key: cord-0733008-65mwvbjn
authors: Tomovski, Igor; Basnarkov, Lasko; Abazi, Alajdin
title: Endemic state equivalence between non-Markovian SEIS and Markovian SIS model in complex networks
date: 2022-04-30
journal: Physica A
DOI: 10.1016/j.physa.2022.127480
sha: f0606e250656fc8059422cb72a7af5fae13dfff2
doc_id: 733008
cord_uid: 65mwvbjn

In the light of several major epidemic events that emerged in the past two decades, and emphasized by the COVID-19 pandemics, the non-Markovian spreading models occurring on complex networks gained significant attention from the scientific community. Following this interest, in this article, we explore the relations that exist between the mean-field approximated non-Markovian SEIS (Susceptible–Exposed–Infectious–Susceptible) and the classical Markovian SIS, as basic reoccurring virus spreading models in complex networks. We investigate the similarities and seek for equivalences both for the discrete-time and the continuous-time forms. First, we formally introduce the continuous-time non-Markovian SEIS model, and derive the epidemic threshold in a strict mathematical procedure. Then we present the main result of the paper that, providing certain relations between process parameters hold, the stationary-state solutions of the status probabilities in the non-Markovian SEIS may be found from the stationary state probabilities of the Markovian SIS model. This result has a two-fold significance. First, it simplifies the computational complexity of the non-Markovian model in practical applications, where only the stationary distributions of the state probabilities are required. Next, it defines the epidemic threshold of the non-Markovian SEIS model, without the necessity of a thrall mathematical analysis. We present this result both in analytical form, and confirm the result trough numerical simulations. Furthermore, as of secondary importance, in an analytical procedure we show that each Markovian SIS may be represented as non-Markovian SEIS model.

SIS, as basic reoccurring virus spreading models in complex networks. We investigate the similarities and seek for equivalences both for the discrete-time and the continuous-time forms. First, we formally introduce the continuoustime non-Markovian SEIS model, and derive the epidemic threshold in a strict mathematical procedure. Then we present the main result of the paper that, providing certain relations between process parameters hold, the stationary- In our recent paper [15] , we introduced the mean-field discrete time non-Markovian SEIS (Susceptible -Exposed -Infectious -Susceptible) model as a basic mathematical non-Markovian form that describes re-occurring spreading processes, taking place on complex networks. In the model formulation, We assumed that status transitions from Exposed (non-Infectious) to Infectious status and Exposed (both non-Infectious and Infectious) back to Susceptible status, follow temporal distribution described with Discrete Time Probability Functions (DTPFs):

• daily manifesting function b(τ ): probability that an Exposed and previously non-Infectious node, becomes Infectious exactly at day τ ;

• manifesting function B(τ ): probability that an Exposed node, is Infectious at day τ ;

• daily recovering function γ(τ ): probability that an Exposed node, recovers exactly at day τ ;

• recovering function Γ(τ ): probability that an Exposed node, is recovered by day τ ;

In this paper, We first extend the discrete-time concept to continuous-time non-Markovian model form. Adequately, the functions γ(τ ), Γ(τ ), b(τ ) and B(τ ) in this scenario are continuous, and are further referred to as Continuous-Time Probability Functions (CTPFs), with γ(τ ) and b(τ ) referred to as instance recovering probability and instance manifesting probability, correspondingly. For the continuous-time form, we derive the epdemic threshold in a strict mathematical procedure. Then, the main result of this paper is presented: that for J o u r n a l P r e -p r o o f Journal Pre-proof each mean-field non-Markovian SEIS model (discrete-time or continuous-time), exists a Markovian SIS model, such that the stationary state probabilities of each node being exposed in the SEIS model, equals the stationary probability that the node is Infected in the SIS model, providing certain relations between process parameters hold. We consider this result to be of at-most importance for the following reason: non-Markovian models, although highly accurate in analyzing natural phenomena, are computationally sufficiently more demanding. Investigating these models with utilization of Markovian analogs (as shown here as possible), significantly reduces the computational complexity in acquiring significant data related to the endemic state of the diseases. The presented analysis directly leads to relations that define the epidemic threshold for the non-Markovian (SEIS) models occurring on complex networks, without the necessity of a thrall mathematical procedure. Similar type of equivalences, between the non-Markovian and the Markovian SIS model, using different settings and approaches, have been established by the authors in [1] and [10] As a result of secondary importance, it is shown that an arbitrary Markovian SIS model occurring on complex networks, may be represented an non-Markovian SEIS model. This equivalence is only vaguely mentioned and numerically illustrated in the Conclusions of [15] ; here we show this feature through a rigorous mathematical procedure. One should note that similar analysis was conducted in respect to the non-Markovian and Markovian SIR model in [16] .

In this Section, we present a short introduction to the processes and models used in the analysis. First, a very brief description of the well known mean-field Markovian SIS model is presented. Then, following [15] , a more detailed description of the discrete-time non-Markovian SEIS process and the accompanied mean-field model is given. Finally, a mean-field approximated continuous-time SEIS model is introduced, and for this model the epidemic threshold is derived.

In this paper, it is considered that all processes occur on a network rep-J o u r n a l P r e -p r o o f Journal Pre-proof resented with the adjacency matrix A. In the general case, the network is directed, weighted, and strongly connected; consequently the matrix A = [a ij ] is asymmetric, with 0 ≤ a ij ≤ 1, and irreducible (Perron-Frobenius theorem for non-negative irreducible matrices holds).

Although well known and widely elaborated in the scientific literature, in this sub-section we give a short introduction to the Markovian SIS model occurring on complex networks. The Markovian SIS model is a status model, in which each node, at given moment, may be in one of two distinctive statuses: Susceptible (S) and Infected (I). A node is infected at time t + ∆τ , if the node was Susceptible at time t, and acquired the spread agent from its Infected neighbours in the time interval (t, t+∆τ ), or was Infected at time t and did not recover in the interval (t, t+∆τ ).

Probability of the recovery, within a unity time interval is defined by the curing rate γ. Probability of acquiring the spread agent by a Susceptible node i, from one of its infected neighbours j is defined with the product a ij β, where a ij is the averaged epidemiological significance of contact between nodes, acting in the j → i direction, within a unity time interval, and β is the infection rate,

i.e. the probability of infection following a certain epidemiological contact with infected neighbour.

Following the formal definition stated above, and assuming statistical independence of joint events, the mean-field Markovian SIS model in discrete-time is described with [17, 18] :

where p I i (t) is the probability that node i is infected at time t. This equation form of the Markovian SIS model is often referred to as microscopic Markov chain approach in the literature [19] [20] [21] . It is the most common form of representation of the SIS process in the study of epidemic diseases.

J o u r n a l P r e -p r o o f

The Markovian SIS model in continuous form is also well known [22] , and may be written as:

Principles on which the mean-field continuous-time Markovian SIS model is derived are similar to those applied in formulation of the microscopic Markov chain approach, i.e. the assumption of statistical independence of joint events.

However, one major difference exist between both formats: due to infinitesimal duration of the model updating period, in the continuous-time model form, it is fairly assumed that a susceptible node may acquire the spreading material from only one of its infected neighbours. This notion transforms the productlike term, that approximates the probability of transition of spread material from infected neighbours to a Susceptible node, into a sum-like term. The presented approach of mean-field modeling the Markovian SIS process is referred to as Quenched Mean-Field (QMF) or N-Intertwined Mean-Field Approximation (NIMFA), in the scientific literature [19] .

Other methods of modeling the SIS process, have been studied in the past 20 years, as well. Early works on the subject, revolved around the degreebased mean-field (DBMF) approximation, introduced by Pastor-Satorras and Vespignani in [23, 24] . Recently, models that take into account the second-order statistical dependence, and therefore significantly improve on the accuracy, but increase the complexity of the mathematical models, were introduced in [25, 26] .

The discrete-time form of the SEIS model analyzed in this paper is originally introduced in [15] . For completeness, in what follows, we re-state the formal definition of the observed process, and for more details we refer the readers to the cited paper.

The SEIS model, as desribed in [15] and in this work, is a status model, which, in respect to the spreading agent, is characterized by two disjunctive statuses, Susceptible (S) and Exposed(E), and by status Infectious (I), that is J o u r n a l P r e -p r o o f Journal Pre-proof a sub-state of the E-status. A node is in status Exposed, at time t, if at the given instance it contains the spread agent; otherwise the node is considered to be Susceptible. Exposed node may be Infectious (manifesting infectiousness) or non-Infectious. Node is Infectious if it contains the agent (is Exposed) and is capable to spread the agent further to its neighbors.

Susceptible node contracts the spread agent from one of its Infectious neighbors and becomes Exposed at time t, depending on two factors. First is the epidemiological importance (on average) of the contact at the given instance, denoted with the probability a ij . Second, is a cluster of unknown variables that determine whether the transfer of spread agent will occur, following a certain epidemiological contact, with potential to evolve into an infectious process within the contracting node, and are described by the well known parameter β, i.e. the infection rate. In this paper, as well as in [15] , it is considered that factors described by the parameter β are purely stochastic and by no means may be referenced to events related to the spreading dynamics; in that sense β is considered constant throughout the process. As an example, one may consider β to represent the fraction of the individuals that have no immunity, or are prone to developing an infection towards certain viral disease; average capacity of an individual to produce the viral load required for infecting others e.t.c. In a number of recent papers, that explore forms of the continuous-time non-Markovian SIS model [1, [4] [5] [6] 10] , parameter β is time dependent (p.d.f) and denotes the probability of the Infected node to transfer the spread agent to its neighbors at given moment. This role in our model is played by DTPFs/CTPFs B(τ ) and b(τ ).

Similar discussion is relevant in respect to parameters a ij : knowledge about potential infectious contact, followed by occurrence of symptoms, alters the contact dynamics of the individual (the spreading node). However, implementing these behavioral changes in the model, by far exceed the scope and will increase the complexity of this paper; for those reasons, in what follows, we consider parameters a ij to be constant, as well. In this sense, the model evolves around the notion that the S → E transition is purely Markovian in charac-J o u r n a l P r e -p r o o f Journal Pre-proof ter. It is the authors impression that, disregarding behavioral changes of the Exposed/Infectious nodes, this setting adequately mimics the reality.

The process of agent contraction by the Susceptible node plays a role of a trigger event (τ = 0): all consequent processes within the node are timereferenced to this transition. Exposed (but non-Infectious node) may become Infectious exactly at time τ after the trigger event with probability b(τ ). Exposed node is Infectious at time τ following the trigger event with probability B(τ ). To stress the difference between b(τ ) and B(τ ), as explained in [15] , the model allows for two different types of Infectiousness manifestation:

• Cumulative manifestation -in this case b(τ ) has a character of a mass probability function in the discrete-time case scenario and density probability function in the continuous-time scenario. Adequately,

, has a cumulative character, with the sign "<" indicating that the Exposed node may not necessarily become Infectious prior to recovery. This type of behavior is typical for epidemic diseases;

• Random manifestation -in this case B(τ ) = b(τ ) has a random character, with 0 ≤ b(τ ) ≤ 1 being the only restriction.

Exposed node recovers and becomes Susceptible again exactly at time τ following the exposure, with probability γ(τ ), and is recovered at time τ with probability Γ(τ ) = Let p E i (t) denote the probability that node i is Exposed, and p I i (t) denote the probability that node i is Infectious, at time t. Considering the definitions stated above, and following [15] , the mean-field discrete-time SEIS model is mathematically defined in the following form:

with P i (t) representing the product-like term:

that denotes the probability that a Susceptible node i will contract the spread agent from its neighbours, at time t. The system of equations (3) is derived relying on the well known assumption of statistical independence of joint events, used as basis in developing microscopic Markov chain models [17, 18, 27, 28] .

For details related to derivation of the equation (3), we refer the readers to [15] .

One should note the difference in the formulation of the non-Markovian SEIS model, as introduced in [15] and presented in this paper, compared to the classical Markovian SEIS model. In the classical version the E-stage refers only to nodes that contracted the spread agent, but are non-Infectious [29] [30] [31] . This distinguishes between E and I statuses as disjunctive, leading to conservation

In the non-Markovian SEIS model, the I status is a sub-state of the E status, consequently the conservation equation creates a unique time-frame in which all other processes that an individual node undergoes, following the contraction of the spread agent, occur. In that sense, one should note that separation of status Exposed (but non-Infectious) and Infectious, as well as other statuses that may occur in extended non-Markovian scenarios (models), into disjunctive statuses is possible (with adequate adjustments to the DTPFs/CTPFs); however that requires introduction of multiple time-frames (each subsequent status should be referenced to the previous). Even further, each of these time-frames should be referenced to the origin of the initial time-frame (i.e. the moment of the S → E transition). This will make the analytical model more complicated.

In this section, we introduce the continuous-time non-Markovian SEIS model, as an extension to the discrete-time model (3) . Prior to the formal introduction of the model, we stress the major difference that exist between discrete-time and continuous-time modeling approach. It is a standard practice in modeling spreading phenomena in continuous time to assume that the transfer of the spread agent, within an infinitesimal time interval ∆τ , may occur from a single sources (neighbour), only (no-multiple infectious events assumption). This notion transforms the product-like term in the following manner:

For sufficiently small ∆τ , the term O(∆τ 2 ) is neglected, and the appropriate sum-like term [22] , obtained.

Bearing in mind the differences, We may now re-write the system of equations (3) as follows:

When ∆τ → 0, providing no discontinuities of first kind exist in Γ(τ ) or Γ(τ )B(τ ), the terms multiplied by ∆τ 2 may be neglected. In what follows, we

show that this term may be neglected even in the presence of finite number of first order discontinuities. The analyses is focused around the second set of N equation in the system (4), related to the p I i (t), variables; by analogy, the same analysis is valid for the set of equations related to the p E i (t) variables. Consider a point 0 ≤ τ i < T , such that a discontinuity of first kind Γ(τ ) or Γ(τ )B(τ ) exists at τ = τ i . Let ∆τ be an integration constant, such that the series τ k = k∆τ provides a proper sampling of Γ(τ ) and Γ(τ )B(τ ). Let k i be an index such that k i−1 ∆τ < τ i ≤ k i ∆τ . Under these assumptions, the following J o u r n a l P r e -p r o o f Journal Pre-proof inequality may be considered:

Let R be a total number of first kind discontinuities of either Γ(τ ) and Γ(τ )B(τ ). Then, in accordance with the relation above:

The preceding analyses indicates that the the terms multiplied by ∆τ 2 in the set of N equations, related to the p I i (t) in (4), may be neglected, since for finite R, N R∆τ may be maid arbitrary small, with the right choice of ∆τ . Similar analysis, leading to the same conclusion, may be conducted for the set of equations related to p E i (t) in (4). Consequently, from (4) and considering ∆τ → 0, one obtains the integral form of equations for the non-Markovian SEIS model occurring on complex networks in continuous-time: and B(τ ) to be smooth around the point τ = 0, and the Heaviside function may be neglected in the system of equations (5).

In this segment, we show that the non-Markovian SEIS model, represented

with (5), may be written in a differential form, as well. Starting from the system of equations (4), one obtains:

Considering that Γ(T ) = 0, by dividing both sides of the equation with ∆τ and by letting ∆τ → 0, the following relation may be written :

Similarly for p I i one obtains: To find the epidemic threshold for the continuous-time non-Markovian SEIS model, we resort to the investigation of the stability criteria of the dynamical system (5), around the point of epidemic origin, i.e. p E i (t) = 0, p I i (t) = 0, for all i. In that sense, we consider the following: Theorem 1. Consider a directed, weighted and strongly connected graph, represented with the adjacency matrix A = [a ij ] that is, consequently, non-negative

the epidemic origin, is a globally asymptotically stable point of equilibrium of the dynamical system (5), providing the following relation holds:

with λ 1 (A) being the leading eigenvalue of the matrix A.

Proof. Consider the system (5). Since p E i (t), p I i (t) ∈ [0, 1], for all i, and B(τ ), Γ(τ ) ≥ 0, the argument under the integral is strictly postitve, so the following relation holds:

In other words, the dynamical behaviour of the system (5) is bounded from J o u r n a l P r e -p r o o f Journal Pre-proof bellow by the epidemic origin, and from above, by the dynamical system:

In that sense, the proof of the global stability of the (epidemic origin of the) system (5), reduces to proof of the global stability of the system (8).

One may notice that the second set of N equations in (8) is self-sufficient.

In that sense the dynamical stability of the system (8) reduces to the dynamical stability of this set of equations only: from (8) 

By conducting a Laplace transform on both sides of each of the equations from the second set of N equations in (8) , following the methodology in [16] , one obtains:

being the Laplace transform of the product B(τ )Γ(τ ), and (9) may be re-written in a vector form as:

leading to a solution in Laplace domain, in the following form:

where (I − βL(B(τ )Γ(τ ))A) is a matrix, which elements are the minors of the

It is a well known result in the dynamical system theory, that the stability of the (origin of the) dynamical system is determined by the position of the poles of the system in the complex plane. If all poles of the dynamical system lie within the left-half of the complex plane, i.e. Re{s} < 0, the dynamical system is globally asymptotically stable. From the equation (12), one obtains that the poles of the system (8) may be determined from the zeroes of the equation:

On the other hand: 

Index k allows for multiple poles associated with a single eigenvalue λ i (A).

From the definition of the Laplace transform of L(B(τ )Γ(τ )), i.e. equation (10) , the following conclusions hold: Re{s i,k } < 0, providing:

for all i and k.

From the Perron-Frobenius theorem for non-negative and irreducible matrices, the leading eigenvalue, λ 1 (A), of the matrix A is distinct, real and largest by module, compared to all other eigenvalues; therefore it minimizes the term 1/β||λ i (A||. For this reasons, providing:

holds, the poles of the second set of N equations of the dynamical system (8) lie within the left-half of the complex plane, resulting in lim t→∞ p I i (t) → 0. This yields lim t→∞ p E i (t) → 0, lim t→∞ p I i (t) → 0, lim t→∞ p E i (t) → 0, and the point of epidemic origin of the dynamical system (5) is globally asymptotically stable.

The Proof is completed.

In accordance with the Theorem, the relation:

defines the boundary between the parametric region related to the state of permanent epidemic presence in the network and the region of epidemic absence.

In that sense, the equation (15) 

By dividing equations in (16):

One should note that relation (17) is not a consequence of the modeling, rather a general feature of the SEIS process, as shown in the Appendix, eq. (A.4). 

with β ef f defined with:

The stationary state solution of the discrete-time Markovian SIS model, may 

and then dividing equations (20) and (18) 

The analysis conducted above, leads to the following conclusion: providing relations (21) and (22) 

with λ 1 (A) being the largest (leading) eigenvalue of the adjacency matrix A.

This relation holds for both discrete-time SIS model [17, 18] , as well as continuoustime SIS model [22, 33] .

SEIS implies that, providing equations (21, 22) in the discrete-time, and (27) in continuous-time SEIS model hold, then: 

J o u r n a l P r e -p r o o f Journal Pre-proof in the discrete-time case, and:

in the continuous case. For the discrete-time case the relation (29) is derived in precise analytical procedure in [15] . The relation (30) is identical to the equation (15) derived in Section 2.3.2.

As presented in this subsection, the stationary-state model equivalence, leads directly to the result for the epidemic threshold of the non-Markovian model.

No thrall statistical or system stability analysis is required to obtain this result.

This feature further emphasizes the importance of the main result of the paper.

In this section we present the results of the numerical analysis, in order to The analyses in the paper are conducted on the following networks:

• Barabási -Albert [34] 

In order to confirm the results for the discrete-time model, following [15] , we consider three different sets of DTPFs. The first two sets, labeled CM1 and CM2, are related to the case of cumulative-like manifestation of Infectiousness.

In the third set, labeled as RM, the manifestation has random character. The sets are presented in Table 1 . Results of the parametric dependence analysis for the discrete-time case are presented in Fig. 1 , with corresponding time series analysis presented in Fig.   2 . Analysis of the epidemic threshold for the different DTPFs and graphs is summarized in Table. 2. The results from the analysis presented in Fig.1 indicate that there is a perfect overlap of stationary state solutions of both the discrete-time non-Markovian SEIS model and the Markovian SIS model, providing relations (21, 22) hold, in all simulated cases. Also, there is a good match between the (meanfield) models result and the reality (stochastic simulations), as well as between the real (stochastic) SIS and SEIS processes, in all, but for the case of the CM1 DTPFs, Figs. 1a and 1c , for the Exposed nodes. These findings are further supported by the results presented in Fig. 2 . The results imply that the (mean- The threshold analysis presented in Table 2 

The CTPF's used in the continuous-time case (cumulative manifestation only) were constructed as follows: it is assumed that the process lasts for total of T = 65 time units. The instance manifestation probabilities were obtained from the Weilbull p.d.f.: w(τ ; α; λ) = αλ(τ λ) α−1 exp (−(τ λ) α ), with parameters α = 2.04 and λ = 0.103, following eq.(2) in [36] (updated in [12] ), normalized to 65 days: b(τ ) = w(τ ; α; λ)/ 65 0 w(τ ; α; λ)dτ . The daily recovering probabilities were obtained from log-normal distribution l(τ ; µ; σ) = 1/(τ σ √ 2π) exp −(ln τ − µ) 2 )/σ 2 , µ = 3, σ = 0.28 normalized to 60 days, Integration constant ∆τ = 0.05 was used in the analyzes.

Results of the parametric dependence analysis of the continuous-time case is presented in Fig. 3 , accompanied by the results for the time series analysis in Fig. 4 . In addition, the epidemic threshold analysis is presented in Table 3 . were performed, with 500 (50%) of the nodes initially set as Exposed at t = 0.

Two parameters were followed during the simulations: the number of non-zero outcomes at the end of each trial (at t = 1000), and the average number of 

Assumption of statistical independence of joint events, widely implemented in mathematical analysis of spreading phenomena on complex networks, is known to impose certain degree of error into analyzed models. Though efforts to assess the degree of error it generates, have been made in analysis of some models (for example see Appendix B in [18] ), there is no definite answer to the size of this error depending on the state of the system. Authors in [25] concluded, that for the SIS process, the model obtained by utilizing this approximation upper-bounds the reality. In our previous experience, and supported by the The results of the analysis are presented in Fig. 5 .

Results of the analysis indicate that, given identical fraction of Infected (SIS) Fig. 3 and Fig. 4 ). and γ(τ ). Illustrative numerical example of this feature for the discrete-time case, has already been presented in [15] . In this article, this feature is theoretically investigated and shown for both model forms in a concise mathematical procedure.

As a note to the readers, in what follows, we avoid the use of label superscripts, since the whole procedure is conducted on the non-Markovian SEIS model, that under the investigated circumstances reduces to the Markovian SIS form.

Consider, as suggested in [15] , B(τ ) = 1, for all τ , and γ(τ ) = γ(1 − γ) τ −1 s(τ − 1), with 0 < γ ≤ 1, with s(τ ) being the Heaviside function. Con 

When T → ∞, the last term in the equation vanishes. Bearing in mind that p E i (t) = p I i (t), the following relation holds: By considering B(τ ) = 1, consequently p E i (t) = p I i (t), and Γ(τ ) = exp(−γτ ) and T → ∞ (see [16] for details) one obtains:

The last relation is identical with the classical formulation of the Markovian Typical time series generated by both models and the co-responding stochastic processes (reality) are presented in Fig. 6 . The figure illustrates a perfect match between the models, with minor discrepancies in respect to the stochastic simulations (reality).

The non-Markovian systems more accurately address the spreading processes and B i (t; τ ) be random variables denoting that the node i that was exposed at time t − τ , is recovered or is Infectious, at time t, respectively.

Following the Appendix in [15] , the non-Markovian SEIS model may be written in the following status form:  We introduce the continuous-time non-Markovian SEIS (Susceptible--Exposed--Infectious--Susceptible) model occurring on complex networks and for this model we derive the epidemic threshold;  For both discrete-time and continuous-time model form, we show that the stationary state status probabilities of the nodes in the non-Markovian SEIS model, may be found from the related stationary state node probabilities in the classical Markovian SIS model;  We utilize the obtained result to confirm the epidemic threshold for both the discrete-time non-Markovian SEIS model and the continuous-time non-Markovian SEIS model on complex networks;

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