key: cord-0427396-kgtuzkb7
authors: Frieswijk, Kathinka; Zino, Lorenzo; Ye, Mengbin; Rizzo, Alessandro; Cao, Ming
title: A mean-field analysis of a network behavioural-epidemic model
date: 2022-03-08
journal: nan
DOI: 10.1109/lcsys.2022.3168260
sha: 4fa6e60a6ed907e1b1b4d3e26e9c8b7a7def31db
doc_id: 427396
cord_uid: kgtuzkb7

The spread of an epidemic disease and the population's collective behavioural response are deeply intertwined, influencing each other's evolution. Such a co-evolution typically has been overlooked in mathematical models, limiting their real-world applicability. To address this gap, we propose and analyse a behavioural-epidemic model, in which a susceptible-infected-susceptible epidemic model and an evolutionary game-theoretic decision-making mechanism concerning the use of self-protective measures are coupled. Through a mean-field approach, we characterise the asymptotic behaviour of the system, deriving conditions for global convergence to a disease-free equilibrium and characterising the endemic equilibria of the system and their (local) stability. Interestingly, for a certain range of the model parameters, we prove global convergence to a limit cycle, characterised by periodic epidemic outbreaks.

Mathematical models of epidemic spreading on networks have been of increasing interest to the systems and control community [1] - [4] . Since 2020, the COVID-19 pandemic has given an extra impetus to such an interest [5] , [6] . In particular, the ongoing pandemic has highlighted the key role of human behavioural response in shaping the course of an epidemic outbreak and how such a response is deeply intertwined with the epidemic spreading process. Some efforts have been made to incorporate human behaviour into epidemic models [7] - [12] , in particular, by adding an alert state, in which individuals take self-protective measures based on factors such as the awareness of the infection prevalence [9] , [12] , communication with neighbours [10] , awareness campaigns [11] , [12] , or by incorporating opinion dynamics mechanisms [13] , [14] . While these models proved useful in capturing some key aspects of real-world epidemics, their inherent oversimplification of the evolving nature of human behaviour limits their practical applicability. social influence, interventions, risk perception, and immediate and accumulated costs of using protection. However, except for the approximation of the epidemic threshold, [21] relies only on numerical simulations, which suggest that the behavioural-epidemic model can reproduce a wide range of behaviours, including eradication of the disease, convergence to endemic equilibria, or periodic oscillations and multiple epidemic waves.

In this letter, we expand on [21] to provide an analytical treatment of the long-term behaviour of a game-theoretical behavioural-epidemic model. To this aim, we propose a continuous-time implementation of the framework proposed in [21] , combined with a susceptible-infected-susceptible epidemic model. Through a mean-field approach [22] , we derive analytical results on the asymptotic behaviour of the system. After having established the epidemic threshold, we analyse the behaviour of the system below and above such a threshold. Below the threshold, we prove global convergence to a disease-free equilibrium (DFE). Above the threshold, we characterise the endemic equilibria (EEs) of the system and their local stability properties. Furthermore, we derive conditions under which the system undergoes periodic oscillations with multiple waves, converging to a limit cycle. Finally, numerical simulations suggest that the locally exponentially stable equilibria are also globally stable, paving the way for future research towards extending our theoretical findings.

Notation: The set of real, real nonnegative, and strictly positive real numbers is denoted by R, R ≥0 , and R >0 , respectively. We say that an event E is triggered by a Poisson clock with (possibly time-varying) rate q E (t), if lim ∆t 0 P E occurs during (t, t + ∆t) /∆t = q E (t).

We consider a population of n individuals V = {1, . . . , n}. Each individual i ∈ V is characterised by a two-dimensional state (x i (t), y i (t)), reflecting their behavioural state x i (t) ∈ {0, 1} and health state y i (t) ∈ {S, I}, at time t ∈ R ≥0 . In particular, an individual i ∈ V either chooses to use selfprotective measures (x i (t) = 1) at time t, thereby preventing any possible contraction of the disease, or to not employ them (x i (t) = 0); simultaneously, the individual can have two different health states: y i (t) = I if i is infected, and y i (t) = S if i is healthy and susceptible to the infection. Each individual is represented by a node in a twolayer temporal network G(t) = (V, E I , E C (t)), illustrated in Fig. 1a 

Since the spreading of a disease typically evolves much faster than social ties do, we assume that the influence layer is time-invariant. Disease transmission from an infectious to a susceptible individual occurs through interactions in close physical proximity, henceforth denoted by contacts, modelled by the contact layer E C (t), where {i, j} ∈ E C (t) iff i and j have a contact at time t ∈ R ≥0 . We assume that contacts are generated according to a continuous-time activity-driven network [23] , in which each individual i ∈ V is assigned an activity rate a i ∈ R >0 , which captures the level of physical activity of individual i. Then, i activates if triggered by a Poisson clock with rate a i and, once active, generates a contact with another individual, selected uniformly at random from V \ {i}.

In the behavioural-epidemic framework proposed in [21] , each individual i ∈ V decides whether to adopt selfprotective measures according to an evolutionary gametheoretic mechanism [24] , depending on social influence, risk perception, costs for adopting self-protective measures, frustration, and government policy interventions. Here, we propose a simplified decision-making mechanism in which the last two factors are omitted. Such a simplification allows the reduction of the number of parameters involved in the system, simplifying its analysis and the presentation of the results, without restricting the broad range of possible emergent behaviours, as we shall demonstrate in this letter.

To capture these factors, we introduce the payoff function

which captures the payoff for adopting self-protective measures (x i = 1), whereȳ(t) := 1 n {i ∈ V : y i (t) = I} denotes the infection prevalence at time t; and π (i)

which captures the payoff associated with not adopting self-protections. The first term, present in both formulae, represents social influence: the more neighbours of i adopt a certain action, the higher the payoff for the corresponding action. The term ζȳ(t), with ζ ∈ R ≥0 , increases the payoff for adopting self-protections as the infection prevalence grows, capturing the risk perception. Here, we assume that people react in a linear fashion in response to the information they receive on the infection prevalence ζȳ(t), but more complex and nonlinear terms may be considered. Finally, the constant c ∈ R ≥0 represents the psychological, social, and economical cost per unit-time associated with the adoption of self-protections, thereby increasing the payoff for not adopting self-protections. Individuals change their behaviour following a stochastic implementation of the classical imitation dynamics mechanism, which is often used in evolutionary game theory [24] , [25] , in which they imitate their peers triggered by Poisson clocks with rate equal to their corresponding payoff functions. Specifically, an individual i who is not adopting selfprotective measures at time t (i.e. x i (t) = 0) will adopt them if triggered by a Poisson clock with rate

and an individual i who is adopting them (i.e. x i (t) = 1) will stop if triggered by a Poisson clock with rate q (i)

Simultaneously, if a susceptible individual i (y i (t) = S) who does not use protective measures (x i (t) = 0) has a physical encounter with an infected individual k (y k (t) = I), then i becomes infected with per-contact infection probability λ ∈ (0, 1]. We assume that self-protective measures are 100% effective in preventing contagion. Hence, if individual i employs protections at time t (x i (t) = 1), then they cannot be infected at time t. Following [26] , we compute that if i is susceptible at time t (y i (t) = S), then i will become infected if triggered by a Poisson clock with rate q (i)

where the first term in the parentheses accounts for the contact initiated by i with infected individuals, and the second accounts for contacts initiated by infected individuals who interact with i. If the disease can be transmitted only in one direction, then only the corresponding term should be considered in Eq. (3) . Note that if individual i employs protection at time t, then q 

The evolution of the state of each individual (x i (t), y i (t)), i ∈ V, is determined by independent Poisson clocks. Hence, the state of the system follows a Markov process on a state space with size growing exponentially with the population size n, making its direct analysis unfeasible. We employ a mean-field relaxation of the stochastic process to derive analytical insight, following the n-intertwined mean-field approach described in [22] . Specifically, we define and study for each individual i ∈ V the probabilities of adopting protective behaviours p 

Also, we introduce the macroscopic variables

which are the average probability that a randomly selected individual is adopting protections and is infected at time t, respectively. Letx(t) := 1 n i∈V x i (t) denote the fraction of adopters of self-protection in the population at time t. In the limit of large-scale populations, n → ∞, the central limit theorem ensures thatx(t) andȳ(t) converge to x(t) and y(t), respectively. Hence, the macroscopic variables in Eq. (5) approximate with arbitrary accuracy the fraction of adopters of self-protective measures and the epidemic prevalence, for any finite-time horizon [27] .

In the rest of this letter, we will make the following simplifying assumption. Assumption 1. We assume that a) the influence layer is complete, i.e. N i = V, ∀ i ∈ V; b) individuals have homogeneous activity, i.e. a i = α ∈ R >0 , ∀ i ∈ V; and c) individuals have the same initial probability of adopting protections, i.e. p (i)

Under item a) of Assumption 1, Eq. (1) reduces to π 1 (t) = x(t) + ζy(t) and π 0 (t) = 1 − x(t) + c, where we have dropped the index i since the payoffs are uniform across the population. Under Assumption 1, we derive a planar system that governs the mean-field evolution of the macroscopic variables (proof in Appendix A) and rigorously analyse it. Proposition 1. In the limit of large-scale populations n → ∞ and under Assumption 1, the two macroscopic quantities in Eq. (5) evolve according to the following planar system:

The following result guarantees that Eq. (6) is always well-defined, i.e. that the variables x and y, which represent fractions of the population, remain within 6) is Lipschitz-continuous. Hence, Nagumo's Theorem can be applied (see [28] ). We are left with checking the direction of the vector field at the boundaries of the domain. We observe thatẋ = 0 for x = 0 and x = 1, whileẏ = 0 for y = 0 andẏ < 0 for y = 1, implying that any trajectory such that (x(0),

In the following, we will make some realistic assumptions on the model parameters. In particular, we want to guarantee that the use of self-protective measures is always preferred when the entire population is infected, while their use is disfavoured in the absence of a disease. To guarantee this, we need to enforce in Eq. (1) that, for any x ∈ [0, 1], π 1 (t) < π 0 (t), if y = 0, and π 0 (t) < π 1 (t), if y = 1. These conditions are satisfied by making the following assumption.

Assumption 2. We assume that c > 1 and ζ > c + 1.

We study the asymptotic behaviour and the equilibria characteristics of the behavioural-epidemic model using the mean-field system in Eq. (6). The following lemma characterises the equilibria of the system in Eq. (6) and their local stability properties. Its proof can be found in Appendix B. i) the DFE (0, 0), which is locally asymptotically stable if λ ≤ µ 2α (with exponential stability if strict inequality holds), and a saddle point if λ > µ 2α ; ii) the DFE (1, 0), which is a saddle point; iii) the protection-free EE (0, 1 − µ 2αλ ), which exists iff λ > µ 2α . When it exists, it is locally asymptotically stable if ζ ≤ 2αλ(1+c) 2αλ−µ (with exponential stability if strict inequality holds) and a saddle point if ζ > 2αλ(1+c) 2αλ−µ . Next, define

The two EEs in the interior are: iv) β + , 1 − µ 2αλ(1−β+) , which exists iff λ > µ 2α(1−β+) and one of the following conditions is satisfied: a)

Lemma 2 leads to the establishment of the epidemic threshold for the system in Eq. (6), i.e. the conditions under which the system converges to one of the DFEs. Proof. Let f (y) = 2αλy(1 − y) − µy. By [2, Lemma 4.1], the solution ofż = f (z) converges to z = 0 if λ ≤ µ 2α . As a consequence of the nonnegativity of y(t) (Lemma 1) and the fact thatẏ ≤ f (y), y(t) converges to 0. If x(0) = 1, convergence to (1, 0) is straightforward, since x = 1 is an invariant manifold. Otherwise, since y(t) → 0 and c > 1, there exists a timet > 0 such that y(t) ≤ c−1 2ζ , for any t ≥t. From Eq. (6), we observe that for any t >t,

Theorem 1 fully characterises the behaviour of the system when the disease is not highly infectious, i.e. λ ≤ µ 2α . In the following, we will consider the opposite scenario λ > µ 2α . We also assume c ≥ 4αλ µ − 3, ensuring that the unique endemic equilibrium only exists for a high enough level of risk perception ζ. Under an upper bound of c, which depends on the other model parameters, Lemma 2 reduces to the following proposition, which clearly illustrates the role of the risk perception ζ. The proof is reported in Appendix C.

Proposition 2. Let λ > µ 2α and 4αλ µ − 3 ≤ c < 32αλ 5µ − 3. Under Assumption 2, the following hold: i) if ζ < 2αλ(1+c) 2αλ−µ , then Eq. (6) has three equilibria: the DFEs (0, 0) and (1, 0), which are saddle points, and the (locally) exponentially stable EE (0, 1 − µ 2αλ ); ii) if ζ > 2αλ(1+c) 2αλ−µ , then Eq. (6) has four equilibria: three saddle points-the DFEs (0, 0), (1, 0), and the EE (0, 1 − µ 2αλ )-and the interior EE (β + , 1 − µ 2αλ(1−β+) ), which is (locally) exponentially stable iff ζ satisfies all of the following three conditions: a)

and c) ζ < αλ 2αλ−µ ((c + 1) 1 + (αλ − 1) 2 + 2µ] + αλ(c − 3) + 2µ).

Proposition 2 focuses on local stability and instability of endemic equilibria. In the following result we establish sufficient conditions under which sustained oscillations with periodic epidemic waves occur, with proof in Appendix D. Example 1. Let c = 3, α = 3, λ = 0.5, and µ = 1. In Fig. 2a , we set the risk perception to ζ = 5, which satisfies the conditions in item i) of Proposition 2. Hence, the only (locally) stable equilibrium of Eq. (6) is the EE (0, 1 − µ 2αλ ). Simulations suggest that all trajectories converge to it. In Fig. 2b , we increase the risk perception to ζ = 8, which satisfies all the conditions in item ii) of Proposition 2. Hence, the interior EE is locally stable: all trajectories converge to it, suggesting globally stability. Finally, we set ζ = 9.5, which satisfies the conditions of Theorem 2. Consistently, all the trajectories in Fig. 2c converge to a limit cycle.

We studied a behavioural-epidemic model in which human behaviour and epidemics co-evolve in a mutually influencing manner. Employing a mean-field approach, we painted an extensive picture of the system behaviour, including a stability analysis of the equilibria and the expression of the epidemic threshold. Furthermore, we explored the role of risk perception in the occurrence of periodic oscillations and established conditions for global convergence to such a periodic solution.

Our promising results pave the way for several avenues of future research. First, the numerical findings suggest that our local stability results might be extended towards obtaining global results. Second, interventions may be incorporated, towards designing control policies to favour a collective behavioural response and mitigate an epidemic outbreak.Third, our theoretical analysis relies on the simplifying Assumption 1. Efforts should be placed towards extending our theoretical findings to more general scenarios, including nontrivial directed networks. Finally, further factors should be incorporated into the model, including limited effectiveness of self-protections, accumulation of socio-economic fatigue, and nonlinear terms to capture more complex risk perception (as in [21] ). This will be key for real-world applications.

A. Proof of Proposition 1

In the limit n → ∞, Eq. (4a) reduces toṗ

Similarly, using item b) of Assumption 1 and Eq. (3), we write Eq. (4b) asṗ

y . Next, observe that, under item c) of Assumption 1, p (i)

x (t) is the same ∀ i ∈ V, so we drop the index i and write p (i)

where the last equality holds for n → ∞. Finally, we combine the expressions above and Eq. (5), to derive Eq. (6).

, where for the latter, µ 2αλ > 0 necessarily requires x < 1 and y < 1 as conditions. Here, (1 − x)(1 − y) = µ 2αλ with x < 1 and y < 1 can be written as y = 1 − 

which for x < 1 has solutions x = 0, and

The EEs are given by (0, 1 − µ 2αλ ) and (β ± , 1 − µ 2αλ(1−β±) ). First, we investigate for which values of ζ we have

which is a necessary condition for β ± ∈ R. Note that the roots of

Next, we study for which values of ζ and c we have β ± ∈ (0, 1), while assuming that the previously identified conditions necessary for β ± ∈ R hold. We start with β + . If ζ < c + 3, then β + > 0. If

which is always satisfied. Hence, β + ∈ (0, 1) if ζ ≥ c + 3 and ζ > 2αλ(1+c) 2αλ−µ , or if ζ < c + 3. Combining this with the conditions for β + ∈ R, yields the regions of ζ for which β + ∈ (0, 1). Note here that if ζ > 2αλ(1+c) 2αλ−µ , then β + ∈ R. The condition on c in a) results from the fact that, for the region to exist, the lower bound must have a lower value than the upper bound. Likewise, we consider β − . Observe

which is satisfied. Combining the above with the conditions ensuring that β − ∈ R, gives the regions of ζ for which β − ∈ (0, 1).

Then, we study local stability. First, we consider the DFE (0, 0). Linearising Eq. (6) about (0, 0), we find that the eigenvalues of the Jacobian matrix are 2αλ − µ and −(c + 1) < 0, where the former is negative iff 2αλ < µ. Thus, the equilibrium is locally exponentially stable (LES) if 2αλ < µ, and a saddle point if 2αλ > µ. The case 2αλ = µ is studied separately, yielding asymptotic stability. Computations are omitted due to space constraints. Next, we consider the DFE (1, 0) . In a similar way, we linearize the system about the equilibrium and we find the eigenvalues of the Jacobian matrix, which are given by c − 1 > 0 and −µ < 0, implying that the DFE (1, 0) is a saddle point. Now consider the EE (0, 1 − µ 2αλ ), with µ < 2αλ, so it exists. Linearising the system about this equilibrium, we obtain a Jacobian matrix with eigenvalues ζ(1 − µ 2αλ ) − 1 − c and µ−2αλ < 0, so the equilibrium is LES if ζ < 2αλ(1+c) 2αλ−µ , and a saddle point if the opposite inequality holds. If the equality holds, Eq. (6) is studied directly (computations omitted), yielding asymptotic stability. Finally, consider the interior EE with x = β + . Let β + ∈ (0, 1) and µ < 2αλ(1 − β + ) for existence. Linearising Eq. (6) about the EE yields the Jacobian matrix

.

By the determinant-trace method, the EE with x = β + is LES if 4αλ ζ (1 − β + ) 2 < µ < 2(1 − β + )(αλ − β + ) and unstable if at least one of the opposite inequalities holds. By replacing β + with β − in the argument above, we complete the proof by obtaining the conditions for the other interior EE.

Item i) follows from Lemma 2. We now prove ii). Under the parameter constraints imposed by the hypothesis of the proposition, it follows from Lemma 2 that Eq. (6) has three equilibria on the boundary: (0, 0), (1, 0) and (0, 1 − µ 2αλ ), all of which are saddle points. Moreover, the interior EE with x = β − does not exist. For existence of the interior EE with x = β + , we need λ > µ 2α (1−β+) , which is equivalent to

, where the right-hand side (RHS) is positive for λ > µ 2α and ζ > c + 3. Note that c + 3 ≤ 2αλ(1+c) 2αλ−µ iff c ≥ 4αλ µ − 3. Squaring both sides, algebraic simplifications yield the equivalent expression µ αλ + c − 1 > 0, which holds. The interior EE is LES iff 4αλ

For λ > µ 2α and ζ > max{c+3, 2αλ(1+c) 2αλ−µ }, both the lefthand side (LHS) and RHS of the equation above are positive. After squaring both sides and some straightforward rewriting, we obtain the equivalent condition (2αλ−µ)ζ 2 −2(αλ) 2 [c− 3+ 1 αλ (c+1+2µ)]ζ +2(αλ) 2 (c 2 +2αλ(1−c)−2µ−1) < 0, where the roots of the polynomial are given bȳ

The leading term of the polynomial on the LHS is positive, so the condition is satisfied iffζ − < ζ <ζ + .

5µ − 3, we have c + 3 >ζ + , so the RHS is positive. Squaring both sides and rewriting yields the equivalent expression −ζ 3 + [2(c − 1) + 25µ 4αλ ]ζ 2 − (c − 1) 2 ζ < 0, where the roots of the LHS are ζ = 0 and ζ ± = c − 1 + 25µ 8αλ ± 5 2 µ αλ (c − 1 + 25µ 16αλ ). Since c > 1 impliesζ − < c − 1, we need ζ >ζ + for stability.

Under the conditions of Theorem 2, Eq. (6) has three saddle points on the boundary: (0, 0), (1, 0) and (0, 1− µ 2αλ ), and the unique fully unstable interior EE

] is attractive and invariant.

We study now the behaviour of the system near the boundaries of [0, 1] × [0, 1 − µ 2αλ ], and examine whether it is possible to reach the boundary of the domain. Consider the boundary x = 1. Let us assume that the trajectory reaches x = 1 − ε, with ε > 0 arbitrarily small at time t 0 . From Eq. (6), we observe thatẋ(t 0 ) = ε(1 − ε)(1 + ζy − c − 2ε) when x(t 0 ) = 1 − ε. Hence,ẋ can be positive only if y > c−1 ζ . We consider the regions R ε := [1−ε, 1]×[ c−1 ζ , 1− µ 2αλ ] and S ε := [1 − ε, 1] × [0, c−1 ζ ], whereẋ can be positive only in R ε , while it is negative in S ε . Furthermore, we derive the following uniform bound for any pair (x, y) ∈ R ε :ẏ = −µy + 2αλyε(1 − y) < −µ c−1 ζ < 0, for ε sufficiently small. Similarly, we observe thatẋ = x(1 − x)(2x + ζy − 1 − c) ≤ k(1 − x), for some constant k > 0. These bounds yield a strict bound on the distance between the trajectory and x = 1 before the trajectory exits the region R ε from the bottom and enters S ε . We define u(t) = 1 − x(t). The uniform bound oṅ x in R ε is equivalent to −u ≤ −k(−u), so by the Gronwall-Bellman inequality [29] , −u(t) ≤ −u(t 0 )e −k(t−t0) , which is equivalent to x(t) ≤ 1−εe −k(t−t0) , for any t ≥ t 0 . Here, we used the fact that x(t 0 ) = 1 − ε. Then, the uniform bound onẏ in R ε is used to derive a bound on the time needed for the trajectory to exit R ε . Specifically, since the length along the y-axis of R ε is equal to (1 − µ 2αλ − c−1 ζ ), and the time-derivative of the trajectory along the y-component is negative and greater in modulus than µ c−1 ζ , then there necessarily exists a timet ≤ ζ(1 − µ 2αλ − c−1 ζ )/(µ(c − 1)) such that y(t 0 +t) < c−1 ζ and x(t 0 +t) ≤ 1 − ε , with ε = εe −kt . Hence, the trajectory will exit from R ε and will enter S ε , in whichẋ < 0 andẏ < 0. This establishes that the trajectory cannot further approach the boundary x = 1, nor re-enter R ε from the boundary between S ε and R ε . Thus, there exists a constant ε > 0 such that [0, 1−ε ]×[0, 1− µ 2αλ ] is positively invariant for Eq. (6) .

A similar argument guarantees that any trajectory that starts from the interior is bounded away from the boundaries y = 0 and x = 0. Since convergence to the boundaries is impossible, the boundary equilibria points cannot be reached if the initial conditions of the system are in the interior of the domain (0, 1) × (0, 1). Finally, we consider the open set (x, y) ∈ (0, 1) × (0, 1 − µ 2αλ ). Since the unique interior EE is an unstable point under the above conditions, there does not exist a homoclinic orbit. It follows directly from the generalised Poincaré-Bendixson theorem [30] that every non-empty compact ω-limit set of an orbit is periodic.

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