key: cord-0428369-p40spjfe
authors: Estrada, Ernesto; Bartesaghi, Paolo
title: From networked SIS model to the Gompertz function
date: 2021-10-17
journal: nan
DOI: 10.1016/j.amc.2021.126882
sha: 0cfa5bbac240f0b03d017fa2176d5315711316f2
doc_id: 428369
cord_uid: p40spjfe

The Gompertz function is one of the most widely used models in the description of growth processes in many different fields. We obtain a networked version of the Gompertz function as a worst-case scenario for the exact solution to the SIS model on networks. This function is shown to be asymptotically equivalent to the classical scalar Gompertz function for sufficiently large times. It proves to be very effective both as an approximate solution of the networked SIS equation within a wide range of the parameters involved and as a fitting curve for the most diverse empirical data. As an instance, we perform some computational experiments, applying this function to the analysis of two real networks of sexual contacts. The numerical results highlight the analogies and the differences between the exact description provided by the SIS model and the upper bound solution proposed here, observing how the latter amplifies some empirically observed behaviors such as the presence of multiple and successive peaks in the contagion curve.

The Gompertz function was first introduced by Benjamin Gompertz in 1825 to describe human mortality curves [1] . Formally, a Gompertz curve can be defined as a non-negative real valued function defined on the open interval 0 < t < ∞ [2] :

where P, Q, R > 0. It can be proved that it is the solution of the differential equation of the form (see [2] for the formal proof):ḟ

A point of inflection for the Gompertz curve is the ordered pair log Q R , P e . This curve is appropriate to describe many physical, biological and man-made processes, which are characterized by a rapid growth in the early The standard logistic growth curve can be interpreted as the solution of the Susceptible-Infected-Susceptible (SIS) model, the epidemiological model characterized by the possibility of re-infection after recovery (see, for instance, [21] for an up-to-date introduction to the SIS model). In Fig. 1 , we illustrate the difference between SIS model and Gompertz model. Blue square dots curves in both boxes refer to the SIS model and red circular dots curves to the Gompertz model. Curves in the left box represent the normalized cumulative number of cases N (t) as a function of time t, whereas curves in the right box represent the first derivative dN (t)/dt. Curves on the left can be interpreted as the cumulative probabilities to find an infected individual at time t and curves on the right as the probabilities that new individuals get infected at time t, or equivalently as the infected population density. All the plots refer to the same set of parameters values. The initial fraction of infected individuals or, more in general, the initial population size is set to N 0 = 0.1 and the asymptotic fraction of the infected individuals or, more in general, the final carrying capacity is set to N ∞ = 0.8. In the SIS model, the value of the infection rate is β = 0.2 and the value of the recovering rate is γ = 0.04. Note that, with these values, we have N ∞ = 1 − γ β = 0.8. In the Gompertz model described by Eq. (1), these parameters correspond to P = N ∞ , R = β 1 − γ β and Q = log N∞ N0 , so that, in particular, the value for the damping rate in the Gompertz model is R = 0. 16 . Blue lines show the typical symmetrical behavior of the SIS model and of the logistic curve with respect to its inflection point; red lines show the typical asymmetrical behavior and the long tail of the Gompertz model with respect to its inflection point. In fact, the inflection point of the logistic curve is in N ∞ /2 = 0.4 at time t = 1 R log N∞ N0 − 1 ≈ 12.16, whereas the Gompertz curve is asymmetrical, with an inflection point lower than N ∞ /2, namely N = P/e ≈ 0.3 at time t = logQ R = log log 8 0.16 ≈ 4.6. Finally, all plots have been extended to negative times to highlight the complete behavior of the curves.

Let G = (V, E) be a simple connected graph on n vertices (nodes). We will use indistinctly graph or network to refer to G. The networked SIS model is described by the following n equations:

where x i (t) represents the probability that node i is infected at time t and x i (0) = x 0i = p, ∀i = 1, . . . , n, for some 0 < p < 1, the initial probability of being infected, equal for all nodes. Elements A ij are the corresponding entries of the adjacency matrix A of G. In matrix-vector form, Eq. (3) becomes:

where I is the identity matrix of the corresponding order and u the vector whose components are all equal to 1. In what follows, we write in general

Let us now rewrite the networked SIS model on the basis of the information content I i (t) that a given node i is not infected at time t. The information content (also known as surprisal, self-information, or Shannon information) is given by

where s i (t) = 1 − x i (t) is the probability of not being infected at time t, i.e., of being susceptible. The information content SIS model (IC-SIS) is now written aṡ

with δ ij being the Kronecker delta and

Let us consider the function f (I) = 1 − e −I , which is increasingly concave, so that we can write

Analogously, because g(I) = e I − 1 is increasingly convex we have

Therefore, we havė

Let us call the right-hand-side part of this equation the worst-case scenario IC-SIS as it represents a clear upper bound to the surprisal that a given node i is not infected at time t. Let us designate it byİ i (t) and taking into account that x 0 = p, I 0 = − log q, e −I0 = q and e I0 = 1/q, we can write it aṡ

is the worst-case scenario surprisal that the node i is not infected at time t. When written in matrixvector formİ

it is easy to realize that this equation has the linear forṁ

where B := qβA − γ q I and b := (p + q log q)βA − (p + log q) γ q I u. As a way of comparison, we will also consider the "standard" linearization of the SIS model around the point 0:

with initial condition x 0 , whose solution isx (t) = exp [(βA − γI) t] x 0 . It is easy to realize that this solution is exponentially unstable. We now formally prove that the solutionx (t) := 1 − exp −Î to the worst-case scenario SIS model is an upper bound -therefore a worst-case scenario solution -to the exact SIS model solution, i.e.

x (t) x (t) , and that it is a lower bound to the divergent solution of the linearized model, i.e.x (t) x (t).

Theorem 1. Let x (t),x (t) andx (t) be, respectively, the solution of the exact SIS model in Eq. (4), of the worstcase scenario IC-SIS model obtained from Eq. (7) and of the linearized SIS model in Eq. (9), with the same initial conditions:

where the second inequality holds if β e := β γ > 1 q 2 kmin , with k min minimum degree of the graph.

Proof. We first prove that x (t) x (t). By the method of variation of parameters, the solution of the worst-case scenario IC-SIS model isÎ

which is equivalent toÎ

Since I i (0) =Î i (0) = − log q andİ i (t) <İ i (t), ∀t ≥ 0, ∀i = 1, . . . , n, following Lemma A.1 by Lee et al. [22] , p. 11,

is an upper bound for the original SIS network model solution x i (t), ∀t ≥ 0, ∀i = 1, . . . , n, and this proves the first inequality. We now prove thatx (t)

x (t). To this purpose, let us assume that qβk min − γ q > 0, which implies qβλ 1 − γ q > 0, where λ 1 is the eigenvalue corresponding to the dominant eigenvector of A. Under this condition, since q < 1, we have βλ 1 ≥ βk min > γ q 2 > γ q > γ and both the solutions of the upper bound problemx (t) and of the linearized problemx (t) are increasing functions in time. Again, by Lemma A.1 in [22] , since the initial conditions are the same for both processes, i.e.,x (0) =x (0) = x 0 = pu, it is enough to prove that

for all t ≥ 0. Beingx (t) = 1 − e −Î , then we have

for all t ≥ 0, where the inequality in Eq. (15) follows from e −Îi < 1 for allÎ i ∈ [0, ∞). By Eq. (12) we have

The last inequality can be justified in the following way. Being A and I commuting matrices, Eq. (16) is equivalent to e qβAt e − γ q It βA − γ q I u e βAt e −γIt (βA − γI) u and, hence, to the inequality

Moreover, qβA 0 and βA 0, with qβA βA element by element, such that

where 0 is the null vector, since we assume βk min − γ q > 0. Therefore, we have e qβAt βA − γ q I u e βAt (βA − γI) u, for all t ≥ 0 and, being e βAt (βA − γI) u e p q γt e βAt (βA − γI) u, for all t ≥ 0, all the components of the vector in the left hand side of Eq. (16) are less than or equal to the corresponding components of the vector in the right hand side. Finally Eq. (15) and Eq. (16) imply Eq. (14) and this ends the proof.

and it is equal to the solution for the SI Model by Lee et al. (see [22] and [23] ). Let us observe that if t = 0:

Let us observe that for t → +∞,

Observe that 0 < 1 − qe p < p, as expected, and that 1 − qe p ∼ p 2 as p → 0.

Remark 3. Let us consider β = 0 and γ = 0. The exponential term in Eq. (12) may be written as

where Λ is the diagonal matrix of the eigenvalues of A and M is the orthogonal matrix whose columns are the eigenvectors of A. As t grows to +∞, the diagonal exponential terms e (qβλi− γ q )t grows to +∞ if (qβλ i − γ q ) > 0. In particular, if (qβλ 1 − γ q ) < 0, no one of these terms grows to +∞ and the epidemic decays. Thus, we can identify a threshold given by the following condition

and we can identify τ = 1 q 2 λ1 as the threshold of the worst-case scenario solution. Then, if β e < τ the epidemic decays; if β e > τ the epidemic grows. Let us observe that, as λ 1 increases (and so does the average degree in the network), condition in Eq. (17) becomes stricter and the spread of epidemics is easier; moreover, in general, τ is greater than 1/λ 1 and, as p decreases, it approaches the lower bound 1/λ 1 .

Let us start by defining a matrix D as

so that B = − γ q D. Then we can rewrite the vector B −1 b − log q u appearing in Eq. (12) in the following way:

In this way, the solution of the worst-case scenario IC-SIS model becomeŝ

We are now in a position to present the main result of the current work.

Theorem 2. Let G be a graph in which there is a IC-SIS dynamics taking place on its nodes and edges. Let ψ ν (i) be the i-th entry of the ν-th (orthonormalized) eigenvector of A, associated with the eigenvalue λ ν , with λ n ≤ · · · ≤ λ 2 ≤ λ 1 . Then, under the condition β e < qτ , the worst-case scenario probabilitys i (t) that a node i is susceptible of getting infected for time t → ∞ is given by a Gompertz function of the form

where

Since D = I − q 2 β e A is symmetric and there exists an orthogonal matrix M whose columns are the eigenvectors of A and, consequently, of D such that

. Now, in order to describe the asymptotic behavior ofŝ i (t), let us consider separately the following three cases:

In particular, for t → +∞ŝ

The last term in Eq. (21) can be interpreted as a Gompertz function. Indeed, it is

. . , n as the eigenvector associated to the dominant eigenvalue λ 1 has components all with the same sign. Moreover, in order to get also Q i > 0, the further condition 1 − qβ e λ 1 > 0, equivalent to β e < qτ , must be satisfied. Let us notice that this condition is stricter than 1 − q 2 β e λ 1 > 0, which is equivalent to β e < τ . 2. qτ < β e < τ : in this case Eq. (21) still holds but it cannot be considered a Gompertz function since α 1 < 0.

3. β e > τ , that is 1 − q 2 β e λ 1 < 0:ŝ

where again ζ 1 (i) > 0 and also α 1 = p−pqβeλ1 1−q 2 βeλ1 > 0 since now 1 − qβ e λ 1 < 0. Therefore, in this case, we have as expectedŝ i (t) → 0 for t → +∞.

Remark 4. According to Theorem 2, when the time is sufficiently large, the solution of the worst-case scenario IC-SIS on a network is a Gompertz function of the type: f (t) = P exp −Qe −Rt , under the condition that the effective infectivity rate is lower than qτ = 1 qλ1 . This is an important result because, as we have seen in the Fig.  1 , the main differences between the SIS model and the Gompertz model occurs not at early times of the disease propagation but when the time is sufficiently large. Additionally, it is also at large time when the two models developed by Frenzen and Murray [7] coincide with a Gompertz curve, which would open some interesting avenues for future exploration of these models in relation to the current one.

Remark 5. Let us notice that, according to Eq. (20) , the probability density can be written as

and the point of inflection of the Gompertz curve is then given by

This inflection point, which is the time at which the probability density of recovering from infection is maximum, grows with the logarithm of ζ 1 (i) and ζ 1 (i) is proportional to the eigenvector centrality of node i. This means that the more central is the node, in terms of eigenvector centrality, the more this maximum is delayed in time and the slower will be the healing process for that node.

Remark 6. Finally, we conclude by observing that Eq. (19) can be given the following interesting expression. When the effective infectivity rate is below the threshold τ , that is if it is satisfied the condition 0 < q 2 β e < 1 λ1 , the inverse D −1 = I − q 2 β e A −1 can be considered a matrix resolvent and

, is the vector whose entries are the so-called Katz centrality indices of the nodes in the network [24] . This index belongs to a family of network descriptors which counts the number of walks in the graph involving the corresponding node, giving more weight to the shorter than to the longer ones [25] . Therefore we have

(t) = p e − γ q Dt − I u − p q c − log q u.(26)

We now present some numerical results from computational experiments carried out on two real-world sexual contact networks. The rationale for choosing networks of this type lies in the fact that they typically host contagion processes that can be described using the SIS model. The spreading of many different epidemics of global interest, like influenza, Ebola, Zika, Chikungunya, and others, has been analyzed by applying this model. For instance, Juher et al. [26] studied in details the Syphilis transmission characteristics and disease evolution on a large sexual network in San Francisco and Zhao et al. [27] investigate, in the same framework, the temporal patterns and transmission potential of ZIKA virus in eight Brazilian states, over the same period.

Here we focus on two specific sexual contact networks. Both networks are connected, unweighted and undirected. The first corresponds to m = 83 heterosexual contacts between n = 82 individuals (35 males and 47 females). In this network there is available information about the sex of the individuals. The network was studied in details in [28] where a subset of individuals were infected with Chlamydia Trachomatis and/or Neisseria Gonorrhoeae in Manitoba, Canada. We consider here the giant connected component of this network, which spans two broad geographic areas, northern Manitoba and Winnipeg. The second network of sexual contacts represents m = 266 homosexual contacts between n = 250 individuals. It is known that 236 individuals (94.4%) were men, of whom 219 were homosexual, and 14 (5.6%) were women. Of the 184 men and 13 women for whom age was known, mean age (at mid-early period) was 29.5 years for men and 25.8 years for women. The sex of every individual was not reported in the dataset available. This network has been previously studied in [29] to explain the modest HIV/AIDS burden reported from Colorado Springs at the earliest days of the epidemic. Accordingly, the risk networks of susceptible individuals have been insufficiently cohesive to sustain substantial endogenous HIV transmission [29] . An illustration of both networks studied here is given in Fig. 2 .

In Table 1 we compare some of the general parameters of both networks (see [30] for definitions and examples): (i) edge density δ = 2m/n (n − 1) where m is the number of edges, (ii) average shortest path lengthl, (iii) average Watts-Strogatz clustering coefficientC (see also [31] ), (iv) degree heterogeneity ρ (see also [32, 33] ), (v) spectral radius of the adjacency matrix λ 1 (see also [34] ), and (vi) degree assortativity r (see also [35] ). As can be seen, the two networks are very similar to each other, except that the network of heterosexual contacts (NHeC) is bipartite and thus it contains no odd cycles, which implies that the clustering coefficient is zero, while the network of homosexual contacts (NHoC) is not bipartite and it has clustering different from zero. Both networks are poorly dense, although the NHeC is more densely connected than the NHoC. The degree heterogeneity of both networks is relatively small, their spectral radii are very close to each other, and both networks are degree disassortative indicating a preference of high degree nodes of being connected with low degree ones. Table 1 : A few topological parameters of the two networks of sexual contacts analyzed: δ is the edge density;l is the average path length;C is the average Watts-Strogatz clustering coefficient; ρ is the degree heterogeneity; λ 1 is the spectral radius of the adjacency matrix; r is the degree assortativity.

The main goal of this section is to compare the networked SIS model and the Gompertz-like solution produced as an upper bound of the SIS model. The SIS model is used for sexually transmitted diseases where individuals can get repeatedly infected. Eames and Keeling [36, 37] provide some specific values for β e for both Chlamydia and Gonorrhoea, according to the different models used to describe the virus spreading in the same network we analyze here. For the purposes of our numerical analysis it is enough to keep a mean value equal to β e = 1.5. Therefore, we will consider a disease propagating through these two sexual networks, for which we use the following parameters: β = 0.03, γ = 0.02 and β e = 1.5. In both cases we consider the same probability p, which for the NHeC is 1/82 and for the NHoC is 3/250. This choice is equivalent to a single initially infected individual in the NHeC; an equal initial probability is then adopted in the NHoC to compare the time evolution of the epidemics in both.

We start our analysis by considering the time evolution of the cumulative number of infected individuals in both networks. In Fig. 3 we illustrate the results of the simulations with the parameters given before using the exact solution of the SIS model (panel (a)) as well as with the Gompertz-like solution of the worst-case scenario (panel (b)). As can be seen in panel (a) the evolution of the infection predicted by the SIS model is practically identical for both networks up to approximately t = 40. From that time the propagation of the infection in the NHoC is faster than in the NHeC. In the steady state the number of infected individuals in both networks is below 65% with a slightly bigger percentage in the NHoC than in the NHeC. In the case of the Gompertz-like model the growing process occurs identically for both networks for a longer time than in the SIS model, i.e., up to t = 50 (see panel (b)). Also, after this time the infection grows faster in the NHoC reaching the 100% of infected individuals at t = 90 while the NHeC reaches the saturation at t = 110.

To gain more insights about the importance of these differences we consider some data observed empirically in [28] . According to this observation the branch formed by individuals Notice that both branches contains the same number of nodes, as well as the same proportion of males/females. As can be seen in Fig. 3 (panel (c) ) the exact solution of the SIS model predicts a faster growth of the infection in branch 1, but when this branch reaches its maximum, which is not 100% of infected nodes, the difference with branch 2 is insignificant. This indicates that using SIS we cannot observe branch 1 completely infected and branch 2 only partially infected. The Gompertz approach (see panel (d)) also predicts a faster growth of the infection on branch 1, but with bigger differences than in the case reported by SIS. For instance, at t = 80, SIS predicts 45.7% of infection in branch 1 and 35.4% in branch 2, while Gompertz predicts 90.9% for branch 1 and only 56.1% for branch 2. More importantly, at t = 100 all the nodes in branch 1 are infected, while the probability of being infected for nodes in branch 2 is below 80%. Therefore, in this case we can reproduce, even without fitting the parameters β and γ, what has been observed empirical in the real-world evolution of a sexually transmitted disease in this network.

We now turn our attention to the time evolution of the population density of infected individuals. As can be seen in Fig. 4 (panel (a) ), according to SIS, the growing period is practically the same in both networks, with the NHoC reaching a higher peak than the NHeC. Both networks reach their maximum number of infected individuals almost at the same time. The number of infected individuals per unit time decays slightly faster in the NHoC than in the NHeC, and both curves finally reach the zero percent of infected individuals at the same time.

In the case of the Gompertz-like model (panel (b)) the differences are more marked between the time evolution of the infection in both networks. Here again the infection has the same growing process for both networks, but from this time the differences of the dynamics in the two networks are more significant. From t = 40 to t = 75 the infection decays faster in the NHeC. From that time on, the decay is faster in the NHoC. Another important characteristic observed in panel (b) is that while the whole decay in the NHoC is smooth, it is irregular for the case of the NHeC.

In order to further investigate the non smooth decay of the probability density of infected individuals in the network of heterogeneous sexual contacts we perform the same analysis by separating the nodes according to the sex of the corresponding individuals. As can be seen in Fig. 4 (panel (c) and (d)) both approaches (SIS and Gompertz, respectively) predict a faster growing process for females than for males, with a more pronounced difference in the Gompertz approach. However, while the SIS approach predicts smooth decays of the number of infected males and females, the Gompertz approach predicts an irregular decay for males with two shoulders at different times, and a clearly non-monotonic decay for females where a second peak is observed at about t = 80.

In the literature appear several reasons for this kind of "multiple peaks" behavior in the dynamics of diseases. In the review paper [38] the author asks the question: "What is the mechanism(s) that is generating the multiple peaks? Nobody knows". In the context of Susceptible-Infected-Recovered (SIR) models the authors of [39] proposed several alternatives, two of which are applicable to the case studied here: (i) changes in pathogen transmissibility and behavioral changes; (ii) population heterogeneity where each wave spreads through one sub-population. In another paper [40] (see also [41] ), the authors reported that highly assortative mixing of the contacts tends to lead to more rapid epidemic growth and can produce multiple peaks in disease incidence. An explanation for the influence of assortative mixing and multiple peaks is provided in [42] by considering that the pathogen tends to move gradually from one sexual activity group to the next, where the multiple peaks then correspond to emerging infection within-group epidemics. A similar finding is reported in [43] . On the other hand, in [44] the authors are able to generate multiple peaks of the infection by tuning the parameters of a multi-compartment model.

We investigate here the causes of the multiple peaks observed when using the networked Gompertz model. First we notice that the NHeC consists of two main blocks which corresponds to individuals living in northern Manitoba and in Winnipeg. We then select nodes representing female individuals from both blocks, namely F10, F37, F1 and F4 from the block of Northern Manitoba (left in Fig. 2 (a) ) and F43 and F45 from the block of Winnipeg (right in Fig. 2 (a) ). In Fig. 5 we illustrate the results of the progression of the infection through these nodes. We have initiated the process by considering that every node has exactly the same probability of getting infected, i.e. p = 1/82. However, as "all roads lead to Rome", node M9, which has the highest degree of the network (degree 21), has a high risk of contagion at earlier times of the propagation of the infection. Then, it is very clear in the figure that among the nodes selected here, node F10, which is connected to M9 and also has a high degree (degree 11), is the first to reach the peak of infection. Then, it comes F37, which is a node separated by only three steps from M9. The process continues by infecting nodes F1 and F4, which are five steps away from M9, with the difference that the infection has two routes to infect F1 (there is a square: F10-M4-F2-M5-F10) and only one to infect F4. Then, the infection jumps to the second block and infects node F43 separated from M9 by seven steps and then F45 separated by nine steps from M9.

This result indicates that the networked Gompertz model obtained here captures the wavefront nature of the propagation of a disease on a network when more than one block or cluster exist. As we have seen before this important feature of epidemic propagation is not captured by the exact solution of the SIS model. Multipeaks like the ones reproduced by the current model are observed in several real-life epidemics, ranging from the dengue fever outbreak in Havana (see Fig. 6 in [38] ) to the swine fever virus outbreaks in the Netherlands during 1997-1998 (see Fig. 7 in [38] and [45] ). Additionally, as the NHeC studied here is degree disassortative, our results show that the condition of assortativity is not necessary for the existence of these multipeaks as previously claimed in the literature [40] .

It is known that the Gompertz function has a remarkable effectiveness in fitting experimental epidemiological and biological data, including the worldwide spread of COVID-19. Here we approach the problem of combining this specific function with the information about the structure of real networks hosting a contagion process. To the best of our knowledge, a Gompertz function on networks has not been proposed in the literature so far. This function should take into account the topological structure of the network and at the same time reduce asymptotically to the scalar Gompertz function for sufficiently large times. The present paper fills this gap, presenting a deduction of a Gompertz function on networks starting from a classical contagion model such as the SIS model, which, as is well known, is characterized by a symmetrical behavior with respect to the inflection point. The search for an upper bound for the exact solution to the SIS model on network has led to the identification of a new class of functions that lack such symmetry but instead exhibit a typical Gompertz behavior. We test this function on networks of sexual contacts that have already been analyzed in the literature as the site of transmission for various infectious diseases, which contemplate the possibility that a recovered individual could become infected again. The main results can be summarized as follows: 1) the function deduced as a worst-case scenario of the SIS model describes effectively the behavior of the exact solution simulated on the network under examination; 2) it allows to grasp the different behavior of separate clusters in the network, such as the two components in a bipartite network or specific subsets of nodes; 3) it allows to amplify and therefore better highlight some behaviors reported in the literature in the number of daily cases, such as the existence of multiple peaks of subsequent infections, or the spread of infection from the core towards the periphery of the network itself. Just as the scalar Gompertz curve finds application in the description of an ever increasing number of growth processes, we are confident that the function proposed here could help in the description of similar processes hosted on specific networks. 

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Figure 2: Illustration of the networks of (a) heterosexual contacts (NHeC) and (b) homosexual contacts (NHoC) studied in the paper. In the NHeC, the nodes are labelled by the sex of the individuals, M for male and F for female. The block of individuals residing in Winnipeg are sketched with a broken line