Analyzing the impact of modeling choices and assumptions in compartmental epidemiological models

Transcription

1 Simulation Special Section on Meical Simulation Analyzing the impact of moeling choices an assumptions in compartmental epiemiological moels Simulation: Transactions of the Society for Moeling an Simulation International 2016, Vol. 92(5) Ó The Author(s) 2016 DOI: / sim.sagepub.com Özgür Özmen 1*, James J Nutaro 2*, Laura L Pullum 2 an Arvin Ramanathan 1 Abstract Computational isease sprea moels can be broaly classifie into ifferential equation-base moels (EBMs) an agent-base moels (ABMs). We examine these moels in the context of illuminating their hien assumptions an the impact these may have on the moel outcomes. Drawing relevant conclusions about the usability of a moel requires reliable information regaring its moeling strategy an its associate assumptions. Hence, we aim to provie clear guielines on the evelopment of these moels an elineate important moeling choices that cause the ifferences between the moel outputs. In this stuy, we present a quantitative analysis of how the choice of moel trajectories an temporal resolution (continuous versus iscrete-event moels), coupling between agents (instantaneous versus elaye interactions), an progress of patients from one stage of the isease to the next affect the overall outcomes of moeling isease sprea. Our stuy reveals that the magnitue an velocity of the simulate epiemic epens critically on the selection of moeling principles, various assumptions of isease process, an the choice of time avance. In orer to inform public health officials an improve reproucibility, these initial ecisions of moelers shoul be carefully consiere an recore when builing an ocumenting an ABM. Keywors Susceptible infecte recovere (SIR), epiemiology, agent-base, event-base, equation-base moels 1. Introuction Computational moels of isease sprea processes can be broaly classifie into two types: (i) ifferential equationbase moels (EBMs) an (ii) agent-base moels (ABMs). For influenza-like illnesses (ILI) an other infectious iseases, computational moels are critical for public health officials to unerstan the effects of isease sprea amist ense urban populations, for preicting the socio-economic effects of major epiemics, an for forecasting the effects of ifferent intervention strategies. 1 3 While both EBMs an ABMs are wiely use in practice, moelers often neglect to examine the impact of ifferent moeling choices an assumptions ecie in the early stages of moel evelopment. EBMs moel the population by segmenting it into ifferent compartments, such as susceptible (S), infecte (I), an recovere (R), with each compartment representing a stage in the progression of the isease. 2 Aitional compartments can be efine base on moel requirements. Transitions between each of the compartments capture the overall ynamics of the epiemic, showing how the population as a whole gets infecte an eventually recovers. EBMs capture aggregate behaviors over the whole population. Iniviuals an interactions between iniviuals are omitte from EBMs, an so these moels can be use with relatively little computational power an relatively small amounts of input. Therefore EBMs can capture large-scale (incluing country-wie/continent-wie panemic) isease sprea with relatively little effort. On the other han, ABMs inclue iniviual level information to capture how people interact an participate in ifferent activities (work, school, exercise, etc.). 4,5 Although the term agents in epiemiology is use to refer to infectious agents, in this stuy we use ABMs to 1 Health Data Sciences Institute, Oak Rige National Laboratory, TN, USA 2 Computational Sciences an Engineering Division, Oak Rige National Laboratory, TN, USA * SCS member Corresponing author: Laura L Pullum, P.O. Box 2008 Oak Rige, TN 37831, USA. pullumll@ornl.gov

2 460 Simulation: Transactions of the Society for Moeling an Simulation International 92(5) escribe moels that incorporate iniviual entities (or agents that can be autonomous, heterogeneous an/or aaptive) an are evelope by emphasizing the progression of the iniviual (an his/her behavior) over the course of the isease. While iniviuals within a community unergo a similar isease progression as in EBMs, the outcomes from the ABMs are largely etermine by the interaction between these iniviuals an the ata that characterize the interactions. Previous research on comparing epiemiological moels has largely focuse on the comparison of ifferent moels an their outcomes without consiering the unerlying assumptions an implementations specific to these moels. Rahmana an Sterman 6 examine the choice between EBMs an ABMs from the perspective of heterogeneity in iniviual populations an the network topology of interactions, incluing fully connecte, ranom, small-worl, scale-free, an lattice networks. The authors foun that the EBM an ABM implementation base on the ifferent network moels iffere significantly in terms of their outcomes incluing iffusion spee (the amount of time that passes until the peak loa is observe), peak loa on health services (the maximum infectious proportion of the population observe over time), an total isease buren (cumulative number of people who ie from the isease at the en of the epiemic). Further, the authors speculate that the ifferences in the outcomes coul be a consequence of the ifferences in network topologies, heterogeneity (amongst iniviuals in the population), an iscretization of stochastic interactions between agents. However, they also observe similar ifferences for the homogenous population. In their homogenous population scenario, the ifferences in the output trajectories can originate from stochasticity or moeling assumptions examine in our work. Unless the moel assumptions are explicitly ocumente, we cannot etermine how much stochasticity or moeling assumptions contribute to their conclusions. Ajelli et al. compare an ABM that is base on census ata of Italy with a structure meta-population moel name Global Epiemic an Mobility (GLEaM) 7 that was esigne to preict spatio-temporal aspects of global epiemics by incorporating International Air Transport Association (IATA) ata. 8 Despite the similarity of results regaring the isease propagation at ifferent locations, this comparison i not provie etails on unerlying moeling principles an assumptions in the two istinct moels. Jaffry et al. qualitatively compare the results from an EBM an an ABM an note that the ifferences in the results were a consequence of the sampling variability. 9 Similarly, Macal claime that isagreements between agent-base an system ynamics moels of epiemiology are cause by stochasticity. 10 Kenney et al. applie a variety of valiation techniques to one EBM an one ABM an aresse the importance of knowing the abstraction (i.e., conceptual moel) an implementation ecisions when comparing moels. 11 Cross-valiation techniques, 12 sensitivity analyses, 13 an other statistical techniques 14 have been use to analyze EBMs an ABMs. Although unerstaning how istinct moels can have similar behaviors base on their input parameter values is valuable, it is ifficult, in general, to istinguish whether the results are a consequence of the calibration process or moeling assumptions an the specific implementation use in the moels. In this stuy, we present our analyses of the effect of the moeling approach with ABMs an EBMs using 1918 flu panemic parameters. Our goal is to emonstrate our results via a realistic scenario. The 1918 flu, referre to as the Spanish flu, was an exceptionally ealy an wiesprea panemic which resulte in over 50 million eaths worlwie. 15 With nearly a thir of the worl population infecte, an an increase case fatality over other types of flu, this panemic represents a wiely stuie phenomenon within moern infectious isease epiemiology. Our analysis examines three aspects of susceptible infecte recovere (SIR) 16 moels; these are (i) the choice of an iniviual-base (ABM) or population-base (EBM) approach; (ii) the unerlying assumptions in moeling the isease process; an (iii) temporal resolution. We assess the effects of the aforementione moeling choices on the behavior of large-scale epiemiological simulations for a realistic scenario. For this purpose, we evelop a family of moels that are closely relate to the basic SIR. Each moel highlights a specific choice of EBM versus ABM an the accompanying mathematical assumptions concerning the isease process within an iniviual, interactions between iniviuals, an the temporal resolution with which the moel evolves. Using these moels, we make an explicit stuy of how the unerlying mathematical structures of these moels can result in ifferent outcomes in spite of their share conceptual unerpinnings; specifically, we examine the following. I. The choice of moel trajectories an their attenant temporal resolution. This may be continuous or iscrete-event, which both have systems evolve in time over the real numbers, or iscrete-time, which has systems evolve in time over the natural numbers. II. The nature of the couplings between agents, which may be instantaneous functions (e.g., as occurs in EBM moels an in some iscrete-time an iscrete-event moels) or a elay function (e.g., as occurs in many iscrete-time moels). III. The mathematical process of progressing through the SIR compartments, which may be governe by a linear process (i.e., as in most EBMs) or with a non-linear elay (i.e., as in most ABMs). In a majority of scenarios consiere here, the choice of ABM (i.e., iniviual-base) versus EBM (i.e.,

3 Özmen et al. 461 Table 1. Parameters an their escriptions. Parameter Parameter Description Reference Values pm Probability of eath given that a person has contacte the isease Gani et al tr Time to recover given that the infecte person will not ie Taubenberger an Morens, 15 Three ays Chowell et al., 20 Gani et al. 29 tm Time to ie given that the person with the isease will ie Taubenberger an Morens, 15 One ay Crosby 30 pt Probability of transmission to another person after contact Gani et al re Number of people encountere by an iniviual in a ay Chowell et al. 20 Four people/ay NI Number of people that are initially infectious * 1000 TP Total number of people in the population y 500,000 *Assume values yscale values population-base) moel has no impact on the aggregate (i.e., population-level) ynamics of an epiemic when the moels agree on the mathematical processes for isease sprea (II an III above) an temporal resolution (I above). This suggests that the question of ABM versus EBM is of practical interest only when parameters are at certain limits or some particular use of the moel epens on a quantity mae visible by its agent-base version (e.g., the network of interactions between iniviual agents). However, ifferences between moels in any of the above traits can cause substantive ifferences in outcomes. 2. Methos In this section, we escribe the input parameters an moeling ecisions in etail. To facilitate a meaningful comparison of moels constructe with ifferent tools an using ifferent moeling approaches, we impose the following conitions on all of the moels. (i) Initial conitions: if two or more moels are being compare, then their initial populations are ientical. (ii) Outputting: output from the ifferent moels shows the state of the population at the same instants in time. (iii) Parameterization: for iscrete-time moels, time-relate parameter values are scale in accorance with the step size. Initial parameters an their erive values are iscusse in the following section Parameters for the 1918 influenza panemic A previous stuy of the 1918 flu panemic 17 provies estimates of the isease sprea parameters, an these parameters are use in the current stuy. The parameters were kept consistent across the six EBMs an ABMs use herein an were specific to the isease an population within the Unite States uring the 1918 panemic. A summary of the ifferent parameters an their values are shown in Table 1. Note that we use only a small number of parameters in this stuy in orer to balance the complexity of the moels versus the actual observations of the 1918 panemic. In orer to make the moels computationally feasible, we scale the population of the Unite States in 1918 from 103 million to about 500,000. Note that this is an assumption that is practice in the former analysis 17,18 an the finings are scale to reflect the actual population. Previous stuies estimate the number of ecease within the Unite States to be between 550,000 an 675,000 uring the 1918 influenza outbreak. 15,19 This number can be use to estimate pm (i.e., the probability of eath given that a person has contracte the flu) base on the estimate number of infecte people which correspons to approximately 50% of the population 19 at that time. However, these estimates are likely lower than the actual numbers ue to uner-reporting of incients an long-term fatality rates. Moreover, the methos use to collect the ata can be biase reflecting impacts of emographic an geographic factors. Regarless, the estimates use in this stuy are selecte from the parameter ranges given in previous stuies. An important parameter in epiemiology is the R 0 value. This value is the average number of seconary cases cause by a primary infectious iniviual. 2 Assuming a fully susceptible population, when the basic parameter notation an values (shown in Table 1) are use, the R 0 value is calculate using the following: R 0 = ½((1 pm) 3 tr + pm 3 tm) 3 reš 3 pt ð1þ Accoring to Chowell et al., 20 the R 0 of the 1918 influenza panemic varies between 1:5 to 2:0 epening on which wave of the panemic is consiere. The 1918 influenza occurre in ifferent seasons with istinct peaks observe for each wave of the panemic. This stuy uses the value of R 0 1:8, which is within this range Equation-base isease sprea moels In the previous section, we explaine the choice of parameters use in all the moels evelope. In this section, we escribe the iniviual moels. In SIR moels, the population is ivie into three compartments: susceptible

4 462 Simulation: Transactions of the Society for Moeling an Simulation International 92(5) t Re =(1 tr ) 3 I t I = K an the recovere population at time t is therefore: Re(t)= K 2tr t2 ð3þ ð4þ Figure 1. SIR moel compartments. (S), infectious (I), an recovere (R). The ynamics of the isease sprea process are capture by the interactions between these compartments, which are moele as orinary ifferential equations (ODEs). In the following sections, we escribe variations of the EBMs that mimic the SIR ynamics Simple SIR moel. To account for the large number of eaths cause by the 1918 panemic, we moify the traitional SIR moel to inclue an aitional compartment for the ecease population (represente as D within the moel). The R population is still maintaine to represent immune iniviuals who were sick an have recovere. Together with R an D, we create an aggregate compartment, terme the remove (Re) population. Figure 1 represents the compartments use in the moels. Using those state variables an parameters, the ynamics of this SIR moel is escribe by the following set of equations: 1 pm Re =(pm + ) 3 I t tm tr t D = pm 3 t Re t R =(1 pm) 3 t Re S S = re 3 pt 3 I 3 t t I = t Re t S S + I + R ð2þ The expression for (=t)re in Equation (2) makes a significant simplification that assumes the infectious population is homogeneous. At each instant of time, the moel estimates a fraction (1=tr) of the infectious population will recover regarless of the ay of infection for those population members. To unerstan how this simplification affects the moel s evolution, suppose that Re(t = 0)= 0, I(t = 0)=0, an that everyone recovers (i.e., pm = 0). By fixing (=t)i = K, we may ignore the susceptible population S an assume that the number of infecte people increases at a constant rate K. We may rewrite the moel as: However, if each infecte iniviual recovers in tr ays then we shoul expect the recovere population at time t to equal the infecte population at time t tr. That is, we shoul expect: K 3 (t tr) if tr \ t Re(t)=I(t tr)= ð5þ 0 otherwise Hence the recovere population preicte by the SIR moel an the recovere population that woul actually be generate by the isease process are not the same. To correct this error, we evelop a slightly more elaborate moel that is escribe in the following section SIR moel with substates. The simplifying assumptions in the previously escribe SIR moel can be countere by introucing substates I 1, I 2,., I V with V = btr=dtc, an D 1, D 2,., D M with M = btm=dtc, where Dt is the resolution in time with which the progress of the isease is moele. These substates represent the number of people in their first perio of infection, in their secon perio of infection, an so on. Figure 2 represents the logic behin the ivision of the moel compartments. Assuming that TI is the total infectious population that will recover an TD is the total infectious population that will ie, the EBM moel incorporating these substates note above can be escribe by the following set of ODEs: TI = XV I i i = 1 TD = XM D i i = 1 S S = re 3 pt 3 (TI + TD) 3 t S + R + TI + TD t R = I V t D = D M 8 < t I i = t S 3 (1 pm) I i if i = 1 : (I i 1 I i ) otherwise 8 < t D i = t S 3 pm D i if i = 1 : (D i 1 D i ) otherwise ð6þ

5 Özmen et al. 463 Figure 2. SIR moel with substates. The introuction of the ifferent substates overcomes the simplification in the EBM-SIR escribe previously, but it can run into the issue of iviing the population into a large number of compartments as Dt! 0. To overcome this shortcoming, we introuce a SIR moel that has a elay in the following section SIR moel with elay. In this moel, the substates are replace by a elaye ifferential equation that is the limit of the substate moel as Dt! 0. In this moel, two iscontinuities appear, namely: (i) an event where there is a suen jump at tm of the ecease population (i.e., to account for the eaths of the initially infecte persons) an (ii) an event where there is a suen jump at tr of the recovere population. To overcome these iscontinuities we incorporate two iscrete events. For the first event that occurs at t = tm, we (i) make ecease jump from zero to infectious(t = 0) 3 pm an (ii) remove infectious(t = 0) 3 pm (ecease persons) from the infectious population. For the secon event that occurs at t = tr, we (i) cause recovere to switch from zero to infectious(0) 3 (1 pm) an (ii) remove the recovere from the infectious pool. The equations for this elaye moel with its two iscrete events are: (1 pm) 3 I(t tr) if t. tr R = t 0 otherwise pm 3 I(t tm) if t. tm D = t 0 otherwise S S = re 3 pt 3 I 3 t t I = S + I + R t S + t R + t D ð7þ The SIR moel with elay represents an aitional refinement of the moels presente above. We next escribe the construction of three ABMs reflecting the same processes of isease sprea as those explaine above. Figure 3. Agent processes in the event-base moel Agent-base isease sprea moels ABMs use etaile representations of iniviual entities (i.e., humans) to moel micro-level interactions among iniviuals. However, there are ifferent approaches to avancing time in the moel (event-base vs time-steppe) an hanling the interactions between iniviuals in ABMs. Therefore, in the following sections, three variations of agent-base implementations for the moifie SIR ynamics are introuce Event-base SIR moel. Our first ABM is a iscreteevent moel, an this moel is built by ecomposing the population of the elaye SIR moel into its iniviual agents. In this ABM, the progress of the isease is moele explicitly for each agent an transmission of the isease occurs through aily contact between iniviual agents. Each agent is escribe by the couple iscrete event system (DEVS) specification 21 shown in Figure 3. A DEVS moel is comprise of atomic an couple moels. An atomic moel is a state transition system as efine by Zeigler et al., 21 which has a set of states S, a set of inputs X, an set of outputs Y. An output function l : S! Y efines the output prouce by a moel when in a given state, an this output occurs after an interval given by the time avance function ta : S! R + [f g.

6 464 Simulation: Transactions of the Society for Moeling an Simulation International 92(5) The next state of the moel is given by its internal state transition function int : S! S. It is also possible that the atomic moel will receive input x after some elapse time e prior to its time avance expiring. In this case, the moel generates no output an its next state is given by the external state transition function ext : f(s, e) : 0 4 e 4 ta(s), s 2 Sg 3 X! S. When an input an expiration of the time avance coincie, the next state is given by conf : S 3 X! S. Couple moels escribe how the input an output of atomic moels an other couple moels are interconnecte. Because DEVS is close uner coupling, each couple moel may be reuce to an equivalent atomic moel, an this forms the basis for hierarchical moel construction. A etaile introuction to the DEVS formalism an its simulation proceure can be foun in Zeigler et al. 21 an Nutaro. 22 The DEVS moel efine here has three atomic moels: the exposure process, the isease process, an the infectious process. The exposure process initiates isease in the agent upon receipt of an infect event from some other infectious agent. Upon receipt of its first infect event, the exposure process forwars this event to the isease process an the infectious process. The exposure process then becomes inert, thereby making the agent immune to infection through new contact with other infecte agents. The isease process moels the progression of the isease through its stages, an etermines the outcome for that agent. A sick agent transmits the isease to other agents through its infectious process. The infectious process sens infect events to other agents at a rate etermine by the number of expecte encounters per ay. An infect output generate by the infectious process of agent i becomes an infect input for an agent j 6¼ i selecte at ranom from the set of agents that are alive at the time of the output. The mathematical formulation of these three processes is given below. Exposure process. The exposure process begins a perio of infection for the agent. The exposure process has the single state variable stage that may take the values 0, 1, an 2. The initial value is stage = 1 if the agent begins with the isease an stage = 0 otherwise. If stage = 2, then the agent has alreay been infecte. The ynamics of the exposure process are given by: int (stage)=2 1 if stage = 0 ext (stage, e, x)= stage otherwise con (stage, x)=2 l(stage) = infect 0 if stage = 1 ta(stage)= otherwise ð8þ Disease process. The isease process carries the agent through the stages of the isease. The isease process has a state variable stage that may take the values ecease (D), recovere (R), ying (Dy), recovering (R ), an susceptible (S). The initial state is stage = susceptible. A uniformly istribute ranom variable u with range ½0, 1Š is use to etermine the outcome of the isease for this agent. The ynamics are: D if stage = Dy int (stage)= R otherwise Dy if u 5 pm ext (stage, e, x)= R otherwise con (stage, x)= int (stage) l(stage) = infect 8 >< tm if stage = Dy ta(stage)= tr if stage = R >: otherwise ð9þ Infectious process. The infectious process is responsible for causing other agents to be expose to this agent while the agent is infectious. The infectious process has a Boolean-value state variable infectious; a ranom variable u uniformly istribute in ½0, 1Š is use to etermine if a particular encounter will lea to infection of the encountere agent; an an exponentially istribute ranom variable m with mean 1=re is use to etermine the time between encounters. The initial value of infectious is false. The ynamics of the infectious process are given by: int (infectious)=infectious ext (infectious, e, x)=:infectious con (infectious, x)=:infectious infect if u \ pt l(infectious)= F otherwise m if infectious ta(infectious)= otherwise ð10þ A time-steppe SIR moel with two forms. The moels escribe above incrementally refine the implementation of the isease sprea process. All of the above moels have two features in common: (i) interactions happen instantaneously an (ii) the state of the moel can change at any instant of the real-value simulation clock. The moels escribe in this section eliminate these two features by using a iscrete-time, rather than continuous-time, approach to moel construction. Discrete-time (i.e., timesteppe) moels are a popular type of ABM, an this approach is the basis of most moeling tools for agentbase systems.

7 Özmen et al. 465 Our time-steppe ABM is implemente using the Repast framework, 23 an it eparts from the above moels (EBMs an the event-base SIR moel) in two specific ways. First, this ABM may elay interactions between agents: one variant of the moel uses instantaneous interactions an the other oes not. Secon, this moel has a iscrete time base an the states of its agents change at fixe instants t 0, t 1, t 2,. that are separate by the moel s time increment h (i.e., t n + 1 t n = h). At each step of the simulation, the agents are activate in an orer that is generate at ranom. In Repast, this proceure for activating agents is calle scheuler scramble. 24 Upon activation, an infecte agent raws the number of agents to contact for that time step from a Poisson istribution with mean re. The Poisson istribution is use for two reasons. First, it is necessary to raw the number of people to encounter from a iscrete istribution: an agent cannot encounter part of a person. Secon, inter-encounter times in the event-base moel are rawn from an exponential istribution, an the Poisson istribution is its natural counterpart. Two ifferent approaches, corresponing to two ifferent forms of the time-steppe moel, are use to calculate a new state for the agent an the agents it contacts, as follows. Form 1: Asynchronous interaction. In this approach to upating the agents states, if Agent A infects Agent B at time t, then the state of Agent B is upate immeiately. Hence, there can be two cases: (i) Agent B was activate before Agent A, so Agent B waits until t + 1 to infect other agents, or (ii) Agent B is activate after Agent A, so Agent B starts passing along the infection at time t. Form 2: Synchronous interaction. In this process, if Agent A infects Agent B at time t, Agent B becomes infecte at time t + 1 regarless of the orer of agent activation. Moreover, agents observe the state of other agents as the state that existe before agent activation began. For example, if Agent A infects Agent B an the next agent to be activate is Agent C, then Agent C perceives Agent B as susceptible, not infecte. This process allows an infecte agent to pick an alreay infecte agent to pass the infection along Obtaining comparable outputs from EBMs an ABMs For ABMs escribe here, the state variable stage of the isease process moel is use to construct the quantities ecease, recovere, infectious, an susceptible. For ABMs, the quantities ecease an recovere in the respective EBMs correspon to the count of agents with stage = D an stage = R, respectively. The quantity susceptible in EBMs correspons to the count of agents with stage = S in ABMs. The quantity infectious in the EBMs correspons to the sum of the number of agents with stage = Dy an stage = R in the ABMs. In the timesteppe ABMs, similar to the event-base ABMs, the number of agents that match the corresponing state are counte at the en of each time tick. The only exception is the quantity ecease; ecease is just the number of agents that are remove from the environment in the timesteppe moel. 3. Results In this section, we start with comparisons of point estimators. Subsequently, we focus on the evolution of outputs over time. The ifferences between the moel outputs are elineate in both quantitative an qualitative manners Peak loa, iffusion spee, an isease buren Three metrics in particular are important for policy makers when consiering how to respon to an outbreak of influenza. These metrics are peak loa on health services, efine here as the largest infectious population observe over time; iffusion spee, efine as the amount of time that passes from the initial outbreak until the peak loa is observe; an total isease buren, efine as the cumulative number of people who have ie from the isease at the en of the epiemic. Table 2 compares these metrics across the six moels with parameters for the 1918 flu epiemic as previously escribe. At the 95% confience level, the EBM-elaye an event-base ABM are inistinguishable. The iffusion Table 2. Peak loa, iffusion spee, an total buren of ifferent moels. Moels Peak Loa ( ± 95%CI) Diffusion Spee ( ± 95%CI) Total Buren ( ± 95%CI) EBM-SIR 55, EBM-substates 86, EBM-elaye 114, Event-base 114,800.7( ± 241.7) 13.4( ± 0.2) ( ± 19.4) Time-steppe-asynch 95,749.0( ± 196.7) 13.1( ± 0.1) ( ± 22.1) Time-steppe-synch 87,343.9( ± 165.9) 18.0( ± 0.0) ( ± 23.2) influenza ata EBMs of this stuy are eterministic, an hence o not have variability. The 1918 influenza ata value is foun by scaling the real-worl estimates own base on the population size use in experiments. Confience intervals (CI) assume normality.

8 466 Simulation: Transactions of the Society for Moeling an Simulation International 92(5) spee an total buren of the EBM-elaye moel is within the 95% confience interval for the iffusion spee an total buren of the event-base ABM. The peak loa for the event-base ABM is slightly higher than that of the EBM-elaye moel, but only by 79 persons out of 500,000 people in relation to the lower boun on the 95% confience interval. The ifference in the peak loa measures from the two moels is just 0:07% of the 114,480 (out of 500,000 as the initial population; Table 1) of the EBM-elaye moel, implying that these moels iffer very little with respect to this metric. It is also interesting to note that the other moels iffer significantly with respect to the peak loa values, with the EBM-SIR moel estimating the lowest peak loa compare to the other moels. This is also expecte since the EBM-SIR moel incorporates a significant simplification in the recovery/ eath process as aforementione. However, one can observe that the comparisons with other moels are not the same. While there is significant agreement between the event-base moel an the timesteppe-asynch moel, the time-steppe-synch moel iffers consierably with respect to the iffusion spee. As such, the event-base an time-steppe-asynch moels suggest that the initial rise in the number of sick people is similar in both cases, implying similar isease propagation. However, for the time-steppe-synch moel, the isease spreas more slowly reaching a lower peak loa. The variation of iffusion spee is also true for comparisons between the time-steppe moels an the EBM-elaye moel an comparisons between the three EBMs. The total buren measures an equilibrium state of the moel. Specifically, for the three ifferent EBMs, the total buren measures the overall number of ecease people an, for the ABMs, inicates the maximum number of ecease agents when the moel becomes quiescent. Table 2 suggests that the equilibrium states for the six types of moels are very similar, in spite of not sharing any common isease progression trajectories. The two other observables, namely the peak loa an iffusion spee, measure transient aspects (see the next section) of the moels, an therefore epen on measurements of the state variables when the erivatives are far from zero (for the EBMs) or the agents are active (for the ABMs). Therefore, we can conclue that the transient ynamics for these six moels are quite ifferent with the exception of the event-base an EBM-elaye moels. For these two moels, we shoul expect similar transient behavior because the two moels agree with respect to peak loa an iffusion spee Transient behavior To probe the changes in the transient behavior, we performe aitional analyses on the moels. Figure 4 shows trajectories for each of the six moels with parameters for the 1918 flu epiemic. This figure shows the (S) susceptible, (I) infectious, (R) recovere, an (D) ecease population as a proportion of the total population at the en of each simulate ay. The EBMs are eterministic an so the trajectories shown are the moel s only possible output for the given parameters an initial conitions. The ABMs are stochastic, an the trajectories shown in Figure 4 represent the mean values an the 95% confience interval to inicate typical evolution of the moels. There is a general agreement in terms of the overall shapes of trajectories generate by each moel. While the trajectories closely resemble one another in terms of the final outcomes (as observe previously in terms of the equilibrium measures), there are significant ifferences regaring their transient ynamics. In Figure 4 (S), it is quite clear that EBM-SIR (orange line) shows S ecreasing much more slowly than o the other moels. Similarly, in Figure 4 (I), (R), an (D), the isease propagation in the EBM-SIR is slower. This iscrepancy between the EBM- SIR an the other moels is anticipate because of the simplification in the EBM-SIR moel. The subsequent corrections to the EBM-SIR moel, first by aing substates an then aing elays, brings the trajectory of the EBM- SIR moel very close to that of the event-base ABM. It is much more ifficult to explain the ifferences between the EBMs an ABMs. The substantial ifference between the time-steppe-synch an -asynch ABMs is particularly noteworthy. The choice of a synchronous versus asynchronous upating strategy has a large effect on the moels trajectories. The synchronous strategy spreas the isease more slowly, an we hypothesize that this observation is ue to the elay between the exposure of an agent to the isease an that agent becoming infectious. This notion is reinforce by the trajectories of the time-steppe ABM moel that uses an asynchronous upating strategy, for which such a elay oes not exist. Inee, the timesteppe-asynch moel much more closely approximates the trajectories of the EBM-elaye an the event-base ABM moels. We teste the aforementione hypothesis with simulations of both time-steppe moels using successively smaller time steps. The results are presente in Figure 5(a) an (b) for asynchronous upate an in Figure 5(c) an () for synchronous upate moels. Specifically, in Figure 5(a) an (c), it is clear that the iscrepancies between curves graually iminish as the time steps are successively reuce (h = 1:0, 0.5, 0.2, 0.1, an 0.05 ays). This iminishing size of the time step has two effects: first, it causes the iscretization of time in both of the time-steppe moels to better approximate the continuous time use in the EBM an the event-base moel; secon, it causes the one-step elay between exposure an infection in the synchronous moel to more closely approximate the instantaneous infection that occurs in the other moels.

9 O zmen et al. 467 Figure 4. Evolution of main compartments over time. (S) Susceptible, (I) infectious, (R) recovere, an (D) ecease. Figure 5. The comparisons of ifferent time-step ecisions for time-steppe moels. (a) Infectious: time-steppe-asynch, (b) ecease: time-steppe-asynch, (c) infectious: time-steppe-synch, an () ecease: time-steppe-synch. The lines represent the mean values.

10 468 Simulation: Transactions of the Society for Moeling an Simulation International 92(5) These two effects are both significant in etermining the outcome of the moel. As the time step is reuce, the trajectories of the time-steppe moels resemble one another more closely, approaching the trajectories of the EBM-elaye moel, an the trajectories of the eventbase moel. This suggests that the selection of a time base, either continuous or iscrete, has important consequences for a moel s ynamic behavior. Moreover, if the time base is iscrete, then the size of the time step also has a substantive effect on the moel s behavior. Hence, the problem of choosing a representation for time is an important, but often overlooke, aspect of builing an using ABMs, most of which use a iscrete time base an rely on the moeler s intuition to choose a time step. The choice of an asynch or synch upating strategy has important ramifications. The behaviors of both timesteppe moels are quite istinct when their time step is one ay. This ifference only iminishes as the time step becomes smaller an, consequently, the ifferences between the upate strategies become less significant. This section highlights the importance of choosing when an how agents interact within a moel (see Section 4) Sensitivity to small encounter rates The number of opportunities for a sick person to encounter healthy persons is a major factor in the rate an extent of an epiemic. In our moels, this is capture by the encounter rate. The encounter rate, re, is one of the epiemicrelate parameters that has a high impact on the R 0 value. Small values of re are of particular interest for comparing ABMs an EBMs. If re is small enough, the EBM will always inicate that the epiemic falters an ABMs might generate highly variable outcomes. The cause for this is clearly apparent in the term for (=t)i in the SIR moel presente in Equation (2). Near the start of the epiemic, we may take S=(S + I + R) 1 an so write (=t)i as: t I = pm 1 pm + tm tr (re 3 pt) 3 I ð11þ EBM-SIR. Note that a similar argument hols for both EBM-substates an EBM-elaye. Although the absolute lower boun for the isease to propagate within the ABMs is 0, note that with re 4 2:27 the isease in an ABM is most likely to sprea to a very small number of persons before the epiemic ies out. Hence, the infectious population will grow slightly before shrinking to zero in the event-base moel. This expecte behavior is ifferent from the behavior of the EBMelaye where the number of infectious people is strictly iminishing. Moreover, with the ABMs there is a small but efinite chance that the infectious population will become quite large an the isease will sprea very rapily. This is a scenario that cannot be capture by the EBMs. Inee, numerical experiments show that the eventbase ABM an EBM-elaye moels agree for relatively large values of re. However, for simulations with re 4 2:27 the agreement between EBM an ABM is much less pronounce 25 even though isease propagation is mitigate in both moels. This fact is illustrate in Figure 6 which shows the mean absolute percentage error (MAPE) for the number of ecease when comparing the EBMelaye moel an event-base ABM. The MAPE is efine by: M = 1 T X T t = 1 j O t E t O t j ð14þ where M is the MAPE, t is the number of ays that the isease sprea continues, an O t an E t are the cumulative number of ecease at ay t for the EBM-elaye an event-base moels, respectively. In Figure 6 it can be seen that the level of isagreement increases significantly from which it follows that I grows if an only if: pm 1 pm + (re 3 pt). 0 ð12þ tm tr an rearranging this expression shows that growth requires: re. 1 pm 1 pm + ð13þ pt tm tr For the parameters given in Table 1, the above expression inicates that re. 2:27 is require for growth in the Figure 6. EBM-elaye vs event-base moel: MAPE between the cumulative numbers of ecease.

11 Özmen et al. 469 when the re value falls below 2:27, an outcome that was anticipate by the above analysis. 4. Discussion The primary goal of our stuy is to unerstan the extent to which three moeling choices, namely (i) the choice of moel trajectories an temporal resolution (continuous versus iscrete-event moels), (ii) coupling between agents (instantaneous versus elaye interactions), an (iii) progress of patients from one stage of isease to the next affect the overall isease progress an temporal ynamics/outcomes from these epiemiological moels. Regaring ifferential equation moels, error an uncertainty cause by numerical calculations (relate to (i)) are well-stuie. 26 Although there are stuies that have aime to generate the exact same behavior from system ynamics moels an their agent-base counterparts, 27,28 there are no clear guielines on specific choices in the moel evelopment processes involve. To eluciate the effect of these choices, we introuce six variations of isease sprea moels that mimic the moifie SIR ynamics. First, we iscusse the specific implementation routes taken for each moel. The six moels evelope here capture key features within alreay-establishe influenza moels. By using moel parameter values erive in previous stuies for 1918 influenza, we generate a realistic test case for comparing moel outcomes. We showe that both ABMs an EBMs reach similar equilibrium states, however, their transient ynamics may iffer significantly base on the moeling principles chosen. In particular, within time-steppe moels, the ynamic behavior has a strong epenence on the choice of time step an type of interaction between the agents (i.e., synchronous or asynchronous). However, these strong epenencies are rarely consiere when constructing ABMs, for which an a hoc selection of the time step an interaction moel is most typical. Our results suggest that an a hoc selection is not appropriate in the general case. Inee, if a strong justification exists for a specific choice of time step or type of interaction, either on theoretical grouns or on the basis of available ata, then this justification is a crucial part of the moel s unerlying theory. If this selection is a hoc, then the time step an type of interaction shoul be subjecte to sensitivity stuies like those applie to the moel s other parameters. We emonstrate that while ABMs an EBMs can agree for a wie range of scenarios, this agreement can break own in cases where the isaggregate population in the ABM enables ynamics that are not realizable within EBMs. These emonstrate results concur with other stuies. 6,10 An example of this was emonstrate for small values of re in our influenza moels. When re is sufficiently small, an thus encounters between agents are relatively infrequent, the EBM imposes an irreversible ecline in the number of infectious persons. Conversely, the ABM permits small rises in the number of infections uring the course of a long-term ecline. An experimental comparison of MAPEs reveals that this effect causes a noticeable ifference in the moel trajectories at re 2:3 when using parameters from the 1918 influenza panemic. Our moel evelopment process also reveals why the previous works 6,10 were not able to generate system ynamics results of SIR ynamics for homogenous populations an imprecisely conclue that stochastic variability cause the iscrepancy among the trajectories: when a continuous istribution (i.e., exponential) is use for tr in the ABM an the time step truncates samples of this istribution (iscretizing), the mean of the istribution is ifferent in these moels. Our results suggest that if one nees to reprouce the results of an EBM by a time-steppe ABM, then iscrete counterparts of continuous istributions shoul be use an all time-epenent variables shoul be efine in terms of the time step. Even then, for relatively large time steps, we shoul not expect the timesteppe moel to agree with continuous time moels. Aitionally, a small number of initial infectious people (etermine as 1 by Macal 10 ) an lower R 0 values contribute to the isagreement between EBMs an stochastic ABMs. This work aime to elineate some of the funamental, but often overlooke, assumptions that are ecie in the earliest stages of moel evelopment, an to ientify the impact of these assumptions on the behavior of the resulting moel. These initial ecisions shoul be explicitly an carefully consiere an recore when builing an ocumenting an ABM. The results of our stuy suggest that, if agreement between an ABM an EBM is esire (e.g., for the purpose of moel verification), then it is important for the ABM to use continuous time an instantaneous interactions. These help the ABM better mimic the continuous evolution of the EBM. Perhaps surprisingly, this is true espite the iscrete states of the ABM. For iscretetime moels, the moels shoul be carefully constructe with iscrete istributions. However, we cannot give specific guielines for selecting an appropriate time step ue to its omain an problem efinition epenence. But our results suggest that this is an important topic an a venue for future research. Acknowlegements This manuscript has been authore by UT-Battelle, LLC uner contract number DE-AC05-00OR22725 with the U.S. Department of Energy. The Unite States Government retains an the publisher, by accepting the article for publication, acknowleges that the Unite States Government retains a nonexclusive, pai-up, irrevocable, worl-wie license to publish or reprouce the publishe form of this manuscript, or allow others

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