Comparing AB Model andEmerging Network forEpidemic SpreadingI. Iacopini, L. Lenzi students at the University of Turin

IIn this paper we design and implementan agent-based epidemiological simulationsystem, using the NetLogo programming

language. The epidemic model is an ex-tension of the standard compartmental SIRmodel (widely studied in literature and ana-lytically solvable). The movement of agentsis based on a random walk modified by localforces that represent geographical travel re-strictions, and furthermore by the epidemicdiffusion itself. We analyze the global emerg-ing system dynamics as a result of the com-bination of these induced mechanisms. Wealso generate the aggregated contact networkcompared with an equivalent random net-work created using the Erdos Renyi method.

1 Introduction

Nowadays mathematical models represent a compul-sory tools for the studying of spreading phenomenain a wild variety of different fields dealing with, forexample, rumors and epidemics spreading, opiniondynamics or social interactions in general. In factepidemics modeling is a very active field of researchwhich crosses different areas, involving computerscientists, epidemiologists and social scientists, in-vestigating the diffusion dynamics of viruses, knowl-edge, culture, innovation and language. Despite theirapparent diversity, all of these problems can be in-vestigated starting from the well known complexityparadigm; complexity in point of fact should be con-ceived as a condition in which local interactions and

agent behavior combine to create new macro out-comes, that could not be predicted from the knowl-edge of individual behaviors and features of the localinteractions alone.

1.1 ABMs and Networks

In social epidemiology two approaches are partic-ularly useful: network analysis and agent-basedmodeling.

• Networks are a powerful framework for the com-bined representation of the social structure re-spect to local and global interactions. Locallinks cover the description of the human geo-graphical interactions of everyday life and anapproximated contact network for every singleperson. In a similar way, global links coverthe description of diffusion processes of popula-tion through the global infrastructural network(railways, highways and air transport). For acomplete and faithful representation these lastlinks should be developed on different scales:international, intra-nation and intra-city.

• Agent-Based models strongly use stochastic sim-ulation of individuals in a simulated space (oftenassociated with a dual social space). The powerof this modeling approach has been to break sim-plistic assumptions required for mathematicaltractability, e.g. homogeneity, ignoring interac-tion (Axelrod, 1997). One of the most importantAB model’s feature, often not included but al-most mandatory in the field of complex systems,

Page 1 of 9

is the agent’s autonomy. This approach allowsus to understand how the micro-level behaviorsand movements of these independent agents leadto emerging macro-level disease and prevalencedistributions.

1.2 Epidemic Models: SIR and SIS

The simplest class of epidemic models is the compart-mental model, that assumes that the population canbe divided into different classes depending on thestage of the studied disease. The minimal standardframework involves two or three different compart-ments:

1. Susceptible (denoted by S, those who can con-tract the infection).

2. Infectious (I , those who have contracted theinfection and are contagious).

3. Recovered (R, those who have recovered fromthe disease).

The dynamics of the individuals between the dif-ferent compartments depends on the specific char-acteristics of the disease. The transition from acompartment to the other is triggered by a reactionrate. Usually there are two possible types of elemen-tary processes ruling the disease dynamics. At thebeginning there is a little number of seed agents atthe state I, while all the other agents begin in theS state. The transition from the S to the I state isgoverned by the probability β (triggered by the ef-fective proximity of two or more individuals). Everyinfected agent has a probability µ of spontaneouslymoving to state R (for the SIR model) or to returnto the state S (for the SIS model).In network analysis the outcomes of these two com-partmental epidemic models (e.g. the epidemicthreshold and the prevalence curves) are stronglyinfluenced by the specific topology of the considerednetwork.

2 Developing an extendedcompartmental model

We focus our attention on a more realistic, but stillbasic, compartmental model (Figure 1).Let introduce four classes:

1. Susceptible (denoted by S, those who can con-tract the infection).

Figure 1: Diagram of the compartmental model.

2. Latent (L, those who have contracted the infec-tion and are contagious, but don’t exhibit anysymptoms).

3. Infectious (I , those who have contracted theinfection. They are contagious and do exhibitsymptoms).

4. Immune (U, those who have recovered from thedisease and are no longer susceptible).

In our model we start with a population of s =N − u − l agents in state S, u agents in state Uand a little number l of agents in state L (l << N).When a latent agent gets in contact (we will defineafter what it means) with an S agent he can infecthim with probability β. The same thing happenswhen an agent S gets in contact with an agent I.The transition between the latency phase and the Iis spontaneously governed by the decay probability1τ (where τ is the mean life time of L state). Aninfected agent decays with a transition rate µ andfurthermore the chance to move into an S state isα1, while the chance to move into an U state is α2,such that α1 + α2 = 1. Finally, the total probabilityfor the transition to an S or U state is µα1 and µα2,respectively.

The transition rules between different classes are:

S + Lβ−→ 2L (1)

L1τ−→ I (2)

S + Iβ−→ L+ I (3)

Iµα1−−→ S (4)

Iµα2−−→ U (5)

Page 2 of 9

In the next section we will implement our modelusing the NetLogo platform with an AB approachgrounded on a random walk subjected to particu-lar restrictions and enforcements, depending on theagent’s class and on the geometry of the problem.

2.1 A-B Modeling

We divide the NetLogo grid of patches into M con-tinents, using rigid unassailable barriers containingsome crossing gates with variable width. We startthe simulation populating the boxes with a numberN of agents (instantiated with a random angle), sodivided:

• s susceptible agents randomly distributed.

• u immune agents randomly distributed.

• l latent agents randomly distributed in a singlebox.

At each tick of the simulation the agents obey thefollowing general rules:

Not infected agents randomly sort an angle anda step-length (drawn from an fixed range) andmove.

Infected agents act in the same way of the others,but with a lowered mobility caused by theirsickness state. The step-length is so drawn fromthe previous range, properly rescaled.

Contagion: both L and I agents infect each neigh-boring S agent that stands in his contagion re-gion (a circle of fixed radius) with probabilityβ.

Boundary Condition: when an agent get closeto a barrier there’s an inelastic scatter that pro-hibits the crossing of the barrier.

In order to restrict the randomness of the agent’sdynamics and stress the model towards a little bitmore realistic behavior we also impose the prescrip-tions explained in the following paragraphs.

2.1.1 Infectious Repulsion

When a not infected agent finds an infected agentin his proximity region (a circle with a fixed radiusR patches representing the human vision) he turnsaround of 180 plus a little random extra and movesaway with raised mobility.

2.1.2 Flocking Dynamic

The two macro-classes of not infected agents andinfected agents obey to the same flocking dynamic[9], but only within its own class. Normally this kindof modeling is referred to swarm dynamics, but inthis case it provides an appropriate description ofthe “general idea” that not infected agents locallybias each other, escaping from the infected ones. Theagents follow three rules: alignment, separation, andcohesion.

Alignment means that an agent tends to turn sothat it is moving in the same direction thatnearby agents are moving. In particular nearbymeans a circle of a defined radius r = R

2 centeredon the agent. When an agent finds more thanone agents in its circle, it select the nearest andtry to align themselves with him; we set themaximum turning angle at 15.

Separation means that an agent will turn to avoidanother agent which gets too close (with respectto a minimum-separation threshold dmin = 1patch).

Cohesion means that an agent will move towardsother nearby agents (unless another agent is tooclose).

When two agents are too close, the separation ruleoverrides the other two, which are deactivated untilthe minimum separation dmin is achieved. The threerules affect only the agents heading. Each agentalways moves forward with the step-length of its ownclass.

2.1.3 Global Restrictions

In our simulations we set a number M = 4 of con-tinents (Figure 2). For the sake of simplicity eachcontinent corresponds to a quadrant of the NetL-ogo grid. We can think of barriers built over theCartesian axis (from the end of the grid to the cen-ter) as geographical restrictions to inter-continentagent’s mobility. Given that the agent’s movementis essentially a random walk, the gate width is pro-portional to the net flux of agents between each pairof continents. In this way barriers may represent thetravel restrictions imposed by global authorities inpresence of an epidemic outcome. In order to geta meaningful behavior we need a functional formfor the gate-width w, depending on the number ofinfected agents i, so that little values of f lead to a

Page 3 of 9

(a) Closed barriers

(b) Opening barriers

Figure 2: Barriers dynamics.

notable w decrease. A satisfying function for thispurpose is the Fermi’s Function:

w = A

[1− 2

ebi + 1

](6)

where A is a constant tuned on the NetLogo grid.

3 Simulations and Results

During the simulations we focus our attentionsmostly on two particular features:

• Prevalence Curves They show at each time-step the number of agents that belong to eachspecific compartment.

• Lifetime prevalence (LTP) The fraction ofagents that have experienced (at least one time)the infection condition during the simulations.

3.1 Key Parameters

We are interested in the analysis of these two fea-tures under variations of the following infection keyparameters (previously introduced):

• β, which is the infectivity.

• τ , which controls the mean life of L state.

• µ, which controls the mean life of I state.

• α1, which controls the probability for a decayingstate I to move into a state S.

The remaining parameters of the model are alwaysfixed: s = 700, u = 10, l = 15, R = 4.

3.2 Parameters Investigation

(a) α1 = 0.9

(b) α1 = 1

Figure 3: Prevalence curves for two different α1 values.The other key parameters have values: β =0.68, τ = 7, µ = 0.10.

Each key parameter is investigated in turn, whilethe remaining parameters are fixed at the defaultvalues (determined by empirical testing). Despitethe fact that this does not catch the correlationsbetween the key parameters, it allows to understandthe statistical effect on the prevalence curves.We focus only in epidemic processes with a transientstate, which means that the system always reachesan absorbing state with only S, L or U agents, and

Page 4 of 9

(a) LTP curves as function of time

(b) Total LTP as function of τ

Figure 4: LTP curves as function of time, under vari-ation of parameter τ (a); LTP as function ofτ (b). The other key parameters have values:β = 0.68, µ = 0.10, α1 = 0.94.

no more infected. This request is always satisfied forα1 < 1 (Figure 3(a)). Otherwise the system reachesa stationary state, practically acting as a SIS model(Figure 3(b)).

3.3 Results Investigation

Empirical analyzes show the central rule of the τparameter as dynamic controller (Figure 4(a)):

i For a small τ the spreading does not involve theentire “world”, but only affects a fraction (notmore than two continents). Furthermore, theepidemic rapidly goes towards extinction (Figure5(a)).

ii For a high τ the epidemic wide spreads all overthe continents and the mean life of the disease issignificantly longer than in the previous case (i)(Figure 5(b)).

iii There is an intermediate little range of values forτ where the spreading dynamics is subjected to

(a)

(b)

Figure 5: Two different scenarios for different valuesof parameter τ . The red lines are traces of thepassage of I agents.

large stochastic fluctuations, covering the major-ity of possible outcomes between the two previousregimes (i)(ii).

This simply happen because the epidemic spread-ing is mainly driven by the latent agents that stillhave a complete mobility and are not recognized bythe S agents as a danger, but have the same infectiv-ity of the infected ones. Therefore, the L mean life (τ)is crucial: low values prevent L agents from diffusethe disease at large distance, and so the barriers areable to contain the disease in one continent; on theother side, for higher values the action of the barrierscan no longer contain the spreading. In the inter-mediate phase, all scenarios are accessible and therandomness dominates the system dynamics (Figure

Page 5 of 9

Figure 6: Overlapping the plot of different simulationsfor the total LTP as a function of τ . Theother key parameters have values: β = 0.68,µ = 0.10, α1 = 0.94.

Figure 7: Mean total LTP as function of τ . The otherkey parameters have values: β = 0.68, µ =0.10, α1 = 0.94.

6). The analysis of the total LTP (total means thatthe LTPs are evaluated at the end of simulations) asfunction of τ (Figure 4(b)) permits immediately tovisualize the total epidemic diffusion under variationof τ . In this way is easier to recognize the intermedi-ate region, where the stochastic fluctuations are wellevident.

In order to better understand the behavior of themodel in the intermediate region (iii) we repeat fiftytimes the same simulation, with fixed parameters,and take the mean value of the total LTP for each τ .The resulting plot is shown in Figure 7, where the redline comes from the fitting of the data. The figureclearly confirms that for τ ≤ 4 the total LTP remainsunder the symbolic value of 0.25, which implies thatthe epidemic spreading is isolated only to a singlecontinent. On the other hand, τ ≥ 6 implies a globalspreading all over the continents (pointed out bytotal LTP values, which is always over 0.75). In theintermediate region there is a strictly growing trendthat marks the transition between the two completelydifferent scenarios.

Figure 8: LTP curves as function of time, under varia-tions of parameter β(%). The other key param-eters have values: τ = 7, µ = 0.10, α1 = 0.94.

Summary of the empirical results:

• τ ≤ 4 leads to regime (i).

• τ ≥ 6 leads to regime (ii).

• 4 < τ < 6 leads to regime (iii).

A decrease in the β value (infectiousness parame-ter) corresponds to a decrease in the epidemic spread-ing (Figure 8). This is due to the lower capability forthe infected and latent agents to vehicular the diseaseduring the contacts with their neighbors. Qualita-tively speaking, this implies that as the β decreases,it takes bigger and bigger values of τ in order tomake the infection able to reach every continents (i.e.regime (ii)).

The effect of the variation of the µ parameteris simply limited to the temporal duration of thesimulation (Figure 9). This happens because smallµ values only imply an increase in the mean life ofthe I agents, but, as we previously explained, thespreading dynamics is dominated by the behaviorof the L agents, and so the only consequence is adecreasing in the transition rate between I and S/Uagents states and finally an increase in the globaldynamics time.

Unlike the other parameters, the effects of the α1

variations are no longer unambiguously interpretable(Figure 10). A decreasing in α1 (remembering thatα1 + α2 = 1) leads to a more rapid growth in thenumber of immune agents; theoretically this shouldimplies a depletion of the reservoir of susceptible andso a more rapid epidemic extinction. We observean unexpected rapid and homogeneous diffusion forthese low α1 values and a “zigzagged” growth forhigher values. This last effect is due to the well

Page 6 of 9

Figure 9: LTP curves as a function of time, under varia-tions of parameter µ(%). The other key param-eters have values: β = 0.68, τ = 7, α1 = 0.94.

Figure 10: LTP curves as a function of time, undervariations of parameter α1(%). The otherkey parameters have values: β = 0.68, τ = 7,µ = 0.10.

known progressive and modular epidemic diffusionthrough continents. The difference between thesetwo observed behaviors needs further investigations.

4 Emerging Network

In this final section we are going to create the emerg-ing contact network of the agents. Considering that,the basic agent dynamics without disease appearanceis practically a random walk, the naturally generatedcontact network should belong to the random graphsclass (i.e. Erdos Renyi). The expected functionalform for the degree distribution is a Poissonian:

P (k) =e−µµk

k!(7)

We are interested in understanding how the additionof the epidemic spreading affects this basic dynamic,the network evolution, and the degree distributionat last.

4.1 Network Creation

In order to create the network we consider two dif-ferent type of contacts:

Proximity contact: when the distance betweentwo agents is less than radius r we define a prox-imity contact. The first time this happens anundirected edge is created between the involvedagents, with weight set up to 1. For every ulte-rior contact the weight is increased by one.

Repulsive contact: when two agents undergo to ainfectious repulsion we define a repulsive contact.For each contact of this type we reduce by twothe weight of the shared edge; if the updatedvalues is less than one the link is deleted.

The former contact represents a sort of social re-lationship between individuals, while the latter rep-resents the weakening of the social interaction dueto the appearance of the disease symptoms (a roughsketch of the “survival instinct”).We store all the weights into an adjacency matrix A,where each element aij represents the edge weightbetween agents i and j:

A =

a1,1 a1,2 · · · a1,Na2,1 a2,2 · · · a2,N

......

. . ....

aN,1 aN,2 · · · aN,N

A is clearly symmetric and zero-diagonal. The

degree of an i-agent ki is given by:

ki =N∑j=1

aij =N∑i=1

aji (8)

4.2 Network Analysis

We produce networks over simulations tuned onregime (i) and (ii). From A it is possible to computethe degree distribution. In the first case (τ = 3, (i))there are some noticeable differences between theobtained degree distribution and the Poissonian one(Figure 11(b)). In particular there is a slight densifi-cation on the left side of the mean value, while theright-skewed tail represents the presence of an exigu-ous number of hubs. Figure 11(a) shows the finalscreen-shot of the simulation, with a lower boundover the weights cw for a better visualization. We cannotice a barely appreciable condensation of links onthe bottom left corner (continent number 4), belong-ing to the only continent that has not been infected.

Page 7 of 9

(a) Screenshot

(b) Normed histogram

Figure 11: Final screen-shot (cw = 7) and histogramof the degree distribution for τ = 3. Theother key parameters have values: β = 0.68,µ = 0.10, α1 = 0.94.

In the second simulation (τ = 7, (ii)) this last be-havior is much more evident. Figure 12(a) shows anintermediate screen-shot of epidemic diffusion, wherethe bottom left corner (4th) has an high density ofheavy weighted edges. In fact, we see from the en-closure plot 12(b) that the infection has not reachedthe fourth continent (the blue curve is at zero level).These heavy weighted edges are also represented bya long right tail observing the associated degree dis-tribution in Figure 12(c).The final degree distribution in this (ii) regime (Fig-ure 13(b)) has the same features already describedfor the regime (i).

(a) Screenshot

(b) Puntual Prevalence Curves

(c) Normed histogram

Figure 12: Intermediate screen-shot (cw = 10) andrelated punctual prevalence curves for eachcontinent. The key parameters have values:τ = 7, β = 0.68, µ = 0.10, α1 = 0.94.

Page 8 of 9

(a) Screenshot

(b) Normed histogram

Figure 13: Final screen-shot (cw = 10) and histogramof the degree distribution for τ = 7. Theother key parameters have values: β = 0.68,µ = 0.10, α1 = 0.94.

5 Conclusions

We developed an AB model for epidemic diffusionover a grid. The random work that rules the move-ment of the agents is altered by the spreading ofthe disease that introduces new local paired interac-tions between agents (flocking dynamics, infectionrepulsion). Furthermore, the inter-continents mo-bility is modified by the epidemic prevalence itself,according to an exponential law. The alteration offour parameters leads the system to three differentglobal scenarios. Finally, we build the aggregatecontact network and from the combined analysis ofthe degree distribution and the punctual prevalencecurve we show the deviation of the emerging net-

work from a standard Erdos Renyi undirected graph.In conclusion, the prevention mechanism involvingmobile barriers turns out to be efficient only for aspecific set of disease parameters; in the other casesthis constraints are not able to confine the spreading.Nevertheless, it is worth remarking that, in a realscenario, the closing of barriers implies a significantinterruption of the inter-continent mobility, whichrepresents a severe measure unlikely to be adopted.

References

[1] Shah Jamal Alam and Armando Geller: Networksin agent-based social simulation. In Agent-basedmodels of geographical systems, pages 199–216.Springer, 2012.

[2] Reka Albert and Albert Laszlo Barabasi: Statis-tical mechanics of complex networks. Reviews ofmodern physics, 74(1):47, 2002.

[3] Alain Barrat, Marc Barthelemy, and Alessan-dro Vespignani: Dynamical processes on complexnetworks, volume 1. 2008.

[4] Vittoria Colizza, Alain Barrat, Marc Barthelemy,and Alessandro Vespignani: The role of the air-line transportation network in the prediction andpredictability of global epidemics. Proceedings ofthe National Academy of Sciences of the UnitedStates of America, 103(7):2015–2020, 2006.

[5] Robert De Caux, Christopher Smith, DominicKniveton, Richard Black, and Andrew Philippi-des: Dynamic, small-world social network genera-tion through local agent interactions. Complexity,2014.

[6] Jill Bigley Dunham: An agent-based spatially ex-plicit epidemiological model in mason. Journalof Artificial Societies and Social Simulation, 9(1),2005.

[7] Magda Fontana and Pietro Terna: From agent-based models to network analysis (and return):the policy-making perspective.

[8] Mark EJ Newman: The structure and function ofcomplex networks. SIAM review, 45(2):167–256,2003.

[9] U. Wilensky: Netlogo flocking model. Center forConnected Learning and Computer-Based Model-ing, Northwestern University, Evanston, IL, 1998.

Page 9 of 9