# A Bistable Phenomena Induced by a Mean-Field SIS Epidemic Model on Complex Networks: A Geometric Approach

^{1}School of Information, Shanxi University of Finance and Economics, Taiyuan, China^{2}Complex Systems Research Center, Shanxi University, Taiyuan, China

In this paper, we propose a degree-based mean-field SIS epidemic model with a saturated function on complex networks. First, we adopt an edge-compartmental approach to lower the dimensions of such a proposed system. Then we give the existence of the feasible equilibria and completely study their stability by a geometric approach. We show that the proposed system exhibits a backward bifurcation, whose stabilities are determined by signs of the tangent slopes of the epidemic curve at the associated equilibria. Our results suggest that increasing the management and the allocation of medical resources effectively mitigate the lag effect of the treatment and then reduce the risk of an outbreak. Moreover, we show that decreasing the average of a network sufficiently eradicates the disease in a region or a country.

## 1 Introduction

Mathematical modeling plays a crucial role in fighting against large scale infectious disease such as Tuberculosis, HIV, COVID-19, etc., Compartment models have been used to anticipate the progression of diseases and evaluate the effect of interventions on disease spread [1]. One of such models separates the total population into two distinct categories with respect to disease status. People who have not gotten the disease are labeled “susceptibles”; while those who have been infected by a certain disease are called “infectives”. This kind of compartment model is denoted by “an SIS epidemic model” [2–7], which has been extensively used to address the dynamics of those diseases, describing an individual infected by a disease as having no immunity, thus becoming a susceptible again.

Most of the existing models assume that all the individuals are well-mixed and they have homogeneous mixing of surfaces, which implies that each individual has the same probability to contact other individuals and ignores the degree of social heterogeneity induced by age, household, spatial structures, and social spheres, etc. Generally, the social interactions of individuals generate a certain pattern based on social preferences, which contributes to transmission heterogeneity. Indeed, such factors may play a decisive role in the disease transmission and they also may help health policymakers to take more effective control measures for curbing the disease spread [7, 8]. Epidemic models on complex networks incorporate such contact heterogeneity and take account for how the structures of the networks affect the disease prevalence. A popular degree-based SIS epidemic model has been built [9] and it exhibits threshold dynamics [4, 5]. Since then, many factors including vector-borne [10, 11], infective media [12], awareness reaction [13], and the gene diversity of pathogens [14], etc., have been incorporated into the study of the co-evolution of networks and epidemics.

In epidemiology, the basic reproduction number,

with transmission rate *β* and treatment rate

where *α* denotes a hysteretic effect due to medical limitations. Apparently, *I* is small enough and it tends towards to *I* is large enough. The constant

In view of such epidemiological models incorporating a saturated treatment function, it is not hard to find that most of them enable such models, essentially changing their dynamics. Once a saturated function has been introduced, there always exists a backward bifurcation, which implies that even if some certain control measures make

There are three main contributions in this paper. First, we propose a degree-based SIS epidemic model with a saturated function to study its long-term behaviors. Second, to overcome the difficulty of high dimensions for a network, we adopt an edge-based compartmental approach to lower the dimensions of an SIS epidemic model. Such an approach changes a complete degree-based model to a degree-edge-mixed model, and hence, it lowers the dimension of such a model from

The organization of this paper is as follows: In Section 2, a degree-based SIS epidemic model on complex networks with a saturated function is proposed. Furthermore, we adopt an edge-compartmental approach to rewrite it as a degree-edge-mixed model. Section 3 gives a geometric approach to study the local stability of each equilibrium. In Section 4, we conducted some numerical simulations to illustrate our theoretical results. We give a brief discussion in the last section.

## 2 Model Formulation

In this paper, we focus on the complete stability of each equilibrium by a novel approach. Let us assume that the maximum contact number of an individual is *n* and then the degree set is *t* and degree

where *β* denotes the transmission rate and *γ* represents the treatment rate; *α* stands for the lag effect of the treatment due to the limitation of medical resources. From epidemiological view of points, the term

denotes the probability of a given node connecting to an infected node at time *t*. Hence, it can be considered as a density of an *t*. Following the steps [10], model (1) can be rewritten as a degree-edge-mixed mode.

with initial condition

Lemma 2.1. If

where

Obviously,

This leads to a contradiction with the claim. Therefore,

Remark 2.1. *Lemma 2.1 ensures that the solution of system* (2) *is strictly positive if**, which supports that**for all**and**The strong connectivity of a network guarantees the positivity of the solution as long as**or there exists at least**such that*

Therefore, for all

is positively invariant associate with system (2) and

## 3 Stability of Equilibria

In this section, we will consider the local stability of system (2) by a novel-geometric approach, which resolves such a matter once and for all. First, we try to give the basic reproduction number

Solving Eq. 5 by a constant variation method, one drives a renew equation

So that the basic reproduction number is calculated in form of

The epidemiological meaning of

Theorem 3.1. If

Let us assume that system (8) has the solution with exponential forms, i.e,

If

Theorem 3.2. If

Differentiating V along the solution of system (2) leads to

here we have used the fact that

The equality holds if and only if

From the first equation of (10), we get

Substituting Eq. 11 into the second equation of (10) and canceling

Apparently, if

Consequently, we conclude the following theorem on the existence of system (2).

Theorem 3.3 If

Proof If

From Theorem 3.3, we assert that system (2) has the existence of the endemic equilibrium, but it does not guarantee the uniqueness of the positive solution when *α* is too small or large enough, it is easy to find that the function *F* is a decreasing function associated with

Now, we are concerned with the endemic curve which bifurcates backwards at

Calculating the derivative of Eq. 14 with respect to

where

After a simple computation, we have that

Lemma 3.4. Suppose

Proof This is a direct result from Section 2.3 [26].Next, let us move our attention on to the local stability of equilibria if they exist, which is a challenge issue for a degree-based epidemic model due to the complex structure. We will propose a geometric approach to deal with such an issue.Linearising system (2) around

If

Replacing

Recall that

and hence,

Alternatively, taking the derivative of F with respect to

Plugging Eq. 23 into Eq. 22, we have that

If

Theorem 3.5. Let

(1) *If**, then**is locally asymptotically stable;*

(2) *If**, then**is unstable.*

Proof To address the stability of case (1), we rewrite

where

Hence, if

This, together with the Intermediate Value Theorem, ensures that Eq. 21 has at least one positive real solution. Hence, the endemic equilibrium

Theorem 3.6. If

Theorem 3.7. *Suppose**If**or α is large enough, then the endemic equilibrium**is globally asymptotically stable if*

Proof Let us pick up a candidate Lyapunov function by

where

where

Taking the derivative of

On the contrary, differentiating

Adding Eqs 30, 31 together, one derives that

If

here we have used the fact that

Consequently, the largest invariant set of

## 4 Numerical Simulation

In this section, we will proceed with some numerical experiments to validate our theoretical results. We account for an epidemic spreading on a scale-free network. Hence, we assume that the degree distribution of that network is

First, we fix

**FIGURE 1**. The evolution of the densities of infected edges with different initial values **(A)** With **(B)** With

Second, we fix *α* and consider the existence and the uniqueness of the endemic equilibria. If we choose *α* gradually advances the arrival times and increases the sizes of the associated endemic equilibria. This indicates that enriching the adequate medical resources effectively reduce the risk of an infection.

**FIGURE 2**. The evolution of the densities of infected edges with different initial values **(A)** With **(B)** With

Third, if we fix the structure of the network, then a key value

**FIGURE 3**. The evolution of the densities of infected edges with different initial values **(A)** with **(B)** with

Finally, we want further insight into the existence of endemic equilibria when *α*. In this case, *α* and *α* from 4.8 to 6.8 with step 0.5. Figure 4 suggests that there always exists a backward bifurcation; In addition, the lag effect *α* controls the depth of the backward bifurcation; the larger *α*, the bigger the depth of the bifurcation. *α* is bigger in case of strength of lag effects of medical resources. Figure 4B displays the distributions of the positive equilibria in a given region

**FIGURE 4**. Backward bifurcation in system (2). **(A)** The figure of function **(B)** The diagram of bifurcation phase on the

## 5 Discussion

In this paper, we considered a mean-field degree-based SIS epidemic model with a saturated treatment function. First, we adopted an edge-compartmental approach to simplify a pure degree-based model to a degree-edge-mixed model. Second, we proposed a novel method-a geometric approach to completely study the stability of each feasible equilibrium. The proposed model exhibits a backward bifurcation, i.e,

Compared with the results in [5, 25], a degree-based SIS epidemic model on complex networks shows a threshold dynamic, in the sense that, if

The basic reproduction number *α.* However, it is a vital value which determines whether or not a backward bifurcation occurs. In view of most existing results about that phenomena, they usually showed the existence and did not point out the stability of each equilibrium except some simple models with lower dimensions [2, 3, 18, 19]. In this paper, we proposed a geometric approach to completely solve such an issue. Our results suggest that whereas

Generally, the contact magnitude of an outbreak is characterized by the average degree

**FIGURE 5**. Time series of the densities of **(A)** With **(B)** With **(C)** With

From an epidemiological viewpoint, the occurrence of a backward bifurcation implies that those control measures enabling

However, there are some limitations of this paper. First, we do not incorporate the population demography into the modeling process because introducing birth and death nodes essentially changes the topology of a network [27, 28]. This makes the model become too complex and then it has been become an unresolved issue to analyze its dynamical behaviors from mathematical view of points. Second, we do not couple individual contact data with some reported data for a realistic disease to study its evolutionary behaviors [29]. Third, we do not consider the convolution of information spread and epidemic transmission on multi-layered networks [30]. To carry out such a project, it may provide some valuable control suggestions for policymakers and public health government. We leave these for our future works.

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

## Author Contributions

JY and XW conceived of the presented idea. JY developed the theory and performed the computations. XW verified the analytical methods. All authors discussed the results and contributed to the final manuscript. JY designed the modeling process and analyzed theoretical results. XW conceived of the study and helped to draft the manuscript. All the authors read and approved the final manuscript.

## Funding

This work is partially supported by the National Natural Science Foundation of China (No.12001339, No.61573016), and the Shanxi Province Science Foundation for Youths (No. 201901D211413). Shanxi University of Finance and Economics Youth Research Fund Project (QN-2019017).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: complex networks, an edge-compartmental approach, a geometric approach, backward bifurcation, global stability

Citation: Wang X and Yang J (2021) A Bistable Phenomena Induced by a Mean-Field SIS Epidemic Model on Complex Networks: A Geometric Approach. *Front. Phys.* **9**:681268. doi: 10.3389/fphy.2021.681268

Received: 17 March 2021; Accepted: 03 May 2021;

Published: 17 June 2021.

Edited by:

Mahdi Jalili, RMIT University, AustraliaReviewed by:

Gui-Quan Sun, North University of China, ChinaChengyi Xia, Tianjin University of Technology, China

Lin Wang, University of Cambridge, United Kingdom

Copyright © 2021 Wang and Yang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Junyuan Yang, yjyang66@sxu.edu.cn