Global Stability of a Variation Epidemic Spreading Model on Complex Networks

Epidemic spreading on networks becomes a hot issue of nonlinear systems, which has attracted many researchers’ attention in recent years. A novel epidemic spreading model with variant factors in complex networks is proposed and investigated in this paper. One main feature of this model is that virus variation is investigated in the process of epidemic dynamical spreading. The global dynamics of this model involving an endemic equilibrium and a disease-free equilibrium are, respectively, discussed. Some sufficient conditions are given for the existence of the endemic equilibrium. In addition, the global asymptotic stability problems of the disease-free equilibrium and the endemic equilibrium are also investigated by the Routh-Hurwitz stability criterion and Lyapunov stability criterion. And the uniform persistence condition of the new system is studied. Finally, numerical simulations are provided to illustrate obtained theoretical results.


Introduction
The research of infectious disease has always been a hot issue of nonlinear systems with applications.The popular dynamics on complex network is the epidemic spreading, which describes how infections spread throughout a network [1].In recent years, much research work has been done about the viral dynamics of epidemic spreading [2,3].These results are helpful for preventing and controlling most emerging infectious diseases like SARS, HIV/AIDS, H5N1, and H1N1.They are also meaningful to provide important information for the research in the field of rumor spreading [4][5][6][7], traffic dynamics [8][9][10], computer viruses [11,12], biology mechanism [13], and medicine developing [14][15][16].
In real world, the population size is large enough such that the mixing of individuals can be considered to be homogeneous.Social and biological systems can be properly described as complex networks with nodes representing individuals and links mimicking the interactions [17,18].Suitable mathematical models of the infectious disease spreading in complex homogeneous networks are of great practical value to analyze the detailed spreading process.Because epidemic spreading usually brings great harm to society, it is very urgent to establish accurate propagation models considering the infection contagion spreading problems.In past decades, complicated  models were formulated from different perspectives of epidemiology [19,20].In these models, , , and  denote, respectively, the number of individuals susceptible to the disease, the number of infectious individuals, and the number of individuals who are recovered from being infectious.The process of epidemic spreading can be further modeled with differential equations, such as , , and  model [21][22][23][24][25][26].In these research works the network topological structure is simplified presumptively to regular network or sufficient mixing homogeneous network, where the relationship between the network structure and the epidemic spreading is discussed.Kephart et al. [27] established a virus spreading model based on a homogeneous network by characterizing the average degree as the network metrics and obtained a virus spreading threshold   = 1/⟨⟩, while Pastor-Satorras [28][29][30][31] studied the epidemic outbreaks in complex heterogeneous network, which chose the degree and average degree as the network metrics and obtained the epidemic spreading threshold   = ⟨⟩/⟨ 2 ⟩.Moore and Newman [32] applied the percolation theory to analyze the epidemic spreading behaviors in small-world network and showed the differences of spreading action between smallworld network and regular network.These research results illustrate that the different network topological structure can affect the epidemic spreading.
Apart from the network topological structure, one of the most important characteristics of epidemic spreading models is the dynamical stability which can reflect the development of the spreading behaviors of infectious disease.Hence, the stability problems of these epidemic models need to be investigated.Kuniya [21] applied a discretization method to prove the global asymptotic stability of the  model with the age structure.Zhang and Feng [22] deal with the global analysis of a dynamical model describing the spread of tuberculosis with isolation and incomplete treatment.Lahrouz et al. [23] studied a nonlinear  model with saturated birth and death rates, and the global asymptotic stability of the model is also discussed.Xu et al. [24] analyzed a time-delayed  model with temporary immunity, and some conditions for the globally asymptotically stability of the disease-free equilibrium and the endemic equilibrium are given.Besides, for the epidemic model with time-delay Kang and Fu [25] presented a new  model with an infective vector on scale-free networks and the global stability of equilibrium is proved.The influences of treatment and vaccination efforts on a dynamic disease model in presence of incubation delays and relapse are studied and sufficient conditions for the local stability of the equilibrium are derived [26].
However, few papers are available in the literature to consider variant factors in the epidemic spreading from a systematic framework.In real world, certain variants exist in the infectious disease transmitting, resulting from some factors including gene mutation and cell division environment.Viruses evolve rapidly because they have strong ability of propensity for genetic variation and short generation time, which leads to evading human immunization response and obtaining drug resistance.For example, influenza viruses can be classified into three major types (A, B, and C).There are many different virus forms because of mutation; type A infects many animal species including humans, while type B and type C viruses are mainly human pathogens.If individuals are affected by viruses, not all infected individuals can be recovered.Some of them may suffer from other diseases because virus variation or the infectious individuals contacted with variants.In fact, some infectious persons, who may be infected by some diseases, would have certain probability to become variant members of another group.It is necessary to propose a new model considering this condition.How to build models with variant factors in the epidemics spreading becomes a challenge.Therefore, the paper presents a novel  epidemic spreading model considering variant factors, where  stands for the susceptible and , , and  stand for the infectious, the variant, and the recovered, respectively.
Given the mechanism of the  model in a homogeneous network, which is only composed by blank nodes initially, the entire population can be divided into four groups described by the symbols of , , , and , respectively.They denote four epidemiological statuses: susceptible, infectious, variant, and recovered.All new individuals are supposed to be blank nodes in complex networks.When a susceptible individual contacts the other infected individual, this individual may be infectious with certain probability.Then an infectious individual would have only three states including infectious, variant, and recovered.The infectious individuals would become the variants with certain probability affected by some factors such as gene mutation and the indeterminacy of cell division.Similarly, an infectious person may become a variant with certain probability after contacting with a variant.Usually, human body can be protected by one's immune system.Some infectious individuals with recovery probability may become the recovered, while others will keep the infectious status.We assume that the four groups have the same mortality rate.
In this paper, a novel  epidemic spreading model with virus variation in complex homogeneous network is proposed and investigated.The rest of this paper is organized as follows.In Section 2, the propagation mechanism of the  model in complex networks is presented, and mean-field equations are used to describe the dynamics of epidemic spreading model with virus variation.The existence of endemic equilibrium is considered.Section 3 is devoted to discuss the global stability of the disease-free equilibrium, which is followed by the discussion of the system uniform persistence in Section 4. Then the proofs of global stability of an endemic equilibrium are presented in Section 5.In Section 6, numerical simulations are performed to illustrate obtained theoretical results.Finally, conclusions are given in Section 7.

Epidemic Spreading Model and Its Property
2.1.The SIVRS Model.As described above, in the paper a new  model is established.The model involves a new variant group which is caused by the infectious variation.Assume the number of nodes is  in a closed complex network, which includes four statuses susceptible , infectious , variant , and recovered  as well as some initial blank nodes.All new nodes produced from blank nodes are susceptible.
The flow chart of epidemic spreading is shown in Figure 1.Assume in a homogeneous network only composed by blank nodes initially the susceptible individuals are produced by the blank nodes; the probability is characterized as .Others come from the recovered group with the probability .A susceptible individual will become infectious with probability  if he/she contacts the infected individual.Then an infectious individual may perhaps become one variant when he/she has tight relation with variants or is affected by other factors.We assume that the variant probability is  when contacting with a variant.In the process of epidemic spreading, an infectious individual may become the variant with internal probability .Some infectious individuals with recovery probability  may recover and others will keep the infectious status.In this paper, the four groups are supposed to have the same mortality rate .Then the  epidemic spreading rules can be summarized as follows.(1) Apart from the four groups in a closed network, there are blank nodes which exist in initial network.
The blank nodes may become susceptible ones with probability , namely, crude birth rate.
(2) The susceptible individual becomes infectious with probability  when contacting with an infectious one, namely, infection rate.
(3) An infectious individual can be recovered with probability , namely, recovery rate.
(4) The variants coming from some of the infectious nodes at a variation rate  (internal variation rate) can reflect the variation factors.When an infectious individual contacts with a variant, the individual may become a variant with probability  ( contact variant rate).
(5) The recovered node turns into susceptible with probability  after a period of time due to the loss of immunity.For the four groups in the network, all individuals will become blank with probability , namely, natural mortality rate.
A closed and homogeneous network consisting of  individuals is investigated in this paper.Individuals in the network can be represented with nodes and the contact between different individuals can be denoted by edges.Then the network can be described by an undirected graph  = (, ), where  and  denote the set of nodes and edges, respectively.Therefore, a differential equation model is derived based on the aforementioned rules and the basic assumptions: where ⟨⟩ denotes the average degree of the network.

Existence of Equilibrium.
The system (1) has a diseasefree equilibrium (DFE)  0 , where Denote the basic reproduction number parameter as The following theorem summarizes the parameter restrictions on the existence of equilibrium.
Theorem 1.If  0 > 1 and the inequality is satisfied, there are two endemic equilibria for system (1).
The Jacobian matrix at the disease-free equilibrium  0 of system (1) is Obviously, if  0 < 1, all eigenvalues of matrix ( 16) are negative.Then the disease-free equilibrium  0 is locally asymptotically stable in Σ.Moreover, if  0 > 1, there is one positive eigenvalue and  0 is unstable.Theorem 4. If  0 < 1, the disease-free equilibrium (DFE)  0 is globally asymptotically stable in Σ and if  0 > 1, the diseasefree equilibrium (DFE)  0 is unstable in Σ. <  ( +  + ) ( 0 − 1) < 0. ( is positive definite and L is negative definite.Therefore, the disease-free equilibrium (DFE)  0 is globally asymptotically stable in Σ; the following result can be given.

Uniform Persistence
In this section, the uniform persistence of system (1) will be discussed when the basic reproduction number  0 > 1.
According to Lemma 6, the following result is obtained.

Global Dynamics of Endemic Equilibrium
From the previous analysis, the disease dies out when  0 > 1; then the disease becomes endemic.In this section, Lyapunov asymptotic stability theorem is used to investigate the globally asymptotic stability of the endemic equilibrium  * when  0 > 1.

Numerical Simulation
To demonstrate the theoretical results obtained in this paper, some numerical simulations will be discussed.In this paper, the hypothetical set of initial values (IV) and parameter values will be given as follows.
( Figure 2 shows that if  0 < 1, all solutions in Σ would be attracted to the disease-free equilibrium  0 regardless of the initial values of system (1), which illustrates the validity of Theorem 4. Similarly, it can be seen from Figure 3 that all solutions in Σ ∘ would be attracted to the endemic equilibrium  * regardless of the initial values of system (1) if  0 > 1 and the conditions of Theorem 8 are satisfied, which is obviously the content of Theorem 4.Moreover, the relationship between the values of equilibrium can be verified as shown in (24), which is coincident with the theoretical results:  0 +  0 +  0 +  0 =  * +  * +  * +  * = −   +  .(24)

Conclusion
The stability of the  epidemic spreading model with virus variation in complex networks has been discussed in this paper.The model involves a new variant group which is caused by the infectious variation.By analyzing the model, the disease-free equilibrium  0 is proved to exist when the basic reproduction number  0 is less than 1.The analysis result reveals that the infectious disease dies out when  0 is more than 1 and it becomes endemic.The existing conditions of endemic equilibrium related with the variation rate and the network nodes degree are obtained.Besides, the global asymptotically stability condition of the disease-free equilibrium is obtained by the Routh-Hurwitz stability criterion and the Lyapunov stability criterion.And the condition of the system uniform persistence is also given.The proof of the stability of endemic equilibrium is also illustrated.Finally, a numerical simulation is given to illustrate the correctness of the disease-free equilibrium and the endemic equilibrium results.

Figure 1 :
Figure 1: A schematic representation of  epidemic spreading model.
step The values of S(t), I(t), V(t), R(t) (a) The system states of endemic equilibrium I ( t ) + V ( t ) (b) The evolution for the endemic system states