Overview - Pennsylvania State University
Disease Dynamics Algorithm Synopsis
Overview
The algorithms described in the following sections aim to explore the relationships between disease dynamics and the underlying social dynamics of the contact network on which the disease spreads. By simulating a population that grows and changes according to salient real-world statistical rates (for New York City, circa 1998 [1]) and that is, itself, a conglomerate of family groups, working groups, school groups, and individuals, we have the ability to examine the topological properties of the contact network at any given time during the spread of a disease, which provides us unique insight into how and why the social topology of the contact network influences the propagation of the disease through the population.
Algorithms
The full algorithm consists of two subsets: social processes and disease processes. In general, the social processes are updated on a yearly basis, while the disease processes are updated on a weekly basis (i.e. 52 times for each social update); however, with a few exceptions, updating of the social and disease processes is done simultaneously and synchronously for all individuals in the population. Although adjustments could be made on a by-disease basis, for computational ease, all social processes that do not directly and/or immediately affect connectivity among children (i.e. marriages, formation of/changes to workplaces) are updated in advance of the disease update in the current version of the algorithm. The justification for this protocol in the case in question (measles) is that aggregation of adults is unlikely to have a significant impact on the ability of measles to spread through the population, since most adults are immune to the disease [2, 3].
The basic process of spread on the contact network follows the SIR model with age-defined transmission rates (really the product of a transmission probability and a contact rate). The disease is assumed to pass from infected individuals to susceptible individuals via social links. For measles, the infectious period is assumed to be two weeks.
Social Processes Algorithms
• Birth: Women between the ages of 15 and 45 are eligible to have children, and the number of children born to women of each age i in this range is determined according to a rate per 1000 women of age i. The week at which each baby will be added to the population is randomly chosen, as are the women who will become new mothers.
• Death: The number of people of age i who will die in the current year is determined according to a rate per 1000 people of age i. Individuals are randomly selected for death (unless their age exceeds 95, in which case they are automatically removed from the population), and the week of their removal is randomly determined.
• Age Distribution and Aging: The initial age distribution was adapted from [1], and has been linearized in order to determine values for each age between 0 and 95 (Figure 1). The age of each individual in the population is incremented yearly. For simplicity, all individuals age simultaneously.
Figure 1: Population distribution by age in number per 1000.
[pic]
• Immigration: The immigration rates were adapted from [4], and have been linearized in order to find a rate for each age between 0 and 95. The number of people of age i who will be added to (or subtracted from) the population at time t is equal to the product of the number of people of age i-1 at time t-1 and the immigration rate for age i:
[pic].
The rates of [4] have been further adapted to satisfy the condition that the age distribution is stable, although the total population is able to grow or shrink (current inputs yield annual growth of roughly 0.6%); i.e. the ratio of the number of people moving into age group i from age group i-1, minus the number of people moving into age group i+1 from age group i, minus the number of people of age i who die in the current year, plus (minus) the number of people who immigrate to (from) age group i in the current year to the total population (divided by 1000) is a constant for each age group i:
[pic].
In order to make these rates applicable to many large populations (and not solely to New York City), we found a set of characteristic rates by imposing a threshold of .01 on the derived values; rates whose absolute value was less than .01 were assumed to be 0. This cutoff has the effect of limiting in-migration to age groups i=20 and older. We see predominantly in-migration of individuals in their 20’s and early 30’s (possibly young professionals moving to a large city for employment), out-migration of individuals in their late 30’s through their late 60’s (possibly more (financially) well-established people moving to the suburbs—also reflected by an out-migration of (their) children, aged 11-15), and finally an in-migration of the very elderly (possibly people moving into retirement homes in the city proper). It is required that individuals under the age of 18 leave the population with a family group, and to the extent that it is possible to satisfy the immigration-by-age distribution, entire family groups are moved out of the population if any family member is randomly chosen as an emigrant. The week at which an individual will enter or leave the population is randomly determined; however, if the individual is moving with a family, his or her entire family group will move at the same time.
• Marriages and Families: Approximately 54% of the population over the age of 18 will be paired with a person of the opposite gender at any given time. Marriage is not necessarily correlated with breeding in this simulation, since single women can have children. Family groups are fully connected and remain so until the children turn 18, at which point only the mother and father will remain connected. Because it occurs very infrequently, children whose parents die are not reassigned to new families. The average family size is 3-4.
• School Groups: All children between the ages of 6 and 18 are included in the school subnetwork. This subnetwork consists of fully-connected, age-specific classes of maximum size equal to 50, that are, in turn, interconnected randomly (from a truncated normal distribution) according to the following rule: children between the ages of 6 and 13 can have a maximum of 0.5*(size of their class) connections to other classes in this age range; children between the ages of 14 and 18 can have a maximum of 0.75*(size of their class) connections to other classes in this age range; all children have at least .25*(size of their class) connections to other classes. Each year, school edges are severed for an eight week period and are then reestablished, to simulate summer vacation and the beginning of a new school year (i.e. school-term forcing).
• Workplaces: Individuals have the option to enter the workforce at age 18, but must exit from the workforce at age 65. Each year, approximately 8% of current 18-year olds will remain unemployed, and all others will enter the workforce, if college has not been selected as an option for the population. If college has been selected as an option for the population approximately 28% of people between the ages of 18 and 25 will be in college at any given time, and therefore, 18-year olds will be added to maintain the colleges’ capacities; of the remaining 18-year olds, 8% will be unemployed, and the rest will enter the workforce. The initial number of workplaces is approximately 1% of the total population size, and is allowed to grow over time. Connectivity in the workplace is such that if there are more than 6 people in the initial workplace, the initial workplace is connected as a Barabasi-Albert (BA) graph with seed size 7 (where the seed begins as a graph of 7 nodes and no edges), and if the initial size is greater than 3, but less than 7, the initial workplace is connected as a BA graph with seed size equal to the workplace size. As new workers are added to the workplaces, they are attached randomly to a minimum of three and a maximum of all other workers in the workplace.
• Colleges (optional): Colleges are an optional addition to the social network simulation, and are absent from the current round of measles simulations because of their (likely) small impact on the spread of measles through the population. However, the option exists to choose the number of colleges to include in the population, and from there, to establish subnetworks whose size will be equal to the total population size, divided by (100*number of colleges). Colleges are always maximally filled with individuals between the ages of 18 and 25 and individuals are randomly connected to between 10% and 50% of the other students in their college.
Disease Processes Algorithms
• Initial Immunity (Susceptibility) by Age: The immunity profile (by age) (Table 1) was adapted from [2] and is a set of input parameters at time t=0. The susceptibility of immigrants (i.e. of in-migrants over the age of 20, since there are no in-migrants under the age of 20) is assumed to be higher (3.5%) than it is in the native population (1%). This immigrant-susceptibility percentage is a number that will likely have to be determined on a by-disease basis, since it was derived from measles-specific statistics. First, we find the fraction of immigrants due to internal migration (to New York City), and the fraction due to immigration from each of the (historical) measles infector regions (Western Europe, Japan, and Africa). We assume that the internal migrants and international migrants from non-infector regions will have the same susceptibility as the native population (by age). To find the susceptible fraction of migrants from each of the infector regions, we multiply the fraction of migrants coming from each region by the vaccine coverage in that region. The aggregate susceptibility of migrants (internal, international from non-infector regions, international from infector regions) is 3.5%.
Table 1: Immunity profile by age.
|Age |Fraction Immune |
|0 |0 |
|1 |.033 |
|2 |.133 |
|3 |.266 |
|4 |.4 |
|5 |.533 |
|6 |.65 |
|7 |.71 |
|8 |.78 |
|9 |.85 |
|10 |.9 |
|11-95 |.99 |
• Loss of Maternally-Acquired Immunity: All newborns are assumed to be immune to the disease for (in the case of measles) up to six months after the week of birth. The specific week during this nine month period at which maternally-acquired immunity will be lost is randomly chosen from a Gaussian distribution with mean equal to 12 weeks and sigma equal to two weeks.
• Sparking: Because computational restrictions limit us to, at most, Type II populations (100, 000-300,000), it is often impossible to achieve true endemism of the disease in the simulated population. Therefore, to ensure that there is always some chance of infection, we introduce a sparking process, whereby, with some rate (which is chosen as an input parameter), susceptible individuals will become infectious without contact with an already infected individual in the population. The implicit assumption is that the “spark” has traveled outside of the native population, or has otherwise had contact with an infected individual from outside of the native population, has become infected, and has introduced the disease into the native population. The spark is chosen at random from all current susceptibles in the population.
• Force of Infection by Age: The age-specific mixing and transmission (WAIFW) matrix, β, was derived from the force of infection values by age group (Table 2), the aggregate proportion of the infection by age group (Table 2), and the mixing matrix given in [5]:
[pic].
Although [5] reports values for pertussis, the force of infection profile by age is very similar to that of measles, peaking in the 5-10-year old age class. We find the entries of the mixing matrix by solving the equation, [pic], where [pic] is the fraction of individuals in age class j who are infected (when endemism is reached) , [pic] is the force of infection (per year), [pic] is the mixing and transmission coefficient (per week) . If N is the total population and [pic] is the fraction of total infections constituted by infection in age class j, and if [pic] is the number of susceptible individuals in age class j (where [pic] is a rate per 1000 total), then [pic] can be expressed as:
[pic]. The values for [pic] were obtained from [5] and are given in Table 2; the values for [pic] were obtained from [1] and are also given in Table 2.
Table 2: By age group: number of susceptibles (at endemism) per 1000 ([pic]), aggregate proportion of infected individuals ([pic]), fraction of the group that is infected ([pic]), force of infection ([pic], per year), mixing and transmission coefficient ([pic], per week). The population has been divided into five age groups (j=1…5): [pic].
|Age Class (j) |[pic] |[pic] |[pic] |[pic] |[pic] |
|1 |0.00070352 |.66 |0.093813964 |.22 |0.030253 |
|2 |0.0007044 |.3 |0.042589438 |.48 |0.129717 |
|3 |0.00066649 |.025 |0.003750977 |.18 |0.024653 |
|4 |0.00049767 |.005 |0.001004688 |.04 |0.005444 |
|5 |0.00742792 |.01 |0.000134627 |.045 |0.006125 |
Topological Features of the Social Network
• Total Cumulative Degree Distribution: The total cumulative degree distribution covers a range from k=0 (isolated individuals) to [pic] (school children with large families), irrespective of graph size, and can be partitioned into two regions (Figure 3). The distribution is dominated by nodes for which k10 is only about 20%. The largely uniform region between k=22 and k=50 is due predominantly to school connectivity and secondarily to work connectivity and will be discussed in greater detail in later sections. The region of the cumulative distribution to the right of k=50 is separated from the region to the left by a segment that becomes increasingly switch-like as population size grows. This segment is a product of school connectivity and will be discussed in more detail below.
Figure 3: Cumulative total degree distribution. The probability, P(K>k), is plotted for each degree in an averaged set of typical simulated networks of size 100, 000 and 10,000. The distributions are virtually identical, except for the slope of the transition region between k~40 and k~55.
[pic]
• Cumulative Family Degree Distribution: Family degrees are exponentially distributed ([pic]) between 1 and 10 (Figure 4). The probability of having a family degree greater than 3 is extremely small (k), is plotted for each degree in an averaged set of typical simulated networks. P(K>0) is plotted as a reference.
[pic]
• Cumulative School Degree Distribution: The cumulative school degree distribution is switch-like in appearance for large populations (100, 000), but decreases more gradually for smaller populations (Figure 5), and can be superimposed on the transition region between 4012 is small (k), is plotted for each degree in an averaged set of typical simulated networks.
[pic]
Disease Dynamics
• Susceptibility Over Time: Over time, the fraction of the social network that is susceptible to infection will either grow or fluctuate around its mean starting value (~5%) , depending on the size of the network, on the rate of sparking, and on the transmission (and recovery) rates being used. In general, if the transmission rates are fixed at the values reported earlier, increasing the rates of sparking will counteract the effects of immigration of susceptibles and “disease bypassing”, whereby an individual progresses through school (the primary years of infectivity) without contracting the infection. Since immigration and disease bypassing both tend to deposit susceptibles into the adult age group where transmission rates are low, these processes can result in an accumulation of older susceptibles in the population, and thus in a gradual rise in the susceptible fraction of the population.
• Infection Profiles as a Function of Social Graph Size: While, for all graph sizes, the number of infections on a time scale less than or equal to the infectious period of the disease varies greatly from one simulation to the next (Figure 7a), the aggregate number of infections, when binned in one- or two-month intervals, tends to show increasingly deterministic behaviour both as the size of the population grows and as the rate of sparking increases. For example, at the same sparking rate (~30 sparks/year), it is likely that the infection profiles associated with a population of 100,000 individuals will show much less variability both in the temporal patterns of epidemics, as well as in the average size of epidemics, than will the profiles associated with a population 10 times smaller (Figure 7b,c).
Figure 7 : Infection profiles (4 each) for a population of (a) 100,000 individuals on a weekly basis, and (b,c) 100,000 and 10,000 individuals, respectively, on a two-month basis. Random sparking has occurred approximately 30 times per year in each simulation. The interepidemic time ranges from ~1 year to ~3 years for both graph sizes
(a)
[pic]
(b)
[pic]
(c)
[pic]
The greater uniformity in the (aggregate) infection profiles of the larger population stems from two key features of this population’s topology: (1) it has a much higher abundance of full-capacity schools (as is indicated by the switch-like behaviour of its cumulative school-degree distribution at k~50), and (2) the average inter-class degree of this population is high ( ~12). The large intra- and inter-class degrees work synergistically to synchronize infection in multiple classes within the larger network. If the infection makes its way into a class in the large network, it will quickly establish itself within that class, but in doing so, a large number of children in other classes will also be infected in rapid succession via the abundance of inter-class links emanating from the class in which the infection nucleated. Although the number of individuals infected in the nucleus class of a large population may be exactly the same as the number of individuals infected in the nucleus class of a small population, the size of the subsequent epidemic and the patterns inherent to other epidemics in each population will be significantly different because of the difference in class size, and therefore, in the average number of inter-class links.
• “School-Hopping”: The fact that rapid spread in the larger population occurs as the disease “hops” from one school to another across the plentiful inter-school links is further evidenced by a comparison of the number of schools infected each week in a large population to the number infected on a weekly basis in a small population (Figure 8): the number of schools infected simultaneously in the large population dwarfs the number of simultaneously-infected schools in the smaller population.
Figure 8: A comparison of the number of schools infected on a weekly basis in a population of 10,000 individuals and a population of 100,000, over a period of ~8.5 years.
[pic]
Furthermore, Figure 9 demonstrates that the greater number of infected schools at each time step in the larger population is truly indicative of a more systemic infection of the population, since the number of infected schools, as a fraction of the total number of schools at each time point, is generally higher for the large population than it is for the small population—in some cases, substantially so.
Figure 9: Number of infected schools as a fraction of the total number of schools in the population. This is the same data as is depicted in Figure 8, but it has been binned into two month intervals, and the aggregate data has been divided by the average number of schools over the two month period.
[pic]
• Subpopulation (subgraph) Infections: In addition to being grouped by membership in a work, family, or school group, individuals in the simulated population can be further classified based on combinations of their social factors—namely, their ages and the specific types of social links they possess. Thus, there are either eight or seven nonintersecting subpopulations (subgraphs) to which an individual can belong, depending on whether or not the simulation incorporates colleges: individuals under the age of 6 are classified as preschoolers (psch); those people between the ages of 6 and 18 are classified as being in school (isch); individuals can simultaneously be parents and in the work subnetwork (iwhk) or they can be in the work subnetwork without children (iwnk); individuals can be nonworkers with children (niwhk), or they can be nonworkers without children (niwnk); individuals can attend college (college); or if an individual does not fall into any of the previously described categories, he or she will be classified as other (other). With these classifications, it is possible to track the disease as it spreads through different social and demographic strata of the society, and thus to gauge the role each subpopulation plays in the spread of the disease. As can be seen in Figure 10 a,b, the disease is (not surprisingly) concentrated in school children (Figure 10a), and the epidemic peaks in this subpopulation are echoed by peaks in the preschool subpopulation (Figure 10b). This is likely the effect of infected school children passing the infection to their preschool-aged siblings via family links.
Figure 10: Infection profiles for (a) school (isch) and (b) all other subpopulations in a population of 100,000 individuals with an average sparking rate of ~40 sparks/yr.
(a)
[pic]
(b)
[pic]
Although large epidemics are almost always triggered by the direct sparking of school-aged child, or by the sparking of a preschooler who then infects a school-aged sibling, occasionally, a parent (usually an immigrant) will set off an epidemic when he or she acts as a spark and subsequently infects his or her school-aged children. Although this is a rare effect, it is noteworthy, since it demonstrates a possible side-effect of immigration, and since it illustrates one way in which even childhood diseases are not entirely limited to a “childhood arena”.
• Effects of School-Term Forcing: Although it is always the case that severing school links will prevent or curtail an epidemic (for measles), it is not necessary for school links to be severed—i.e. it does not appear necessary to have school term forcing—to ensure that the disease does not burn through the entire susceptible population at one time. Figure 11 shows cases in which epidemics have been curtailed by the onset of summer vacation (red circles), seen as a sudden drop in infection mass from an epidemic peak to a value near zero, but it also illustrates natural fadeouts of the disease in which the disease has simply exhausted its local supply of susceptibles, given the current topology of the social network (blue circles). This type of local exhaustion can occur, for example, if the disease spreads quickly and easily through classes of children, aged 5 through 9, but then fails to gain significant footing in populations of older children. In such situations, however, the disease has not exhausted the complete supply of susceptibles within the population; at any given time, between 5% and 15% of the total population will likely still remain susceptible. Additionally, Figure 12, for which school ties are never severed, still displays an interepidemic period of between 1 year and 3 years.
Figure 11: Aggregate infection profile for 100,000 individual population. Examples of epidemics curtailed by the onset of summer vacation are circled in red; examples of naturally diminishing epidemics are circled in blue.
[pic]
Figure 12: Aggregate infection profile for 100,000 individual population, without school-term forcing. In this case, school edges are always present. The 1 year to 3 year interepidemic period is still present.
[pic]
Discussion
• Comparison to Pre-Vaccination Measles Data for England and Wales: In populations of roughly 100,000 people, actual case reports indicate that measles epidemics occur either annually or biennially (Figure 13a), and typically last for between 30 weeks and one year. Epidemics, which generally peak around 750 newly infected individuals in a given month, and which rarely peak below 100 newly infected individuals in a given month, may occasionally, though rarely, be followed by epochs during which the disease is extinct. In populations ten times smaller, however, case reports demonstrate that epidemics occur sporadically, as many as three to four years and as few as one year apart, with frequent fadeouts (Figure 13c). Typically, epidemics in populations of 10,000 individuals peak between 75 and 125 newly infected individuals per month, lasting between 25 weeks and one year. Figure 13a also indicates that the TSIR model nicely captures the sizes, durations, and frequencies of epidemics in the population of 100, 000 individuals. However, as can be seen in Figure 13c, the TSIR model somewhat underestimates both the size and frequency of large epidemics in the population of 10,000 individuals. This is perhaps an area in which the relative strength of the network model lies: the network simulations are able to capture the general size, duration, and frequency (with proper tuning of the sparking rate) not only for populations of 100, 000 individuals, but also for populations 10 times smaller (Figures 13b,d). Perhaps this is to be expected, since the heart of the network model is heterogeneity in contacts-- differences that become amplified as populations shrink, and that therefore make the application of fully-mixed models invalid at small population sizes.
Figure 13: (a,c) Reported (red circles) and TSIR model predictions (black) for measles cases in two English cities—one with a population of roughly 100,000 and the other with a population of roughly 10,000—for the pre-vaccination period 1944-1964 [3]. In (a), the y-axis is in units of thousands. (b,d) Epidemic simulations for a population of 100,000 (b) and 10,000 (d); the units of the y-axis in (b) are again, thousands.
(a) (b)
[pic][pic]
(c) (d)
[pic][pic]
References
[1] 'Vital Statistics of New York State 1998 Tables', in 'Information for a Healthy New York, Health, N. Y. S. D. o., Ed., 1998.
[2] Edmunds, W. J., Gay, N. J., Kretzschmar, M., and Pebody, R. G.: 'The pre-vaccination epidemiology of measles, mumpes and rubella in Europe: implications for modelling studies', Epidemiology and Infection, 2000, 125, pp. 635-650.
[3] Grenfell, B. T., Bjornstad, O. N., and Finkenstadt, B. F.: 'Dynamics of measles epidemics: scaling noise, determinism, and predictability with the TSIR model', Ecological Monographs, 2002, 72, pp. 185-202.
[4] Scardamalia, R.: 'The Face of New York-- The Numbers', 2001.
[5] Grenfell, B. T. and Anderson, R. M.: 'Pertussis in England and Wales: An Investigation of Transmission Dynamics and Control by Mass Vaccination', Proceedings of the Royal Society of London. Series B, Biological Sciences, 1989, 236, pp. 213-252.
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