied in [38], and it indicates that discrete system has more complex dynamical behaviours at the endemic equilibrium. However, the theory analysis of the discrete system is not enough, and the discrete epidemic model on networks has just germinated. Therefore, there are many problems to be solved in the discrete-time epidemic model on the networks. Inspired by the above discussion, the dynamics of the endemic equilibrium of the discrete SIS model on networks is studied. The main contribution is highlight as follows. For the discrete SIS epidemic model, the boundedness and positivity of the solutions are studied, and sufficient condition of the bifurcation phenomenon is provided. The basic reproduction number of the SIRS model is calculated and one provides the stability conditions for the equilibrium points. Some immunization strategies for the discrete SIRS epidemic model are discussed. The organization structure of this paper is as follows. In Section 2, dynamical analysis of the discrete SIS epidemic model is provided. Section 3 proposes a discrete SIRS epidemic model, the solutions of the model is analyzed, the basic reproduction number is calculated and the stability of the equilibrium points are discussed. Some immunization strategies are studied in Section 4. In Section 5, some numerical examples are illustrated and Section 6 is the conclusion.

Dynamics of the endemic equilibrium

According to the research results on [38], discrete SIS epidemic system has a disease-free equilibrium E0 (S10 , S20 , · · · , Sn0 , I10 , I20 , · · · , In ) = (N1 , N2 , · · · , Nn , 0, 0, · · · , 0), 2

Jo

i and the basic reproduction number is R0 = βhk rhki . It has been proved that the disease-free equilibrium is globally asymptotically stable if R0 < 1. As the simulation results in Ref. [38] shows, when R0 > 1, the endemic equilibrium may unstable and the bifurcation phenomenon occurs. In this section, the stability and bifurcation of endemic equilibrium are studied. 1 }. Lemma 1. Solutions to the discrete-time SIS model are positive and bounded if h < min{ r1 , βk

Proof. According to the discrete SIS epidemic system , one has Ik (1) = Ik (0)(1 − rh) + βhkSk (0)Θ(0) ≥ Ik (0)(1 − rh). 2

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If 0 ≤ Ik (0) ≤ Nk , one obtains Ik (1) ≥ Ik (0)(1 − rh) ≥ 0. By induction, Ik (t) ≥ 0 for all t ≥ 0 if 0 ≤ Ik (0) ≤ Nk . Similarly, it can be obtained that Sk (1) = Sk (0)(1 − hβkΘ(I(0))) + hrIk (0), ≥ Sk (0)(1 − hβkΘ(I(0)))

of

≥ Sk (0)(1 − hβk).

1 1 If h < βk , it has Sk (1) ≥ 0. By induction, Sk (t) ≥ 0 for all t ≥ 0 if h < βk . Thus, with the initial condition 0 ≤ Ik (0) ≤ Nk , it indicates that Sk (t) + Ik (t) = Nk and 0 ≤ Sk (t), Ik (t) if 1 }. This completes the proof. h < min{ r1 , βk

p ro

Based on the relationship Sk (t) + Ik (t) = Nk , the discrete SIS epidemic system can be written as Ik (t + 1) = Ik (t)(1 − rh) + βhk(Nk − Ik (t))Θ(t),

t = 0, 1, 2, · · · .

(1)

Pr e-

According to the results in Ref. [38], system (1) has an endemic equilibrium I ∗ = (I1∗ , I2∗ , · · · , In∗ ) if R0 > 1, ∗ Nk where Ik∗ = βkΘ r+βkΘ∗ . Applying the linearization method, one obtains the Jacobian matrix A of system (1) at I ∗ ,   βh βh ··· 1 − rh + Nβh   hki g(1, 1) N hki g(1, 2) N hki g(1, n)       βh βh βh   g(2, 1) 1 − rh + g(2, 2) · · · g(2, n)   N hki N hki N hki     A= ,  . . .  ..   . . .   .  . . .       βh βh · · · 1 − rh + Nβh N hki g(n, 1) N hki g(n, 2) hki g(n, n) n×n

where g(a, b) = aNa − aN hkiΘ∗ − abIa∗ . Clearly, the local stability of system (1) at I ∗ is determined by ρ(A), where ρ(A) represents the spectral radius of matrix A. According to the matrix theory, one gives the following result.

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Theorem 1. When R0 > 1, the endemic equilibrium I ∗ is locally asymptotically stable if ρ(A) < 1, unstable if ρ(A) > 1.

urn

Theorem 1 is the stability result of the endemic equilibrium I ∗ of system (1), and matrix A can be obtained after calculating the values of I ∗ and Θ∗ . Although Theorem 1 provides a exact way to ensure the stability of I ∗ , it requires huge calculation and not concise enough in judging stability. In [38], it has showed that there exists bifurcation phenomenon at the endemic under the condition R0 > 1, but for the continuous system, the endemic equilibrium is globally asymptotically stable when R0 > 1. In [24, 25], the authors investigated the stability of the endemic equilibrium and proved it, but these results are not suitable for system (1) especially for the bifurcation situation. To investigate the bifurcation of the endemic equilibrium of system (1), one introduces a new method and provides a sufficient condition to ensure the bifurcation phenomenon of system (1).

Jo

Theorem 2. When R0 > 1, the endemic equilibrium I ∗ is unstable and the bifurcation occurs if the following condition hold n βh X 2 Nk 2 − Ik∗ 2 k p(k) 1 + rh(R0 − 1) − > 1. hki Nk k=1

3

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Proof. For Θ(t) =

Pn

kIk (t) k=1 N hki ,

Θ(t + 1) =

one gets

n X kIk (t + 1) N hki k=1 n X

(2)

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1 [(1 − rh)Ik (t) + βhk(Nk − Ik (t))Θ(t)] N hki k=1 Pn Pn Pn βhΘ(t) k=1 k 2 Nk βhΘ(t) k=1 k 2 Ik (t) k=1 kIk (t) = (1 − rh) + − N hki N hki N hki n 2 βhΘ(t) X 2 βhhk i Θ(t) − k Ik (t) = (1 − rh)Θ(t) + hki N hki =

k=1

n βhΘ(t) X

= (1 + rh(R0 − 1))Θ(t) −

N hki

k 2 Ik (t).

k=1

Pr e-

If the endemic equilibrium Ik∗ satisfies Ik (t + 1) = Ik (t), then applying the definition of Θ, one has that Θ(t + 1) = Θ(t). If system (1) is stable at the endemic equilibrium I ∗ , system (2) will be stable at the endemic equilibrium I ∗ . In turn, if equation (2) is unstable, then system (1) is unstable. ∗ Nk It is calculated that Ik∗ = βkΘ r+βkΘ∗ , thus, one has n βhΘ∗ X 2 βkΘ∗ Nk Θ = (1 + rh(R0 − 1))Θ − k N hki r + βkΘ∗ ∗

∗

k=1

n βkΘ∗ βhΘ∗ X 2 = (1 + rh(R0 − 1))Θ∗ − k p(k) . hki r + βkΘ∗ k=1

Clearly, Θ∗ is the equilibrium point of equation (2). On the other hand, it is easy to obtains lim

βkΘNk = Ik∗ . r + βkΘ

al

Θ→Θ∗

urn

k Thus, Ik = βkΘN r+βkΘ can be used as a special approximation function. In this study, let Ik = can be written as follows

Θ(t + 1) = f (Θ) = (1 + rh(R0 − 1))Θ −

βkΘNk r+βkΘ ,

system (2)

n βhΘ2 X 2 βk k p(k) . hki r + βkΘ k=1

Then, it is calculated that

Jo

f ′ (Θ∗ ) = 1 + rh(R0 − 1) − = 1 + rh(R0 − 1) −

n

βh X 2 2rβkΘ∗ + (βkΘ∗ )2 k p(k) hki (r + βkΘ∗ )2 βh hki

k=1 n X k=1

k 2 p(k)

Nk 2 − Ik∗ 2 . Nk

If |f ′ (Θ∗ )| < 1, Θ(t + 1) = f (Θ(t)) is locally asymptotically stable at Θ∗ . If |f ′ (Θ∗ )| > 1, Θ(t + 1) = f (Θ(t)) is unstable at Θ∗ . Thus, system (1) is unstable if |f ′ (Θ∗ )| > 1. This completes the proof. For the continuous system, the results in [24, 25] indicate that the endemic equilibrium is globally asymptotically stable if R0 > 1. For system (1), when R0 > 1, bifurcation phenomenon occurs under the appropriate conditions. It is easy to see that discrete SIS epidemic on heterogeneous networks provides complex dynamic properties. 4

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3

Dynamics of a SIRS model

of

In this section, the population is divided into three subgroups: susceptible individuals, infective individuals and recovered individuals. The infective individuals will recover from the disease and become recovered individuals with temporary immunity. The discrete SIRS model on networks is formulated as  Sk (t + 1) = Sk (t)(1 − βhkΘ(t)) + µhRk (t),     I (t + 1) = I (t)(1 − δh) + βhkS (t)Θ(t), k k k (3)  R (t + 1) = R (t)(1 − µh) + δhI k k k (t),    0 ≤ Sk (0), Ik (0), Rk (0) ≤ Nk ,

p ro

where Rk is the numbers of the recovered individuals and Nk = Sk (t) + Ik (t) + Rk (t). δ is the recover rate of infective ones form the disease, and µ is the rate of lose immunity. 1 1 1 Lemma 2. The solutions of system (3) are non-negative and bounded if h ≤ min{ βk , δ , µ }.

Proof. The initial conditions is that 0 ≤ Sk (0), Ik (0), Rk (0) and Sk (0) + Ik (0) + Rk (0) = Nk . Then, it gives

Pr e-

Sk (1) = Sk (0)(1 − βhkΘ(0)) + µhRk (0) ≥ Sk (0)(1 − βhk).

1 If h ≤ βk , one has Sk (1) ≥ 0. Similarly, it can be obtained that Ik (1) ≥ 0 if h ≤ 1δ . It also has that Rk (1) ≥ 0 1 1 1 1 , δ , µ }. if h ≤ µ . By induction, Sk (t), Ik (t), Rk (t) ≥ 0 for all t ≥ 0 if h ≤ min{ βk On the other hand, the number of nodes is a positive constant and Sk (t) + Ik (t) + Rk (t) = Nk for all t ≥ 0. 1 1 1 , δ , µ }. This completes the proof. Thus, one has that the solutions are bounded if h ≤ min{ βk

System (3) has a disease-free equilibrium E0 , and it is calculated as E0 = (S10 , S20 , · · · , Sn0 , I10 , I20 , · · · , In0 , R10 , R20 , · · · , Rn0 ) = (N1 , N2 , · · · , Nn , 0, 0, · · · , 0, 0, 0, · · · , 0). 2

al

i 1 1 1 Theorem 3. Supposed that h ≤ min{ βk , δ , µ }, then there exists a basic reproduction number R1 = βhk δhki . Moreover The disease-free equilibrium E0 of system (3) is locally asymptotically stable if R1 < 1 and unstable if R1 > 1.

urn

Proof. Substituting Rk (t) = Nk − Sk (t) − Ik (t) into the first equation of system (3), one has    Ik (t + 1) = Ik (t)(1 − δh) + βhkSk (t)Θ(t), Sk (t + 1) = Sk (t)(1 − µh − βhkΘ(t)) + µhNk − µhIk (t),   Rk (t + 1) = Rk (t)(1 − µh) + δhIk (t).

(4)

where

Jo

Applying the method of computing the basic reproduction number R0 of discrete epidemic, one obtains the Jacobian matrix of system (4) at E0    F +T 0    J = ,  A C 3n×3n             F =          

βhN1 N hki

··· .. .

βhkN1 N hki

··· .. .

βnhN1 N hki

βhkNk N hki

.. .

··· .. .

βhk2 Nk N hki

.. .

··· .. .

βnkhNk N hki

βhkNn N hki

···

βhnkNn N hki

···

.. .

.. .

5

.. . .. .

βn2 hNn N hki

                      

n×n

,

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··· .. . ··· .. . ···

. 1 − δh

1 − βhkN N hki .. . 2 Nk − βhk N hki − µh .. . βhnkNn − N hki

..

.

,

n×n

··· .. . ··· .. .

1 − βnhN N hki .. . k − βnkhN N hki .. . βn2 hNn − N hki − µh

···

p ro

 βhN1 − N hki − µh     ..    .    βhkNk   −  N hki    ..  .  A=    βhkNn  − N hki      δh        

..

        

δh

 1 − µh     C=   

..

.

,

2n×n

.

2n×2n

Pr e-

1 − µh

        

                                    

of

 1 − δh      T =  

According to the result in [39], if matrices F and T are non-negative, F + T is irreducible, ρ(C) < 1 and ρ(T ) < 1, the basic reproduction number R0 for system (3) is defined as R0 = ρ(F [E − T ]−1 ), where E is the n × n identity matrix. 1 1 1 , δ , µ }, it has F + T is irreducible, and ρ(C), ρ(T ) < 1. Then, one has that If h ≤ min{ βk βN1 δN hki

.. .

βkN1 δN hki

··· .. .

.. .

2

··· .. .

βnN1 δN hki

.. .

βkNk δN hki

.. .

··· .. .

βk Nk δN hki

.. .

··· .. .

βnkNk δN hki

βkNn δN hki

···

βnkNn δN hki

···

βn2 Nn δN hki

al

F [E − T ]−1

            =          

.. .

                      

,

n×n

It is clearly that the rank of F [E − T ]−1 is 1. Thus, the basic reproduction number is calculated as n

urn

R1 = ρ(F [E − T ]−1 ) = tr(F [E − T ]−1 ) =

β X 2 βhk 2 i k p(k) = . δhki δhki k=1

Then, one obtains that E0 is locally asymptotically stable if R1 < 1 and unstable if R1 > 1. This completes the proof. 1 1 1 Theorem 4. If µδ R1 < 1 and h ≤ min{ βk , δ , µ }, the disease-free equilibrium E0 of system (3) is global asymptotically stable .

Jo

Proof. Considering the following Lyapunov function V (Sk (t), Ik (t), Rk (t)) =

n

n

n

k=1 n

k=1

k=1

X X X 1 1 1 k(Nk − Sk (t)) + kIk (t) + kRk (t) 2hN hki 2hN hki 2hN hki

1 X = k(Nk − Sk (t)). hN hki k=1

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There introduces the difference operate ∆. For a difference function x(t), it has ∆x(t) = x(t + 1) − x(t). Then, one has that ∆V (Sk (t), Ik (t), Rk (t)) = V (Sk (t + 1), Ik (t + 1), Rk (t + 1)) n n 1 X 1 X = k(Nk − Sk (t + 1)) − k(Nk − Sk (t)) hN hki hN hki

1 N hki

k=1 n X

k=1

of

=

1 hN hki

k=1

k(Sk (t) − Sk (t + 1))

k(βkSk (t)Θ(t) − µRk (t)).

p ro

=

k=1 n X

Substituting Rk (t) = Nk − Sk (t) − Ik (t) into ∆V (Sk (t), Ik (t), Rk (t)), one has n

∆V (Sk (t), Ik (t), Rk (t)) =

1 X k(βkSk (t)Θ(t) − µNk + µSk (t) + µIk (t)) N hki k=1 n X

n n n µ X µ X µ X kNk + kSk (t) + kIk (t) N hki N hki N hki k=1 k=1 k=1 k=1 ! ! n n 1 X 2 µ X = Θ(t) k Sk (t) − µ + µ k(Sk (t) − Nk ) N hki N hki

Θ(t) N hki

k 2 Sk (t) −

Pr e-

=

k=1

It is clearly that Sk (t) − Nk ≤ 0 and n

k=1

n

1 X 2 1 X 2 k Sk (t) − µ ≤ k Nk (t) − µ N hki N hki k=1

al

k=1

βhk 2 i = −µ hki   δ =µ R1 − 1 . µ

urn

1 1 1 , δ , µ }, ∆V = 0 if and only if Sk (t) = Nk , then E0 is One obtains that ∆V ≤ 0 if µδ R1 < 1 and h ≤ min{ βk global asymptotically stable. This completes the proof.

Theorem 5. System (3) has a endemic equilibrium E ∗ if R1 > 1, where E ∗ = (S1∗ , S2∗ , · · · , Sn∗ , I1∗ , I2∗ , · · · , In∗ , R1∗ , R2∗ , · · · , Rn∗ ),

and

µδ Nk , δµ + δβkθ∗ + µβkθ∗

Jo

Sk∗ =

Ik∗ =

µβkθ∗ Nk , δµ + δβkθ∗ + µβkθ∗

∗ RK =

δβkθ∗ Nk . δµ + δβkθ∗ + µβkθ∗

Proof. The endemic equilibrium of system (3) satisfies that Sk∗ (t + 1) = Sk∗ (t), Ik∗ (t + 1) = Ik∗ (t) and Rk∗ (t + 1) = δ ∗ Rk∗ (t), then, one gets that Rk∗ = µδ Ik∗ and Sk∗ = βkΘ ∗ Ik . considering that δ ∗ δ I + I ∗ + Ik∗ µ k βkΘ∗ k δβkΘ∗ + µδ + µβkΘ∗ ∗ = Ik µβkΘ∗ = Nk .

Rk∗ + Sk∗ + Ik∗ =

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∗

∗ ∗ It is clearly that Ik∗ = δβkΘ∗µβkΘ +µδ+µβkΘ∗ Nk . Thus, there exists a positive Ik if Θ is positive and satisfies P ∗ n Θ∗ = N 1hki k=1 k δβkΘ∗µβkΘ +µδ+µβkΘ∗ Nk . One defines a function as follows n

1 X µβkΘ k Nk . N hki δβkΘ + µδ + µβkΘ k=1

of

F (Θ) = Θ − It is calculated that F (0) = 0 and

n

1 X µβk k Nk N hki δβk + µδ + µβk k=1

p ro

F (1) = 1 −

n 1 X µβk =1− kp(k) hki δβk + µδ + µβk k=1

> 0.

Obviously, if F ′ (0) < 0 and F ′′ (Θ) > 0, 0 ≤ Θ ≤ 1, equation (2) has a positive root 0 ≤ Θ∗ ≤ 1. It is calculated that n

Pr e-

1 X µ2 δβk kp(k) F (Θ) = 1 − , hki (δβkΘ + µδ + µβkΘ)2 ′

k=1

and

n

F ′′ (Θ) =

1 X 2µ2 δβk(δβk + µβk) kp(k) > 0. hki (δβkΘ + µδ + µβkΘ)3 k=1

F ′ (0) = 1 − R0 , if R1 > 1, it has F ′ (0) < 0, then equation (2) has a unique positive root 0 ≤ Θ∗ ≤ 1. System has a endemic equilibrium E ∗ if R1 > 1. This completes the proof.

urn

al

Next, one considers the stability of endemic equilibrium E ∗ of system (3). Applying the relationship Rk = Nk − Sk − Ik , system (3) can be written as    Sk (t + 1) = Sk (t)(1 − uh − βhkΘ(t)) + µhNk − uhIk (t), Ik (t + 1) = Ik (t)(1 − δh) + βhkSk (t)Θ(t), (5)   0 ≤ Sk (0), Ik (0).

The Jacobian matrix of system (5) at E ∗ is as follows    B11 B12   B= ,   B21 B22 2n×2n

Jo

where

B11

 1 − uh − βhΘ∗          =        

..

. 1 − uh − βhkΘ∗

..

. 1 − uh − βhnΘ∗

8

                  

,

n×n

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βhS1∗ N hki

βhkS1∗ N hki

+ uh . . . .. .. . . ∗ βhkSk ... N hki .. .. . .

∗ βhnSn N hki

... .. .. . . βhk2 Sk∗ N hki + uh . . . .. .. . . ∗ βhnkSn ... N hki

...

B21 = (1 − uh)E − B11 , B22 = (1 + uh − δh)E − B12 . Applying the matrix theory, the following result can be obtained.

      ..   .    ∗  βhnkSk    N hki   ..    .   2 ∗  βhn kSn  + uh N hki βhnS1∗ N hki

,

n×n

of

B12

            = −           

4

p ro

Theorem 6. When µδ R1 > 1, the endemic equilibrium E ∗ of system (3) is locally asymptotically stable if ρ(B) < 1, unstable if ρ(B) > 1.

Immunization strategies

Pr e-

Immunization control is an important subject in the dynamics of epidemic models on networks. In this section, based on system (3), one considers some immunization strategies [26, 28]: uniform immunization, proportional immunization and acquaintance immunization. Uniform immunization is a method to immunize susceptible nodes with certain proportion λ(0 < λ < 1), which ignores the heterogeneity of networks, the proportion of susceptible nodes can be infected becomes 1 − λ. In this case, the transmission function λβhkΘ(t) of system (3) becomes (1 − λ)βhkSk (t)Θ(t).

According to the calculation of basic reproduction number of system (3), one obtains a new basic reproduction number R1U = (1 − λ)R1 . Considering the heterogeneity of networks, in the way of proportional immunization, the proportion of the immunized nodes with degree k is λk (0 < λk < 1), and it satisfies

al

β(1 − λk )k = c,

where c is a positive constant. Then one obtains

urn

λk = 1 −

c , kβ

it indicates that the degree k of immune nodes satisfy k > βc . In this case, the transmission function λβhkΘ(t) of system (3) becomes chSk (t)Θ(t)), and the basic reproduction number becomes c X β X 2 R1P = kp(k) + k p(k). δhki δhki c c k> β

For the system that most nodes with degree k >

c β,

k< β

one has

Jo

c R1P ≈ . δ Proportional immunization requires an understanding of the distribution of degrees across the entire network, which is difficult to implement.The effect of the method is depended on the constant c. The basic idea of acquaintance immunization is to select a certain proportion λ of nodes and randomly select one neighbour of each of them as immune objects. The probability of immunization nodes with degree k is calculated as follows λk =

kp(k) λkp(k) × λN = . N hki hki 9

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In this case, the transmission function λβhkΘ(t) of system (3) becomes λkp(k) )kSk (t)Θ(t)). hki The basic reproduction number is calculated as n

n

k=1

k=1

β X 2 λβ X 3 p(k) k p(k) − k p(k) δhki δhki hki

of

R1A =

p ro

β(hk 2 ihki − λhk 3 p(k)i) = . δhki2

Here, one calculates the average immunity rate (Λ) of the susceptible, it is defined as Λ=

n X

kλk =

k=1

n X λkp2 (k) k=1

Then, one has

hki

.

Pr e-

hk 2 λk i Λhk 2 i + σ hk 2 λkp(k)i = = , 2 hki hki hki where σ is the covariance of k 2 and λk . Thus, it has that R1A =

βhk 2 i(1 − Λ) − σ βhk 2 i < (1 − Λ) ≈ R1U . δhki δhki

Numerical simulation

urn

5

al

It indicates that acquaintance immunization is more effective than uniform immunization. Compared to uniform immunization, proportional immunization has different injection ratios for different degrees of nodes, and the immune effect is better, but the operation requirements are higher because you need to know the exact degree distribution of the network. The immune effect of acquaintance immunization is better, and it is easier to operate than proportional immunity in operation. It is not necessary to know the specific distribution of the network, and the effect is good and the operation is convenient.

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In this section, one gives some examples to verify the results in the previous section. Considering a network with 100 nodes, it indicates that N = 100. Supposed that it contains two types of nodes, N5 = 80 and N10 = 20. It is calculated that hki = 6 and hk 2 i = 40. Example 1. Considering the SIS epidemic model (1). Let r = 0.1, β = 0.05. It is calculated that ∗ R0 = 1.667 > 1 and I5∗ = 50.263, I10 = 15.434. The initial condition is I5 (0) = 30, I10 (0) = 20. Let h = 1, it is calculated that ρ(A) = 0.8045 < 1, according to Theorem 1, the endemic equilibrium of system (1) at I5∗ ∗ and I10 are locally asymptotically stable, the result is shown in Fig. 1 (a). Let h = 4.9, it is calculated that ρ(A) = 1.0576 > 1, the endemic equilibrium of system (1) is unstable, the result is shown in Fig. 1 (b). Example 2. Considering the SIRS epidemic model (3). Let β = 0.01, δ = 0.1, µ = 0.1. It is calculated that R1 = 0.6667 < 1, then the disease-free equilibrium of system (3) is locally asymptotically stable. The stability of the disease-free equilibrium is shown in Fig. 2, the initial conditions is S5 (0) = 40, I5 (0) = 15, R5 (0) = 25, S10 (0) = 10, I10 (0) = 2, R10 (0) = 8. Example 3. In this example, one considers the endemic equilibrium of system (3). Let β = 0.1, δ = 0.1, h = 1 and µ = 0.2. It is calculated that R1 = 6.6667 > 1, according to Theorem 5, system (3) has a endemic equilibrium E ∗ , where ∗ ∗ ∗ E ∗ = (S5∗ , S10 , I5∗ , I10 , R5∗ , R10 ) = (15.4496, 2.1376, 43.0336, 11.9083, 21.5168, 5.9541).

10

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(a)

55

(b)

60

50

50 45 I5

I5

40

35 30

I10

30

of

I10

Population

Population

40

20

25 20

10 15

0

0

5

10

15

20 t

25

30

35

p ro

10

0

40

5

10

15

20 t

25

30

35

40

80 70

Population

60 50 40 30 20 10

Pr e-

Figure 1: Stability and bifurcation of the endemic equilibrium I ∗ of system (1), r = 0.1 and β = 0.05. h = 1 in (a) and h = 4.9 in (b).

0 0

S

10

I

5

I10 R5 R10

100

al

50

S5

150

t

urn

Figure 2: Stability of the disease-free equilibrium E0 of system (3), β = 0.01, δ = 0.1, h = 1 and µ = 0.1.

50 45

S5

40

S10 I5

Population

Jo

35

I

10

30

R

25

R10

5

20 15 10 5 0 0

20

40

60

80

100

t

Figure 3: Stability of the endemic equilibrium E ∗ of system (3), β = 0.1, δ = 0.1, h = 1 and µ = 0.2.

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Then, it is calculated that ρ(B) = 0.7625 < 1, applying the result in Theorem 6, the endemic equilibrium E ∗ of system (3) is locally asymptotically stable, it is shown in Fig. 3. Example 4. Considering system (3), let β = 0.2, δ = 0.1, h = 1.9 and µ = 0.2. It is calculated that the basic reproduction number R1 = 13.333 > 1, there exists a endemic equilibrium ∗ ∗ ∗ E ∗ = (S5∗ , S10 , I5∗ , I10 , R5∗ , R10 ) = (7.864, 1.034, 48.09, 12.64, 24.05, 6.322),

of

and ρ(B) = 1.2146 > 1. Applying the result in Theorem 6, the endemic equilibrium is unstable. As can be seen in Fig. 4, bifurcation phenomenon occur at the endemic equilibrium E ∗ .

p ro

80 70 60

Population

50

S I R

40 30

10 0 0

5

Pr e-

20

10

15

20

25 t

30

35

40

45

50

Figure 4: Bifurcation of the endemic equilibrium E ∗ of system (3), β = 0.2, δ = 0.1, h = 1.9 and µ = 0.2.

6

Conclusion

urn

al

This paper investigates the dynamics of discrete epidemic models on heterogeneous networks. The condition is given to ensure the non-negative and boundedness of the solutions of the discrete SIS epidemic model, the stability of the endemic equilibrium is discussed. Theorem 2 provides a new result to study the bifurcation phenomenon of the endemic equilibrium, it shows that discrete SIS epidemic models have more complex dynamic properties than the continuous counterparts. After considering the recovered population, the dynamic analysis of a discrete SIRS epidemic model is considered. Properties of the solutions is studied, the basic reproduction number is calculated and stability of the equilibriums is discussed. Some strategies of immunization on networks are considered. Finally, some numerical examples are illustrated to verify the theoretical results.

Acknowledgments

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This work was supported by the National Natural Science Foundation of China (Nos. 61573008, 61973199, 61973200, 61803235), the Natural Science Foundation of Shandong Province (No. ZR2018MF005), Taishan Scholar Project of Shandong Province of China, the SDUST graduate innovation project (No. SDKDYC190353) and the SDUST Research Fund (Nos. 2014TDJH102, 2018TDJH101).

References

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Jo

urn

al

Pr e-

p ro

of

1. The dynamics of the endemic equilibrium of the discrete SIS model on networks is discussed, a new method is provided to explore the bifurcation phenomenon. 2. A discrete SIRS epidemic model on heterogeneous networks is established, which is more realistic to describe the disease transmission. 3. The basic reproduction number of the SIRS model is calculated and one provides the stability conditions for the equilibrium points. 4. Some immunization strategies for the discrete SIRS epidemic model are discussed.