The Impact of Demographic Variables on Disease Spread: Influenza in Remote Communities

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1 esupplementary Information The Impact of Demographic Variables on Disease Spread: Influenza in Remote Communities Marek Laskowski, Luiz C. Mostaço-Guidolin, Amy Greer, Jianhong Wu, Seyed M. Moghadas This supplementary information provides details of community profiles, model structure and parameters, and the simulator that functions as an in-silico laboratory for conducting simulation experiments aimed at understanding the relationships between demographic variables (i.e., age and household composition) that underlie the emergent phenomenon of infectious disease spread. The results of model simulations for several aspects of the epidemic, including hospitalizations, age standardized infection ratio, household attack rates, and serial interval distributions, are also presented. Model Details There are two basic entities in the model: (i) agents that represent individuals living in the simulated community; and (ii) the environment that represents the community in which agents interact. The interaction of agents is determined by modular behavioral components, each containing state information relative to that behavior. For example, each agent s disease component maintains the state of disease specific to that agent, and movements are driven by schedules, leading to indirect relationships between agents by virtue of being co-located at a particular position and at a particular time step in the community. Disease may spread from infected to uninfected agents that are located at the same position on a particular time step according to the disease state component. Figure A1 illustrates these relationships. Figure A1. Fundamental model relationships. Like other agent-based models, the relationships between entities in our model may be viewed as a bipartite graph [1] with agent nodes and position nodes, and edges between them representing the agent occupying the position for a particular time step. 1

2 The model developed here is comprised of the following interacting modules: Temporal, Spatial, Age Demographic, Agent Schedule, Disease Transmission, and Disease Progression. Below, we provide details of each module, and describe data sources as well as their means of integration. Temporal module In our model, time advances in discrete steps of one hour. The details of the simulation of each time step are described by the pseudocode presented in Listing A1. Listing A1. Pseudocode for the main simulation loop. Precondition: initial state of the environment is set, all agents are created, and have initialized disease and schedule models 1 while there are any remaining infected, exposed, or presymptomatic agents do: 2 advance the disease state of each agent by the time step 3 each agent decides to which location it desires to go 4 model disease transmission between co-located agents 5 agents move to their desired location 6 advance the time of day by the time step 7 end while..do Spatial module The agent s environment is divided into locations that correspond to contact mixing groups. There are four types of locations: households, workplaces, classrooms, and other public spaces in the community. All the locations are arranged in a lattice, and households, workplaces, and classrooms are separated by one public space in each direction. Figure A2 shows details of the layout, which is not meant to reflect the exact topography of the community, centralizes school and work locations while leaving households at the periphery. This layout becomes important when considering random mixing near an agent s current location. The lattice can be viewed as having rows and columns that are numbered starting with (0,0) in the top left corner, with indices increasing towards the bottom and right for rows and columns, respectively. Recent empirical work [2] suggests that homogenous mixing would likely suffice within schools, whereas this model further compartmentalizes them by age or grade. Locations in which the same sets of agents are regularly positioned are analogous to clusters or highly connected nodes in a social network. An agent s location changes based on agent schedules and the time of day as described by the agent schedule module. Given the hourly time step and the relatively small community size, we ignored the travel time and means between locations. 2

3 Figure A2. Spatial layout of the ABM lattice-like community. The number of household mixing groups is based on Statistics Canada community profiles[3], with a total of 436 private dwellings in the remote community (RC). Without any available data on the number of workplaces or workplace sizes in the particular community, the number of workplaces is based on the assumption that the remaining 129 dwellings that are not households serve as workplaces [3]. This assumption leads to reasonable workplace sizes for the experiments that were conducted. The effect of workplace size is not the focus of this work, and the results presented in the main discussion suggest that the effect of workplace size is dominated by the other variables considered here. There are 12 classrooms or school mixing groups, based on student age. The distribution of agents amongst household mixing groups is based on data [4] which specifies the count of households of each size in the community. The specific procedure used to assign agents to households remains the same regardless of the household size distribution used for particular scenarios or experiments in the main text. This random distribution procedure is applied for every simulation run, resulting in a slightly different community structure for each different random seed used. The procedure to assign agents to households is as follows. Agents are separated into two temporary lists: those over 18 (adults), and those younger than 18 (children). Also, all houses in the community are placed in a third temporary list. Beginning with the list of adults, agents are removed from the list and assigned to a uniformly random house from the list of houses. Once the list of adults is exhausted, the distribution procedure continues with the list of children in order to bias against households with only children. During the distribution procedure, the first n i houses to have i occupants assigned are removed from the list of houses in order to satisfy the constraint in the data that there are n i households of size i. However, once there are only houses that will have 6 or more residents, they are not removed from the list of houses, and agents continue to be distributed amongst the remaining houses in the list until the lists of agents are exhausted. For the 3

4 experiments in which community household composition profiles were shifted, a different household size distribution or constraint is used to distribute agents amongst houses rather than the data that was used originally. The first of these alternative household composition profiles ( households of size 4 and 5 ; household composition in Table 1 of the main text) attempts to distribute agents amongst houses as uniformly as possible. That is, using the same number of available houses, to minimize the maximum household size. Given the population of 1895 individuals and 436 households, equal distribution of individuals will result in some households of size 4 and some of size 5 (an average of 4.35 individuals per household). The specific profile used in the scenario has 285 households of size four and 151 households of size five was arrived at by solving the following system of linear equations: 4x + 5y = 1895 x + y = 436 where x and y are the households of sizes 4 and 5, respectively. The second alternative household composition profile used in the experiment scenarios shifted the profile such that the ratio of households of a certain size to all households is the same as in Winnipeg ( Winnipeg-like household composition in Table 1 of the main text). Since the population of RC is much smaller than Winnipeg, there are far fewer households. However, since the average number of persons per household is lower in Winnipeg (2.4) than in RC (4.4), the number of houses will have to be increased from 436. The household composition data for Winnipeg [4] specifies how many households have 6 or more persons so we assumed that all households with more than 6 persons have a size of 7. This is done in order to estimate the ratio of households of particular sizes to the whole. In order to account for all the persons in private dwellings in Winnipeg (622430) [4], 2470 households of size 6, and 3640 households of size 7 are required. Based on these assumptions, the ratios of households of a certain size in Winnipeg are calculated by dividing the counts of households of each size by the total number of households in Winnipeg (261130) and are presented in Table A1. Table A1. Ratio of household sizes in Winnipeg. Household size Ratio The number of houses required for a population of size 1895 to have the same household size ratios as Winnipeg in Table A1 satisfies the following equation: 7 iri x = 1895, i= 1 where x is the total number of houses and i R is the ratio of households with size i to the whole. This yields 795 houses, which implies 359 more houses would have to be placed in the community 4

5 in order to achieve the same household composition profile as Winnipeg. This provides the household size profile used for the experimental scenario, and is summarized in Table A2. Table A2. Community household size profile used in the experimental scenario. Household size Number of households For distributing individuals amongst workplaces, after the number of employed individuals is determined (as explained in the Agent Schedule section below) but before the simulation begins, we first distributed uniformly the individuals that work in schools, estimated at 48 [3], amongst the 12 classroom mixing groups. The remaining working individuals are distributed randomly amongst the locations designated as workplaces in the community with a uniform probability. Distribution amongst the classroom mixing groups was done according to their age such that all students with the same age are in the same classroom. Age demographic module The age profile for RC was based on [3], which provides counts of individuals in five-year age-groups, stratified by gender. The probability of an agent being female is simply the ratio of individuals that are female in the data. Once the gender is determined, the age group of the agent is determined randomly with a weighted probability. The probability of the agent belonging to each age group is the count of agents in that age group divided by the count of all the agents. Once the age group is determined, the age of the agent counted in years is selected randomly within the age range of that group, with a uniform probability. In experiment scenarios where Winnipeg demographic age profiles are used ( Winnipeglike age distribution in Table 1 of the main text), the counts in Table A2 are scaled down to a total population of It should be noted that for these scenarios employment rates were also changed to better reflect the Winnipeg level of activity (details in the Agent Schedule section). These shifts affect workplace mixing group sizes and school mixing group sizes (analogous to classrooms or grades). Agent schedule module There is no data available on individual s time use in remote northern communities. We therefore made some reasonable assumptions of basic activities and schedules in an attempt to capture the main causes for contact within a small community. That is, agents spend time at and around their home, work, or school locations. Before the simulation starts, during the agent creation process after the agent age and gender has been determined (as above), the procedure to determine whether an agent attends school or work is applied as follows. Individuals under 6 years of age are assumed to attend neither work nor school. Individuals 6-17 years of age are assumed to attend school. Agents years of age 5

6 (depending on the gender of the agent) are employed as described in the Employment ratios presented in Table A3. Agents 65 and older attend neither work nor school. Table A3. Employment data for RC [3] and Winnipeg [5]. Number of individuals years of age Number of employed individuals RC Winnipeg Male Female Male Female Employment ratio For the experimental scenarios in which the age profile is shifted to resemble Winnipeg ( Winnipeg-like age distribution in Table 1 of the main text), the Winnipeg employment ratios are used to more accurately capture the behavioral profile of the community. All agents follow the same basic scheduling pattern with variation introduced depending on whether the agent attends work or school (or neither). Also, for a particular agent, there is some random variation as follows: Between 00:00 (midnight) and 08:00, all agents are at their assigned home locations. The simulation begins at midnight so the initial state of all agents is at their respective household. Between 08:00 and 16:00 agent behavior will be different depending on whether the agent attends work, school, or neither. If the particular agent attends work or school, it will remain in their respective work or school location, except for lunch time (12:00 13:00) during which the agent exhibits the random walk behavior in Listing A2. Between these hours, agents that do not attend work nor school, will (with equal probability) either be at their assigned home location, or exhibit the random walk behavior in Listing A2. Between 16:00 and 00:00 agents will (with equal probability) either be at their assigned home location, or exhibit the random walk behavior in listing A2. Listing A2. Pseudocode for random walk behavior. Precondition: A given agent has decided to perform random walk, and the agent has a current location 1 delta_columns := 1 2 delta_rows := 1 3 rand_num := generate a random floating point number between 0 and 1 4 while rand_num < 0.5 do: 5 delta_columns := delta_columns + 1 rand_num := generate a random floating point number between 0 and 1 6 end while..do 7 rand_num := generate a random floating point number between 0 and 1 8 if rand_num < 0.5 then: 6

7 9 delta_columns = delta_columns * (-1) 10 end if..then 11 delta_rows is calculated in a similar manner Postcondition: agent will travel delta_rows and delta_columns on the community grid or lattice (figure A2) from the agent s current location When applicable, the decision to either go to the assigned location or perform the random walk, is made on a per-agent per-time-step basis. The random walk behavior described in listing A2 is relative to the agent s current location. Thus, the agent will tend to stay near to their current location, but occasionally venture farther away (with exponentially decreasing probability) when exhibiting the random walk behavior. Agents that are in the infected disease state (see Figure A3) and are staying home (I s ) will always be at home and not exhibit the random walk behavior while in this state. Listing A3. Pseudocode for disease transmission. 1 for each location L in the model: 2 for each infectious or presymptomatic agent I at L: 3 if I is presymptomatic then: 4 Pt := base probability of transmission / 2 5 else: 6 Pt := base probability of transmission 7 end if-else 8 for each susceptible agent S at L: 9 if age of S is >= 50 then: 10 apf := P1 11 else: 12 apf := P2 13 end if-else 14 rand_num := generate a random floating point number between 0 and 1 15 if rand_num < Pt*(1-apf) then: 16 S transitions to the exposed disease state 17 end if-then 18 end for each susceptible agent 19 end for each infectious or presymptomatic agent 20 end for each location Disease transmission module At the start of each simulation scenario, all agents are initialized to the susceptible (S) state, except for one randomly chosen agent (the initial case), which is initialized to the I k state. Each simulation time-step is considered as an independent Bernoulli trial where transmission is effectively viewed as a "rate", that is transmissions per unit time. The module for disease transmission is presented in Listing A3. When there are n infectious individuals and m susceptible individuals co-located in a particular time-step, n m an independent trial is performed between infectious and susceptible individuals. Once infected, the susceptible individual is removed from the list of susceptible individuals. Disease progression module The module is based on a compartmental type of model that resembles a Markov chain. Typically, agents begin in the susceptible compartment (S). Once exposed (E), the individual will remain in this compartment for the exposed period, T E. After time T E has elapsed, the disease 7

8 compartment will change to pre-symptomatic (P), during which transmission to other susceptible agents can occur without showing any symptoms of disease. The individual will remain in this stage for the pre-symptomatic period, T P. Once T P has elapsed, the infectious individuals is identified (I d ) with probability α, or remains undiagnosed (I u ) with probability 1 - α. Undiagnosed cases will stay home (I s, self-isolation) with probability ω, or follow their normal schedule (I k ) with probability 1 - ω. Diagnosed cases will be hospitalized (H) with probability π, and effectively removed from the simulation without contributing to disease spread. Diagnosed agents that are not hospitalized will move to the I s compartment with probability λ and the I k compartment with probability ρ. The individual will remain in the I k or I s compartment for the infectious period T I (Figure A4). After T I has elapsed the agent moves to the recovered (R) compartment and acquires immunity against reinfection. We assumed a probability of α=0.02 for an infectious case to be identified following symptom onset [6]. The probability of self-isolation was assumed to be 0.8 for children below 18, 0.3 for adults between 18 and 50 years of age, and 0.8 for individuals above age of 50 [7,8,9]. The age-specific probability of hospitalization, once diagnosed, was estimated from pandemic data collected for the Burntwood health region in the province of Manitoba, Canada, where the majority of remote and isolated communities are located. The probability of hospitalization based on age is drawn from aggregate infection and hospitalization data over a subset of communities in the Burntwood region that resemble the community modeled here. Figure A3. Compartmental disease state model. 8

9 Figure A4. Log-normal distribution of the infectious period captured over 1000 simulations, resulting in data points. The mean infectious period is 3.1 days. Model Plausibility In this section a number of measures of emergent model behavior are presented to support the qualitative aspects of the findings reported in the main text. Contact Network The characteristics of the contact network reflect the combined performance of the demographic (number of houses, number of agents, and their ages) and agent schedule. Figure A4 shows the average number of daily contacts that lasted for 1, 6, and 9 hours for four different age groups in the model. This figure shows the variation in contact patterns that are consistent with observed patterns published in the literature [10]. The average number of contacts peaks for individuals around 15 years of age, and decreases in older individuals. Figure A5. Average number of daily contacts by age group that lasted for 1 (black), 6 (dark grey), and 9 or more (light grey) hours during a 24-hour model simulation. 9

10 Transmissibility We assumed a transmission rate the resulted in an average number of 1.9 secondary infections (in the absence of any limitation to disease spread). This was done by setting p 1 and p 2 to zero, and varying the base probability of transmission, P base, until the average number of secondary infections resulting from the initial case averaged over 1000 runs was approximately 1.9. This P base was used for all the experimental scenarios. Comparisons of pandemic model output for several of the experimental scenarios discussed in the main text are presented in figure A6 with p 1 and p 2 set to 0. Each curve represents the average of 1000 independent runs for that scenario. The effects of demographic shifts have a plausible effect on the outcome, considerably changing not only the peak time, but also the magnitude of the outbreak. The peak time for the original demographics scenario is between 20 and 30 days which is consistent with data for pandemic 2009 in the region studied here. Figure A6. Epidemic curves for selected experimental scenarios (corresponding to Table 1 in the main text) with p 1 =p 2 =0. The standard error bars shown on the epicurves represent the degree of variability in the results of 1000 simulation runs. 10

11 Hospitalizations Counts of confirmed cases, counts of hospitalizations, and age-specific probability of hospitalization from data are summarized in Table A4. When counts were small, an assumed hospitalization probability was used in the model, based on the nearest adjacent age-groups. Table A4. Input data for age-specific hospitalization probability. Age range Confirmed cases Hospitalizations Probability of hospitalization if confirmed Probability used in model 0 to to to to to to to to to to to to to and over For each simulated scenario, we obtain the range of the ratios of hospitalization to the final size (total number of infections), and the results are summarized in Table A5. 11

12 Table A5. Range of the ratios of hospitalization to the final size. Simulation hospitalization ratio scenarios 0.5 p 1 0.7; 0.1 p p 1 0.3; 0.5 p Original demographics Age employment shift Egalitarian Scaled Winnipeg Scaled Winnipeg age employment shift School closure four weeks School closure eight weeks Serial Intervals The distributions of serial interval (the time interval between clinical onsets in primary and secondary cases) were recorded for each scenario, as presented in Figures A7-A10. Only cases where there were one or more secondary infections were considered in averaging the serial interval over sample realizations. The average serial interval was estimated from the log-normal curve fit to each distribution. 12

13 Figure A7. Distributions of the serial interval for the original household composition of RC as reported in census data and: (a) the original age distribution of RC (average: 3.31 days); (b) a shift to Winnipeg-like age distribution (average: 3.35 days); (c) the original age distribution of RC with switched levels of pre-existing immunity (average: 3.53 days); and (d) a shift to Winnipeg-like age distribution with switched levels of pre-existing immunity (average: 3.49 days). 13

14 Figure A8. Distributions of the serial interval for the original age profiles of RC as reported in census data and a shift in household size to have: (a) 4 and 5 persons per household (average: 3.43 days); (b) Winnipeg-like composition (average: 3.48 days); (c) 4 and 5 persons per household with switched levels of pre-existing immunity (average: 3.62 days); and (d) Winnipeg-like composition and switched levels of pre-existing immunity (average: 3.63 days). Figure A9. Distributions of the serial interval for a shift to have: (a) Winnipeg-like age distribution and household composition (average: 3.5 days); (b) Winnipeg-like age distribution and household composition with switched levels of pre-existing immunity (average: 3.62 days). 14

15 Figure A10. Distributions of the serial interval with the original demographics of RC as reported in census data and: (a) 4 weeks school closure (average: 3.29 days); (b) 8 weeks school closure (average: 3.3 days); (c) 4 weeks school closure with switched levels of pre-existing immunity (average: 3.52 days); and (d) 8 weeks school closure with switched levels of pre-existing immunity (average: 3.53 days). Fitting a Linear Model to Final Sizes The trends in clinical attack rate reduction according to increase in protection levels of different age groups (p 1 and p 2 ) discussed in the main text (Figures 2-5) can be well modeled by a function linear in p 1 and p 2, i.e. a plane (attack rate = ap 1 +bp 2 +c). Figures A11-A14 shows corresponding fits to the experimental scenarios presented in the main text. The fitting parameters are summarized in Tables A6-A7. 15

16 Figure A11. Fitting of the linear model to the attack rates (corresponding to Figure 2 in the main text) for: (a) the original age distribution of RC; (b) a shift to Winnipeg-like age distribution; (c) the original age distribution of RC with switched levels of pre-existing immunity; and (d) a shift to Winnipeg-like age distribution with switched levels of pre-existing immunity. 16

17 Figure A12. Fitting of the linear model to the attack rates (corresponding to Figure 3 in the main text) for: (a) 4 and 5 persons per household; (b) Winnipeg-like composition; (c) 4 and 5 persons per household with switched levels of pre-existing immunity; and (d) Winnipeg-like composition and switched levels of pre-existing immunity. Figure A13. Fitting of the linear model to the attack rates (corresponding to Figure 4 in the main text) for: (a) Winnipeg-like age distribution with 4 and 5 persons per and household composition; (b) Winnipeg-like age distribution and household composition with switched levels of pre-existing immunity. 17

18 Figure A14. Fitting of the linear model to the attack rates (corresponding to Figure 5 in the main text) for: (a) 4 weeks school closure; (b) 8 weeks school closure; (c) 4 weeks school closure with switched levels of pre-existing immunity; and (d) 8 weeks school closure with switched levels of pre-existing immunity. Table A6. Parameter estimates of the linear model with 0.5 p and 0.1 p Simulation a b c scenarios estimates std. error estimates std. error estimates std. error Original demographics Age employment shift Egalitarian Scaled Winnipeg Scaled Winnipeg age employment shift School closure four weeks School closure eight weeks

19 Table A7. Parameter estimates of the linear model with 0.1 p and 0.5 p a b c estimates std. error estimates std. error estimates std. error Simulation scenarios Original demographics Age employment shift Egalitarian Scaled Winnipeg Scaled Winnipeg age employment shift School closure four weeks School closure eight weeks Age Standardized Ratios of Infection Using model outputs for total number of infections in different age groups and demographic data, we calculated the range of age standardized infection ratio for each simulated scenario. This ratio is the proportion of infected cases in a given age group to the proportion of the population in the same age group, given by: no. of infections in age group i ( total no. of infections in all age groups ) population size of age group i ( total population size ) The ranges for age standardized ratio with 95% confidence intervals, corresponding to simulated scenarios in the main text, are illustrated in Figures A15(a-g).. 19

20 Figure A15. Age standardized ratios for (a) original demographics, (b) age employment shift, (c) egalitarian, (d) scaled Winnipeg, (e) scaled Winnipeg age employment shift, (f) school closure four weeks, and (g) school closure eight weeks. Bars in dark gray correspond to the parameter ranges 0.5 p and 0.1 p 2 0.3, and those in light gray correspond to the switched ranges 0.1 p and 0.5 p The error bars represent 95% confidence interval. 20

21 Attack Rates for Specific Locations (Home, School, Work, Other) and Epidemic Lengths Tables A8 and A9 summarize the ranges of attack rates for specific locations in each simulated scenario. These rates correspond to the ratio of the total number of infections in that specific location to the initial number of susceptible individuals. The ranges of epidemic lengths, presented in Table A10, were calculated by measuring the time from the beginning of the simulation (i.e., the introduction of the initial infectious agent) in scenarios where the final size was at least 5% of the population, until the earliest time at which there were no exposed, pre-symptomatic, or infectious agents encountered in the model outputs. Table A8. Range of attack rates for different locations in simulated scenarios with 0.5 p and 0.1 p Simulation Scenarios Home School Work Other Original demographics Age employment shift Egalitarian Scaled Winnipeg Scaled Winnipeg age employment shift School closure four weeks School closure eight weeks

22 Table A9. Range of attack rates for different locations in simulated scenarios with 0.1 p and 0.5 p Simulation Scenarios Home School Work Other Original demographics Age employment shift Egalitarian Scaled Winnipeg e Scaled Winnipeg age employment shift School closure four weeks School closure eight weeks Table A10. Average Epidemic lengths for simulated scenarios, in which the epidemic final size exceeded 5% of the total population. 0.5 p 1 0.7; 0.1 p p 1 0.3; 0.5 p Simulation Scenarios weeks weeks Original demographics 8 12 Age employment shift Egalitarian Scaled Winnipeg Scaled Winnipeg age employment shift School closure 4 weeks School closure 8 weeks References 1. Stephen Eubank, Hasan Guclu, V. S. Anil Kumar, Madhav V. Marathe, Aravind Srinivasan, Zoltán Toroczkai & Nan Wang. Modelling disease outbreaks in realistic urban social networks. Nature 429,

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