In Response to COVID-19: Configuration Model of the Epidemic Spreading
Alexander I Nesterov, Pablo Héctor Mata Villafuerte, Gennady P Berman
IIn Response to COVID-19: Configuration Model of the Epidemic Spreading
Alexander I. Nesterov ∗ and Pablo H´ector Mata Villafuerte † Departamento de F´ısica, CUCEI, Universidad de Guadalajara, Guadalajara, CP 44420, Jalisco, M´exico
Gennady P Berman ‡ Theoretical Division, T-4, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: July 13, 2020)A configuration model for epidemic spread, based on a scale-free network, is introduced. We obtainthe analytical solutions describing both unstable and stable dynamics of the epidemic spreading,and demonstrate how these regimes can interchange during the epidemic. We apply the model tothe COVID-19 case and demonstrate the predictive features of our model.
Keywords: COVID-19; epidemic spreading; scale-free networks; statistical mechanics; hidden variables
INTRODUCTION
According to the World Health Organization (WHO),the COVID-19 epidemic started in the Chinese city ofWuhan on December 31, 2019. The speed of the epidemicspread is very high, and on March 11, 2020, the WHO de-clared the COVID-19 outbreak a pandemic. Now, morethan 200 countries in the world are affected by the coro-navirus epidemic. COVID-19 presents an immense chal-lenge for the scientific community. The advances in un-derstanding of the disease’s properties and the mecha-nisms of its spreading will allow scientists to better pre-dict the dynamics of the epidemic and the characteristictime of recovery. They will also help to make the rightdecisions for overcoming the pandemic [1–3].Network science has contributed to diverse fields inboth the natural and human sciences. We hope that,due to its intrinsic interdisciplinary nature, the networkapproach will be useful for understanding the dynamicsof the COVID-19 pandemic. One of the reasons for em-ploying network science in the study of this phenomenonis the analogy between the spread of information in socialnetworks and the spread of disease by contact betweenindividuals. For instance, the spread of news, rumors,or gossip through a population has features in commonwith disease spread. The ideas and models for dissemi-nation of information in networks can help us to betterunderstand the propagation of disease [4–11].A critical issue in mathematical modeling of the epi-demic is a choice of the adequate mathematical toolsand a suitable network model, reflecting the features ofreal networks. The fascinating discovery of contempo-rary network science is the universality of network topol-ogy. Many real networks, such as social networks, air-line networks, the World Wide Web, computer networks,the Internet, urban networks, and others, exhibit scale-invariance, so they can be treated as scale-free networks [12]. Their scale-free nature, and other features, makethem an excellent candidate for the mathematical mod-eling of the epidemic spreading [11–16].The idea that a statistical approach is adequate to study complex networks is a natural one since networksare large complex systems, and a deterministic approachcannot describe their collective behavior. Nowadays, themethods of statistical mechanics have become powerfultools for the study and explanation of real-world networkproperties [8, 11, 14, 16–21].In this paper we study the connection between net-work structure and the epidemic spreading using statisti-cal physics methods, and propose a mathematical modelbased on scale-free networks which allows us to extrapo-late the dynamics of an epidemic. We obtain the analyt-ical solutions that describe both unstable and stable dy-namics, and demonstrate how these regimes interchangeduring the process of evolution of the epidemic. We ap-ply the model to the COVID-19 case and demonstratethe predictive features of our approach. To conclude,we discuss the existing issues of our approach and possi-ble future developments. In the Supplemental Material(SM), we present the results of the application of ourapproach for COVID-19.
STATISTICAL DESCRIPTION OF NETWORKS
A network is a set of N nodes (or vertices) connectedby L links (or edges). One can describe the network byan adjacency matrix, a ij , where each existing (or non-existing) link between pairs of nodes ( ij ) is indicated by1 or 0 in the i, j entry. Individual nodes possess localproperties such as node degree, k i = (cid:80) j a ij , and clus-tering coefficient [16, 22–24]. The whole network can bedescribed quantitatively by its degree distribution, P k ,and its connectivity. The latter is characterized by theconnection probability p ij , i.e., the probability that a pairof nodes ( ij ) is connected.The most general statistical description of an undi-rected network in equilibrium, with a fixed number ofvertices, N , and a varying number of links, L , is givenby the grand canonical ensemble. The probability of ob-taining a graph, A , with the energy, E A , and the number a r X i v : . [ phy s i c s . s o c - ph ] J u l of links, L A = (cid:80) ij a ij , can be written as [19–21, 25, 26], P A = 1 Z exp (cid:0) β ( µL A − E A ) (cid:1) , (1)where β = 1 /T , with T being the network temperature,and µ is the chemical potential. The partition functionreads Z = (cid:88) A exp (cid:0) β ( µL A − E A ) (cid:1) . (2)The temperature is a parameter that controls clusteringin the network, and the chemical potential controls thelink density and the connection probability.Let us assign to each edge (cid:104) i, j (cid:105) the link energy, ε ij .Then, the energy of the graph can be written as E A = (cid:80) i
1. Let us assign to each node a hidden variable (cid:15) i as follows: k i = e − β c (cid:15) i , where β c is a constant withdimension of inverse temperature. As above, the linkenergy of the edge (cid:104) ij (cid:105) is given by ε ij = (cid:15) i + (cid:15) j . Thequantities (cid:15) i are distributed according to ρ ( (cid:15) i ) ∼ β c ( γ − e β c ( γ − (cid:15) i , where 0 ≤ (cid:15) i ≤ (cid:15) and (cid:15) = T c ln k [25].Due to the scale invariance of the network’s topologicalproperties, the parameter β c is a dummy parameter andcan be chosen arbitrarily.In the continuous limit, we have ρ ( (cid:15) i ) → ρ ( (cid:15) ), where ρ ( (cid:15) ) = Ce αβ(cid:15) and α = β c ( γ − /β . The constant C is defined by the normalization condition (cid:15) (cid:82) ρ ( (cid:15) ) d(cid:15) = 1.This yields ρ ( (cid:15) ) = αβe αβ ( (cid:15) − (cid:15) / αβ(cid:15) / . (6) CONFIGURATION MODEL OF THE EPIDEMICSPREADING
In our model, each node in the network is identifiedwith an individual. The probability of an individual be-ing infected by an infected individual depends on theconnection degree, susceptibility of the population, andthe connection probability. Since the transmission of theepidemic depends on the edge between two nodes beingoccupied or not, the connectivity is essential for the epi-demic spread. We assume that the disease spreading doesnot take the network out of the thermodynamic equilib-rium.The assumption that the disease may propagate onlyalong the links in the network is usual for the modelsdealing with the spread of epidemics on networks (see,i. e. [11]). We assume that the probability of contactbetween pairs of individuals leading to disease can vary,just as in Ref. [5]. Thus, some pairs can have a higherprobability of disease transmission than others. However,the “connection” (or existing link) does not guarantee thedisease-causing contact. Therefore, we distinguish infec-tive individuals, who may infect others, and susceptibleindividuals, who can be infected. Note that there are atiny number of infectious individuals at the beginning ofa disease outbreak. Since the transmission of infectionis a stochastic process, depending on the pattern of con-tacts between individuals, a description should take thisarrangement into account [27].Consider a pair of nodes, ( i, j ), where one node, i , isinfective, and the other one, j , is susceptible to infection.Suppose that the transmission rate of disease betweennodes is r ij . Then, the average number of infected nodesis given by N c = (cid:80) (cid:104) ij (cid:105) p ij r ij . In the continuous limitthis yields N c = (cid:82)(cid:82) p ( (cid:15) (cid:48) , (cid:15) (cid:48)(cid:48) ) r ( (cid:15) (cid:48) , (cid:15) (cid:48)(cid:48) ) ρ ( (cid:15) (cid:48) ) ρ ( (cid:15) (cid:48)(cid:48) ) d(cid:15) (cid:48) d(cid:15) (cid:48)(cid:48) .We divide the population into classes with a fixed linkenergy of pairs, ε = (cid:15) i + (cid:15) j , and consider only the pairsof individuals in which one is infective, and the other oneis susceptible. We assume that the transmission rate ofdisease depends only on the link energy, and present thisrate in the simplest form: r ij = Cδ ( (cid:15) i + (cid:15) j − ε ), where C is a constant. Now, the average number of infectiousnodes, N c ( ε ), with a given energy, ε , is given by N c ( ε ) = C (cid:90) (cid:90) (cid:15) ρ ( (cid:15) (cid:48) ) ρ ( (cid:15) (cid:48)(cid:48) ) δ ( (cid:15) (cid:48) + (cid:15) (cid:48)(cid:48) − ε ) d(cid:15) (cid:48) d(cid:15) (cid:48)(cid:48) e β ( (cid:15) (cid:48) + (cid:15) (cid:48)(cid:48) − µ ) + 1 . (7)Performing the integration, we obtain N c ( ε ) = Ae αβ ( ε − µ ) e β ( ε − µ ) + 1 . (8)Here the constant A accumulates all constants that theintegral (7) includes.The number of infected nodes, ∆ N t , inside the inter-val β ∆ ε can be written as ∆ N t = N c β ∆ ε . (The inversetemperature, β , is introduced to keep the right dimen-sionality.) Performing the integral, N t = β (cid:82) N c ( ε ) dε ,we obtain N t = Ae αβ ( ε − µ ) Φ (cid:0) − e β ( ε − µ ) , , α (cid:1) + B, (9)where B is a constant of integration, and Φ( z, s, a ) is theLerch Transcendent [28].Now we consider the epidemic spreading. We assumethat the link energy is an increasing function of time.Then from Eq. (9) it follows dN t dt = N i , (10)where N i = κN c is the number of infectious individuals( incidence ) per unit of time (usually daily), and κ ( t ) = β ˙ ε ( t ) is the transmission rate of the epidemic disease.The rate, κ , determines the expected number of peoplethat an infected person infects per time, as the SIR modeldefines it [27, 29, 30].To understand the impact of the parameter α , con-sider the asymptotic solution of Eq. (10). In the limit of ε ( t ) (cid:29) µ , we have N i ∼ e ( α − β ( ε ( t ) − µ ) . Substituting N i in Eq. (10), we obtain N t ∼ − e ( α − βε ( t ) α < βε ( t ) , α = 1 e ( α − βε ( t ) , α > α ≥
1, the total number of infectednodes N t → ∞ when t → ∞ . For any choice of α , theobvious condition N t ≤ N should be imposed.Thus, α is a crucial parameter for the epidemic spread-ing, a critical spreading parameter . It determines thethreshold of the outbreak. When α ≥
1, the system haslost stability, and the process of the epidemic spreadingbecomes uncontrollable.
Dynamical system beyond the network model
Here we present our configuration network model asthe system of ordinary differential equations, and com-pare it with the well-known SIR model [27, 31]. From Eq. (10) it follows that N c satisfies the followingdifferential equation: dN c dt = aN c (cid:16) − N c K (cid:17) , (12)where a = ακ , and K = αAe ( α − β ( ε − µ ) is the carryingcapacity . As one can see, Eq. (12) is just the modified lo-gistic equation, widely used for description of populationgrowth (see, i.e., [27, 31]). When κ = const and α = 1,we have K = const, and (12) becomes the standard lo-gistic equation.Adding to Eqs. (10) and (12) the equation for thecarrying capacity, we obtain the decoupled system of or-dinary differential equations that describe our configura-tion model: dN t dτ = N c , (13) dKdτ =( α − K, (14) dN c dτ = αN c (cid:16) − N c K (cid:17) , (15)where τ = (cid:82) κ ( t ) dt is a dimensionless time. The systemof Eqs. (13)-(15) is completely equivalent to Eqs. (9) and(10). The parameter α now is interpreted as the intrinsicgrowth rate. In Eqs. (13) – (15), N t is the total numberof infected individuals, N c is the number of new infectiveindividuals, and K denotes the carrying capacity of theinfective population. The number of infected individu-als per unit of time can be obtained from the relation N i = κN c . The solutions of Eqs. (13) – (15) should beconsidered for N t ≤ N , where N is the total population.The most commonly used models for epidemic trans-mission are the Susceptible-Infected-Susceptible (SIS)and the Susceptible-Infected-Removed (SIR). While inthe SIS model recovered individuals could again be in-fected, the SIR model assumes that those recovered fromthe disease have immunity, and therefore each individualcan only be infected once [27, 29, 30].The SIR model divides the population into the fol-lowing three classes. Susceptible ( S ): individuals in thesusceptible state may be infected when they encounteran infected individual. Infectious ( I ): the individualswith the disease, meaning they have the disease and canspread disease by infecting others. Removed ( R ): thosewho have recovered from the disease (or deceased) or haveimmunity. The sum of the three numbers is a constant, S + I + R = N , where N is the population. The SIRsystem is described by the following system of ordinarydifferential equations [27, 31]: dSdt = − νISN , (16) dIdt = νISN − λI, (17) dRdt = λI. (18)Here νIS/N is the number of new individuals infectedper unit of time, with ν being the infection rate.Comparison of Eqs. (13) – (15) with Eqs. (16) – (18)of the SIR model shows that N t = I + R , N c shouldbe identified with IS/N and κ with ν . After this iden-tification, one can show that Eq. (13) can be writtenas a combination of Eqs. (16) and (17). Thus, in ourapproach only one equation can be matched to the SIRmodel. Other equations, Eqs. (14) and (15), are new.The SIR model assumes that the population is thor-oughly mixed, the individuals have the same number ofcontacts per day, and that transmission of the diseaseby contacts among individuals takes place with the sameprobability. None of these assumptions is realistic [5].Our model takes into account that the probability ofan individual being infected may depend on the featuresof each pair of individuals, where one is infective andthe other is susceptible, for instance, with a weak/strongimmune system, and so forth. Besides, we omit the as-sumption that the population is mixed. APPLICATION TO THE COVID-19 PANDEMIC
To relate the time-dependent solution (9) to the dy-namics of the epidemic spreading, and extrapolate its fu-ture development, one should adjust the parameters A , B , α , β and µ with the data available for the known timetrial domain. Besides, we have to specify the dependenceof ε ( t ) on time. To proceed, we take a simple but reason-able choice of a linear function, ε ( t ) = κt . This choiceprovides a reasonable agreement of our results with mostof the actual data. However, for the description of theepidemic spreading with a second epidemic wave, as hasoccurred for example with Iran, one can make a mre so-phisticated choice of ε ( t ). We will show how to deal withthis case below.The COVID-19 data for this analysis are obtained froma publicly available database [32, 33]. We made a com-parison of our model with coronavirus data extracted for213 countries from the database [33]. The detailed ac-count of the comparison of our findings with the empiricaldata is presented in the Supplementary Material (SM).For convenience, we define new variables: βε = κt and b = βµ . Then, Eq.(9) can be recast as N t = Ae α ( κt − b ) Φ( − e κt − b , , α ) + B. (19)For each country, the constants A and B are obtainedfrom the initial and final conditions. We understand thefinal condition as the total number of infected individualsat a cut-off date contained in the Coronavirus Pandemic(COVID-19) database [33].We divide all countries into three groups. Group I:The countries where the epidemic is over or almost over.Group II: The countries where the epidemic is in progress, but available empirical data allow us to predict the epi-demic end. Group III: The countries with a high-valueof the critical parameter, α ≥
1, or with high-level fluc-tuations in the daily number of infected individuals. Theinsets show the number of daily new cases.In Fig. 1, we compare the theoretical solutions of ourmodel (solid blue curves) with the empirical data (reddiamonds) for each of the groups. The insets show thenumber of daily new cases. Shadow cyan bands show the95% confidence interval. In Fig. 2, we demonstrate theforecasting features of our model for the UK. When thetime trial domain increases (curves from 1 through 5),the unstable regime of our solution ( α >
1) changes toa stable regime ( α < α >
1) tothe first one ( α < . Country Transmission rate Critical spreadingof epidemic κ parameter α Bangladesh 0.072 1.026Belgium 0.214 0.787Brazil 0.075 1.108Canada 0.135 0.787Chile 0.226 0.229Colombia 0.1 1.39France 0.371 0.9Germany 0.335 0.901India 0.079 1.366Iran 0.05 0.646Italy 0.207 0.8Mexico 0.072 1.15Pakistan 0.202 0.229Peru 0.094 0.746Russia 0.152 0.946South Africa 0.04 2.094Spain 0.25 0.8Turkey 0.386 0.946UK 0.157 0.827USA 0.883 1.004World 0.285 1.04
In Table 1, we present the essential parameters for thecountries from the top 20: the transmission rate of epi-demic κ and the critical spreading parameter α with acut-off date of 4 July 2020. Second wave of COVID-19
As mentioned above, the linear dependence of ε ontime can not describe the epidemic spreading with mul-FIG. 1: Theoretical outcomes of the configuration model: solid blue curves. Shadow cyan bands show the 95%confidence interval. Actual data: red diamonds. Insets show the number of daily new cases.FIG. 2: Example of the country migration from onegroup to another during the epidemic spreading. Actualdata: red diamonds.tiple waves. In order to describe this process, we proposethe following modification: ε = κt + (cid:80) n c n e iω n t , where c n and ω n are additional fitting parameters. In Fig. 3,we compare the theoretical outcomes of our configurationmodel with the empirical data for Iran, the United Statesand Saudi Arabia, with the choice of ε ( t ) as ε = κt + (cid:88) n = − c n e iω n t . (20)One can see that this choice of modulation of ε ( t ) ap-proximates the second wave effect well. CONCLUSION AND DISCUSSIONS
The accurate forecast procedures and predictive pro-tocols for the COVID-19 pandemic spread require theknowledge of many factors, such as human behavior, reli-able tests, and government regulations. In this situation,the importance of the mathematical models is in their“predictive monitoring” [34]. This implies that the pre- dictions should be based on the dynamics of the epidemicspreading and changes in the real-world scenarios.The main problem of forecasting is the absence of reli-able and accurate data on daily and total cases reported.It is important to stress that analytical solutions could bemisleading for long-term forecasting because the param-eters are going to change in the long run. Therefore theextrapolation and prediction should be taken with muchcare and, in most cases, only for a short period (see SMfor details).The mathematical models that describe the epidemicspreading can be divided into two main groups: thestochastic and the deterministic models, written for time-dependent average variables. Our model is an interme-diate one — it is based on a set of differential equationsobtained from the statistical physics of networks. Weused the following assumptions in the derivation of ourmodel: 1) the probability of an individual being infectedthrough the disease-causing contact depends on the in-dividual’s features; 2) the epidemic spreads across thecomplex network in thermodynamical equilibrium.We have obtained analytical solutions which interpo-late the daily and total numbers of infected individuals,and forecast the future epidemic development. These so-lutions describe both the stable epidemic developmentand the dynamics close to instability. This method, asapplied to COVID-19, gives a reasonable picture for thepandemic development. Our approach can help to betterunderstand the network’s role in the epidemic spread-ing. Further progress in this direction will require to useof more advanced approaches, including those based onnon-equilibrium statistical physics.The work by G.P.B. was done at Los Alamos Na-tional Laboratory managed by Triad National Security,LLC, for the National Nuclear Security Administrationof the U.S. Department of Energy under Contract No.89233218CNA000001.FIG. 3: Second wave COVID-19.Theoretical outcomes of the configuration model: solid blue curves. Shadow cyanbands show the 95% confidence interval. Actual data: red diamonds. Insets show the number of daily new cases. ∗ [email protected] † [email protected] ‡ [email protected][1] David Adam, “Special report: The simulations drivingthe world’s response to COVID-19.” Nature , 316–318 (2020).[2] B. Ivorra, M.R. Ferr´andez, M. Vela-P´erez, and A.M.Ramos, “Mathematical modeling of the spread of thecoronavirus disease 2019 (COVID-19) taking into ac-count the undetected infections. The case of China,”Communications in Nonlinear Science and NumericalSimulation , 105303 (2020).[3] Sarah Cobey, “Modeling infectious disease dynamics,”Science , 713–714 (2020).[4] B´ela Bollob´as, Random Graphs , 2nd ed., Cambridge stud-ies in advanced mathematics 73 (Cambridge UniversityPress, 2001).[5] M. E. J. Newman, “Spread of epidemic disease on net-works,” Phys. Rev. E , 016128 (2002).[6] Sam Moore and Tim Rogers, “Predicting the Speed ofEpidemics Spreading in Networks,” Phys. Rev. Lett. ,068301 (2020).[7] Alun L. Lloyd and Robert M. May, “How Viruses Spreadamong Computers and People,” Science , 1316–1317(2001).[8] Albert-Laszlo Barabasi, Network Science (CambridgeUniversity Press, 2016).[9] S. N. Dorogovtsev and J. F. F. Mendes ,
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Mathematical Models in Epidemiology
SUPPLEMENTAL MATERIAL
In this Supplemental Material we explain the calcu-lations, approximations, and intermediate steps of themain text. The total number of infected individuals inour model is described by the the Lerch Transcendent: N t = Ae αβ ( ε ( t ) − µ ) Φ (cid:0) − e β ( ε ( t ) − µ ) , , α (cid:1) + B, (21)and the daily number of infected individuals, N i , is ob-tained by taking the derivative of N t , N i = dN t dt . (22)When the epidemic does not generate multiple waves, alinear dependence of ε on time yields good results. Itis convenient to define new variables: βε = κt and b = βµ . (Note that in the SM we consider only the lineardependence of ε on t .) Then, Eq.(21) can be recast as N t = Ae α ( κt − b ) Φ( − e κt − b , , α ) + B. (23)For each country, the constants A and B are obtainedfrom the initial and final conditions. We understand thefinal condition as the total number of infected individ-uals at a cut-off date within the Coronavirus Pandemic(COVID-19) database [33].A numerical best-fit algorithm has been used to fit themodel to the data by adjusting parameters α , κ , and b above for each country. In principle, both the total case data and the daily case data should yield the sameparameter values, but the former is usually less noisy.The data should begin at the first non-zero case count(i.e., N t > COVID-19 forecast
The accurate prediction of the COVID-19 pandemicspread requires the knowledge of many factors, such ashuman behavior, reliable tests, and government regula-tions. The forecast’s main problem is the absence of reli-able and accurate data on daily and total cases reported.Therefore the prediction should be taken with much careand, in most cases, only for a short period.Our model is deterministic by nature, and therefore theprediction also is deterministic. To introduce the fore-cast’s uncertainty in case numbers, we use the residualstandard deviation ( S res ), available in Maple and
Mathe-matica . We define the 95% confidence interval in the trialtime-domain as ∆ N c = 1 . S res . To obtain the plumediagram, we use the probe function N t ± ∆ N c and fit itto the empirical data in the trial interval.In Figs. 4, 5, we demonstrate the forecasting featuresof our model for China and the United Kingdom. Shadowbands show the 95% confidence interval. Insets show thenumber of daily new cases. In our estimates, we assumethat current interventions will continue indefinitely.In the following pages, we show the algorithmicallydetermined curves and parameters for each country.Though we analyzed the complete dataset for 213 coun-tries, plus the worldwide aggregate, for practical pur-poses, we include here only countries with 50000 totalcases or more, and also exclude some with very volatileor irregular data. As mentioned in the main text, somecountries have begun a second wave of contagions, andthey should be considered in the same way as Iran, SaudiArabia, and the United States.Countries appear in alphabetical order by coun-try code (i.e., GBR for Great Britain, notnot