Modeling Contact Networks and Infection Transmission in Geographic and Social Space Using GERMS

Abstract

Background: 

Stochastic models of discrete individuals and deterministic models of continuous populations may give different answers to questions about infectious diseases.

Goal: 

Discrete individual model formulations are sought that extend deterministic models of infection transmission systems so that both model forms contribute cooperatively to model-based decision making.

Study Design: 

GERMS models are defined as stochastic processes in continuous time with parameters analogous to those in deterministic models. A GERMS model simulator was developed that insured that the rate of events depended only on the current state of model.

Results: 

The confidence intervals of long-term averages of infection level in simulated GERMS models were shown to contain the deterministic model means.

Conclusion: 

GERMS models provide a convenient framework for testing the sensitivity of model-based decisions to a variety of unrealistic assumptions that are characteristic of differential equation models. GERMS especially facilitates making more realistic assumptions about contact patterns in geographic and social space.

ANDERSON AND GARNETT¹ have discussed various needs for using discrete individual simulations in transmission system models. They emphasized that keeping models simple enhances understandability and theoretical utility but that using models for disease control decisions often requires realism that adds considerable complexity. The key to making a model simple enough to advance understanding is to formulate it so it captures only the essential features of the issue under consideration. The key to using models to help with disease control decisions is an ability to test whether decisions made through the use of models are sensitive to realistic complexities that were left out of the model. We present here a model framework and model simulator called GERMS that responds to both needs by allowing the modeler to flexibly add realistic details to deterministic models in a stochastic framework with discrete individuals.

GERMS was designed to pursue the following objectives: (1) to design surveillance systems that can identify where infection is going undetected by analyzing data from infected individuals on where they met their partners, (2) to design control programs like contact tracing, (3) to determine how network conformation affects population transmission potential, (4) to develop statistics that use data from independently sampled individuals rather than contacttraced individuals to describe network conformation, (5) to develop phylogenetic analyses of agents from individuals in different geographic and social settings to determine network conformations affecting transmission, (6) to investigate the effects of stochastic events like the die-out of infection in populations, (7) to investigate how geographic and social space influence contact network structure, (8) to determine how repeat transmissions during ongoing contacts affect infection levels, (9) to develop microbial risk assessment methods for determining the population effects of factors affecting the risk of transmission such as water treatment or filtration methods, and (10) to analyze how multiple serotypes with cross-reactions affect infection patterns.

For their complete pursuit, these objectives require discrete individual models such as the GERMS modeling framework to be described in this article. Our tactic for describing GERMS is to first describe a differential equation model and its corresponding GERMS model. We then demonstrate that the equilibrium prevalence in the GERMS model fluctuates around the endemic equilibrium of the deterministic model. Finally, we describe ways to elaborate the GERMS model toward greater realism.

We choose as our starting point a model with both single and with monogamously paired individuals. It is similar to models presented by Dietz and Hadeler.² Contacts that transmit infection in such models have specified durations while the contacts that transmit infection in differential equation models of populations of pure individuals have no duration. In this model, individuals sequentially form heterosexual monogamous partnerships of modest duration and infection is only transmitted during these partnerships. Infection has a simple course with no incubation period and no induction of immunity.

The Differential Equation Model Form

As with all differential equation models of populations, the differential equation model form has no individuals or pairs of individuals. It has only continuous population segments that behave similarly to populations of large numbers of individuals or pairs of individuals. Since individuals are identified only as being male or female, and infected or uninfected, there are four differential equations for individuals. Pairs of individuals always have one male and one female. Since both the male and female may be infected or uninfected, there are again four differential equations for pairs of individuals. To begin simply, we assume that both genders occur in equal numbers and have the same potential to form partnerships. We modify the original Dietz and Hadeler model,² however, to make explicit an additive approach to handling differences between genders.

Differential Equations for Unpaired Individuals

EQUATIONS (7)–(10) where, I_rs is the number of individuals of gender r (m or f for male or female) in infection state s (i or u for infected or uninfected), P_mf is the number of partnerships where the males (first subscript) and females (second subscript) have infection status i or u for infected or uninfected, σ is the rate of couple dissolution, λ is a single partnership potential value applicable to everyone in this population, and d is the duration of infection.

In each differential equation for unpaired individuals, the first term represents individuals coming from the breakup of partnerships. The second term represents individuals forming partnerships. It corresponds to the first term in differential equations for partnerships below. The first parentheses in the second term represents an average total partnership potential per gender. This average defines the rate of partnership formation for both males and females and keeps the overall partnership rate of males balanced with that of females. The second parenthesis in the second term shows the fraction of the total rate that corresponds to the infection state of the gender on the left hand side of the equation. Infected individuals revert to the uninfected state via the last term in the equations. Individuals who are not in partnerships have no way of getting infected.

Differential Equations for Partnerships

EQUATIONS (11)–(14) where, h is the transmission probability per sexual contact in a partnership, and φ is the rate of sexual contact in a partnership.

The first term in each of these equations has three consecutive large parentheses and represents the rate that males and females in the corresponding states of infection are forming new partnerships. To emphasize the parallelism between the stochastic and deterministic formulations, the three parentheses have been arranged to correspond to the way GERMS generates partnerships. The first parenthesis shows that the overall rate of new partnership formation in a group of mixing males and females is an arithmetic average of the new partnership potential of males and females. The next two parentheses show that infected and uninfected individuals on each side of the partnership are chosen in proportion to their total partnership potential. The second term represents the breakup of partnerships. The next two terms in the first three equations or the next one term in the last equation represent elimination of infection in partnerships. Such elimination causes the status of a partnership to shift between the four partnership compartments. The final terms in the last three equations represent infection transmission.

The Corresponding GERMS Model

GERMS models have discrete individuals as their basic unit rather than the continuous segments of populations of individuals or pairs of individuals as in differential equation models. Like deterministic differential equation models, GERMS models operate in continuous time. Being stochastic, they model a probability distribution of outcomes rather than a fixed outcome value as deterministic models do. The GERMS simulator is a program that realizes one particular outcome from the distribution of outcomes determined by a GERMS model. In this regard, a GERMS model specifies the cumulative distribution of time to the next event for different types of events and the GERMS simulator draws random numbers that are applied to the cumulative distributions to specify the time of the next event. We will present here an overview of GERMS model specifications. Additional formal mathematical specifications of this model can be found elsewhere.^3,4

A GERMS model specifies:

• a distribution of individuals in geographic and social space by characteristics of those individuals such as their frequency of risk behaviors or presence of risk factors
• a distribution of sites in that same space where links such as sexual partnerships are formed between individuals
• the rules that make individuals available for linkage in bins according to their geographic and social position
• the stochastic process governing the formation of ongoing links such as sexual partnership formation that allow for the transmission of infection
• the stochastic process governing the transmission of infection during ongoing links (transmission cannot occur in the absence of such links)
• the stochastic process governing the duration of different stages of infection with specified degrees of contagiousness
• the stochastic process governing the development and duration of immunity after infection

In the GERMS model corresponding to the above deterministic model, the only events are formation of a partnership, breakup of a partnership, transmission of infection in a partnership, and elimination of infection. The geographic and social location of individuals in that model is at a single social site where individuals of the opposite sex pair randomly.

Two of the stochastic processes are always modeled as “memoryless”-the rate of partnership formation and the rate transmission of infection in partnerships between infected and susceptible individuals. Those rates remain constant no matter how long one has been in a state experiencing those rates. This generates a negative exponential distribution of times to next events and allows the GERMS simulator to add up the rates across all such events so that the total rate at which any instance of the next such event can be used to schedule the next such event. At the time of that next event, the individuals or the pairs of individuals experiencing the event are then selected according to their contributions to the overall rate of any such next event. In practice, the GERMS simulator currently selects, first, the site where a new partnership is to be made and, then, the couple to form the partnership.

The times to partnership breakup and to infection elimination or infection stage progression are not restricted to memoryless processes. They may have any possible gamma distribution. To allow for such distributions, the GERMS simulator does not choose the time of the next partnership breakup and then determine which partnership is to breakup. Instead, at the start of each partnership, a time to breakup of that partnership is selected randomly from the gamma distribution and kept on a list of future events. The same applies to time to infection elimination. In the deterministic model considered above, rates of breakup and infection elimination are memoryless. The GERMS simulator, however, still sets a time for each such possible event.

A key aspect of the GERMS simulator is that the time of the next scheduled partnership formation is adjusted if a partnership breakup event comes before the next scheduled partnership formation event. The adjustment as described in Chick et al³ insures that the rate of partnership formation corresponds to the partnership formation potential of the population at any instant as in the continuous model. The continuous model does not schedule any events because there are no discrete events, only continuous processes. That is like having an infinite rate of events so there is never any time between partnership formation and breakup events. In the discrete model, if a partnership breaks up before a scheduled event, that could increase the total partnership formation potential in a population. If the old scheduled time for partnership formation is not adjusted, the rate of current events depends on past states of the model and creates inconsistencies with the deterministic models.

GERMS models partition the partnership potential of each individual among bins where partnerships are formed. There are two classes of bins: one-sided and two-sided. The analogous continuous model structure to a one-sided bin is a structured mixing formulation.⁵ The continuous model presented above corresponds to a single two-sided bin where individuals make partnerships only with other individuals of the opposite gender and only with individuals who do not already have a partner.

This means that male j has a partnership potential ξ_Mj that equals 0 when he is partnered, and that equals the base partnership propensity λ_Mj when he is not partnered. We use this ξ_Mj notation instead of just λ because it helps later when we consider concurrent rather than just monogamous partnerships. Let ξ_Fk and λ_Mk be analogous quantities for female k. In the deterministic model under consideration λ_Mk = λ_Fk = λ. Because partnership status changes through time, the partnership propensities ξ_Mj and ξ_Fk will also change.

The instantaneous rate of partnership formation between j and k is determined by a mixing rate function r_jk that depends on the partnership propensities and the total numbers of individuals, n_Mj and n_Fk, that are eligible to partner with male j and female k, respectively. The function used in GERMS models is the arithmetic mean of the partnership propensities after normalizing for the number of possible partners. Equation (1) Here, “eligible to partner” means that each individual currently has no partner. The sum of the r_jk over all potential couples (j,k) is the overall rate r of partnership formation. Equation (2)

This formulation does not make the chances that a male will form a partnership with different females proportional to the partnership propensity of those females (or vice versa). It has the nice quality, however, of making the overall rate r of heterosexual partnership formation in the population the average of the total partnership potential of each gender.

Because n_Mj and n_Fk change through time, partnership formation is a nonhomogenous Poisson process.⁶ If the partnership propensities are the same (λ = λ_Mj = λ_Fk) for all individuals in a sequentially monogamous population as in the deterministic model examined, then n_Mj = the number of unpartnered males, n_Fk = the number of unpartnered females, and r = (n_Mj)(n_Fk)λ/2.

GERMS Simulation Output

The user can specify specific summaries of data at specified intervals to be output by GERMS. Alternatively, GERMS will output a file documenting every event in a simulation so that analyses may be performed to explore hypotheses as to why different population patterns were observed. Several output graphs can be produced automatically at the end of a simulation run. In Figure 1 we see the prevalence, presented as a 30-day running average, from a typical run.

Fig. 1: Gonorrhea prevalence (30-day running average) for a simulation with monogamous partnerships in a homogenous population with equal numbers of males and females having identical partnership potential and exponential rate processes.

The simulation that generates Figure 1 is of a closed population (no recruitment or departures) of N = 1000 males and N = 1000 females that have potential only for monogamous relationships in a single bin. All individuals have the same partnership potential ξ = 1/14. Partnerships break up at rate σ = 1/14 days. The rate of contact during a partnership is φ = 3/7 (3 per week), with per-contact transmission probability h = 0.3. Infection duration is exponentially distributed with mean d = 55 days, a value that is reasonable for untreated gonorrhea. We chose these parameters corresponding to a high-risk population in order to provide conditions where infection would not die out of the model population during simulation. Such die-out generates important differences between the behavior of deterministic and stochastic simulations. GERMS models will have the same behavior as deterministic models only when such die-out does not occur. Figure 1 demonstrates the stochastic fluctuations that occur even in a large, homogeneous population that many modelers would feel comfortable treating deterministically. When the model population is more realistically divided into smaller segments rather than assuming homogeneity as in this case, such fluctuations would cause infection to die out from some segments. Given parameters generating lower endemic prevalence of infection, stochastic die-out from the entire population would be likely.

Demonstrating the Correspondence of GERMS and the Deterministic Model

Elsewhere we have mathematically analyzed GERMS models to provide theoretical predictions of endemic infection level.³ We demonstrate here that numerical solution of deterministic differential equation models having analogous parameters provides another means for predicting mean GERMS prevalence values at equilibrium. Mean simulation behavior was determined in the manner presented elsewhere.^3,4 The basic approach is to observe long runs where the trend of values over time is not significant given observations at intervals where there is no detectable dependence between outcomes. The confidence intervals for the estimate of the mean are much narrower than the distributions seen in Figure 1 because they are the confidence intervals for an estimate of the mean rather than for the mean at any point in time. A variety of other parameter values are examined in Table 1. Consistency between the deterministic and stochastic models is seen for all cases.

TABLE 1: Comparison of Stochastic and Deterministic Model Results for a Heterosexual Pairing Model With Constant, Homogeneous Populations and Rates

As can be appreciated from Table 1, the system is markedly sensitive to the duration of infection. A less than 10% change in infection duration resulted in a nearly 35% reduction in prevalence. Such sensitivity is consistent with therapy being an effective way to stop gonorrhea transmission. Kretzschmar and Dietz have commented upon similar sensitivities with regard to HIV transmission.⁷

Adding Realistic Complexities to the Model

The ease of adding realistic complexities to GERMS models as compared to deterministic models makes them especially valuable for testing the sensitivity of model based decisions to model assumptions. The most unrealistic assumptions of transmission models usually relate to contact structure. Differential equation models that do not explicitly model pairs assume a point time contact process that corresponds to setting the duration of partnerships in GERMS models to zero. This is unrealistic for any contact process spreading infection. The sensitivity of decisions based on such models to the point time contact assumption could be evaluated using GERMS models with more realistic durations of contact.

Homogeneity of populations is a common assumption of transmission models. If individuals have multiple characteristics of importance, differential equation models can become unwieldy. This is especially the case for differential equation models with contacts having duration like the pair model presented earlier in this article. GERMS models allow for easy assignment of multiple characteristics to individuals that affect either contact processes, transmission of infection given contact, or the course of infection.

Homogeneity of contact processes or random mixing is another highly unrealistic assumption of most transmission models. This is easily relaxed in GERMS models by putting multiple bins in a model and assigning partnership potential differentially to different bins according to the characteristics of individuals. GERMS makes such assignment easy to relate to observable realistic characteristics by allowing one to assign both bins and individuals to geographic and social space.

Differential equation models of pairing with diverse individual characteristics cannot be properly formulated to allow for arbitrarily high levels of concurrency of partnerships. Nonsexual contact is likely to be characterized by high levels of such concurrency. Sexual transmission has been shown to be sensitive to even low levels of such concurrency.^8–10 GERMS handles concurrency quite simply. For the sake of completeness, we formulate concurrency within the context of a structured population with mixing in different bins.

The parameter f_b,Mj is used to describe the fraction of the partnership propensity for male j allocated to activity setting b. Male j is only allowed to form partnerships in activity setting b if f_b,Mj > 0. The partnership propensity ξ_b,Mj of male j in activity setting b is defined to be EQUATION (3). The λ parameter is now a baseline partnership propensity that is translated into a current partnership propensity by this equation. The θ parameter has a value from zero to one. A value of zero generates only monogamous relationships under the convention that 0⁰ = 1 and 0¹ = 0. Any value greater than zero allows for any number of concurrent partnerships but the frequency of high numbers of concurrent partnerships will be low at low values. At a value of one, having current partners does not slow partnership formation at all. Similar notation, ξ_b,Fk, f_b,Fk, is used for female k.

The rates of partnership formation expressed in equations 1 and 2 are elaborated under individual heterogeneity, mixing heterogeneity, and concurrent partnership formation as follows. The instantaneous rate that male j and female k form a partnership in activity setting b is EQUATION (4) Here, “eligible to partner in b” means that (1) j and k are not already partnered to each other, (2) both j and k can form partnerships in activity setting b, and (3) both j and k either allow for partnership concurrency or currently have no partners. It is convenient to define the indicator variables c_b,jk and e_b,MjEQUATION (15) The rate r_b,j that male j forms a partnership in activity setting b is therefore, Equation (5) The factor e_b,Mj insures that an individual with no eligible partners will not form a partnership, and n_b,Mj drops out of the denominator in the first term on the right side of equation 5 because there are n_b,Mj values of k such that c_b,jk = 1. The probability that individual j is selected to participate in a partnership formed in activity setting b is proportional to r_b,j. The rate r_b of partnership formation in activity setting b is EQUATION (6) The overall partnership formation rate is therefore r = r₁ + … + r_B.

Note that the parameter f_b,Mj describes the fraction of the partnership propensity for male j allocated to activity setting b as if an individual could be in several places at the same time. A more realistic assignment of individuals to different settings at different times is cumbersome and is unlikely to affect the uses to which the model will be put. Changing fractional assignments over time can be used to increase realism when appropriate.

The above formulation allows for relaxing restrictive assumptions in the VESPERS stochastic simulation of infection transmission from the Minnesota Micropopulation Simulation Resource.¹¹ This model assigns individuals to social spaces like families, neighborhoods, schools, etc. It does so, however, by assuming all contacts last for the duration of a simulation rather than being formed and dissolved over time as in GERMS. Such contact structures could be reproduced in GERMS using bins corresponding to the VESPERS contact structures and setting partnership potential very high with θ = 1 and very long durations of partnership. The sensitivity of conclusions based on such a model to assuming contact structures are fixed could be assessed using more realistic settings of partnership duration and θ.

Some of the most restrictive assumptions of transmission models relate to the natural history of infection and immunity. No real infections behave in either the susceptible-infected-removed (SIR) or susceptible-infected-susceptible (SIS) fashion characteristic of so many transmission models. Infections often have complex courses such as the stages of HIV infection that are so influential in transmission dynamics.¹² Likewise, immunity is never absolutely complete and everlasting as in SIR models nor completely absent as in SIS models. The GERMS model simulator allows for stages of infection and immunity with fixed transmission probabilities characteristic of each stage. It also allows for stages of immunity with fixed probabilities of transmission.

Additional Insights With GERMS

To demonstrate the effects of the GERMS concurrency formulation, we examined situations where partnership potential was reduced when one had an ongoing partnership but not eliminated as in the above models of monogamy. We set θ to 0.1. When everyone has this level of concurrency, the total number of partnerships is increased by about 20%. In Table 2 we see that this results in a 67% increase in prevalence. When only 10% of the population have concurrent partnerships, the total population number of partnerships is raised by a little more than 2%. This results in a 19% increase in prevalence when partnerships are made at random and a 22% increase when individuals with concurrency potential make half of their partnerships at sites where only other individuals with concurrency potential are found.

TABLE 2: Endemic Prevalence of Infection Given Various Polygamy Scenarios

Using GERMS to Design Contact Pattern Statistics

The fourth objective listed near the beginning of this article involves using GERMS to design data collection and analysis methods for measuring aspects of contact networks. We now provide an example of how GERMS can be used for program design decisions in the face of realistic complexity that makes the design process problematic.

In nonlinear systems like infectious disease transmission systems, the population risk is not just the sum of individual risks.¹³ The arrangement of individual elements in any system determines its behavior. In transmission systems, the arrangement of contacts at the population level determines population levels of infection. Standard epidemiologic analyses that estimate the frequency of infection in individuals by their risk-factor status cannot detect the effects of contact patterns. In fact, valid interpretations of standard risk-factor effect estimates depend upon the assumption that contact patterns have no effect. We have argued elsewhere that this is a fault in the current epidemiologic paradigm that needs to be addressed by recognizing that contact networks are an implicit dimension of epidemiologic data.¹⁴

Standard epidemiologic data analysis methods arrange individuals in rows with both outcome and predictor variables for each individual arranged in columns. It is evident that standard analyses do not view individuals as part of a system because the row in which an individual is situated (the arrangement of individuals) makes no difference to the results of standard analyses.

Social network analysis is performed in a data plane that intersects the plane of standard epidemiologic analyses.¹⁴ The data are arranged as a square matrix with individuals along both axes. The values in the matrix represent intensities of connection between individuals. Transmission models have their key parameters, namely contact rates and transmission probabilities, in this plane. Transmission systems involve not only the relationships in this plane but the effects of risk factors on individuals with regard to infection pathogenesis. Thus, transmission system analysis and risk assessment for infectious diseases must integrate all these dimensions.

Some epidemiologic data intrinsically relate to the contact network plane and much of the value of such data is lost when they are analyzed in the individual risk plane of standard epidemiologic analysis. Data relating to characteristics of sexual partners or to interactions between individuals are prime examples of this. We believe that data on where sexual partners are met, on the interval between meeting and having sex with partners, and on condom use with partners could be particularly powerful when used in calculating statistics describing contact patterns in a population. The extent to which these variables reflect individual risk need not be correlated with their utility for estimating important parameters of contact networks.

Traditions for studying contact patterns are stronger for sexually transmitted infections than for other infections. The natural concern of sexually transmitted disease (STD) control with contact patterns is seen in the long-standing aphorism that it is more important with whom a risk behavior is practiced than what risk behavior is practiced. The concept of “core” groups driving population transmission dynamics arose from the study of STDs¹⁵ and is widely understood as an effect at the population level that transcends individual risk assessment considerations.

STD investigators have studied contact patterns in two ways. They have performed contact tracing to define individual networks,^16–19 and they have performed population surveys and estimated the rate at which partnerships are made with different types of individuals.^20–23 In the first approach, the data are labeled sociometric because they document links between particular individuals. In the second approach, the data are called egocentric since they are collected only from individuals without a determining who is contacted. Recent egocentric studies demonstrate the feasibility of measuring local aspects of contact networks around individuals and that these affect individual risk of infection.^22,23 These studies, however, do not assess population effects of networks that seem very likely to have greater importance than the individual risk effects studied. Ghani et al²⁴ demonstrated theoretically that sociometric network measures designed for social network analyses have the potential to capture strong determinants of population gonorrhea infection level that are missed by egocentric measures. Recent commentary supports the importance of further work measuring contact patterns.²⁵

Both sociometric and egocentric approaches to network measurement have deficiencies that hold back the development of a science of transmission system analysis. Egocentric data cannot directly describe patterns of connection on important variables, such as the sexual behaviors of partners, because the individuals surveyed do not know this data about their partners. Egocentric data are also limited with regard to the various dimensions of contact patterns that can be described. Such data can generate mixing matrices when needed assumptions are made, but cannot describe actual networks. Sociometric data can describe actual networks, but suffers from logistical constraints. Great effort and cost in contact tracing go into describing only small parts of contact networks at the population level. Even in those small parts of the overall network, however, the description often has critical missing data affecting estimates of the network's potential to sustain infection transmission.

We believe that the problems just mentioned can be overcome by using a new approach that estimates population network parameters from egocentric data. This approach would use egocentric data on variables that reflect interactions between individuals and on variables about partners that a subject can directly observe.

Morris and Kretzschmar²⁶ made a major advance by defining network measures related to partnership concurrency. One of their measures can be calculated from egocentric data. These measures only deal with concurrent links. Defining different degrees of temporal relationships between links could strengthen these measures. Duncan Watts²⁷ has defined some dimensions of contact networks that have strong effects on infection levels in populations even when the population is completely homogeneous. His approach to developing network statistics provides for important degrees of mathematical analysis and could be extended to many other aspects of networks relevant to transmission analysis. GERMS model construction falls very naturally into Watts' framework.

Inspired by Watts²⁷ and Ghani and Garnett,²¹ we seek better network measures through the analysis GERMS models. To provide an empirical platform for developing ideas about what aspects a network measure should have to best reflect population transmission potential, we use GERMS to explore the behavior of different measures under different conditions. To test theories about why measures behave as they do, GERMS can be used as an experimental platform on which computer experiments are designed in pursuit of explanations for the empirical relationships observed.

Network Measures Using Sociometric Data

Our first step is to develop new sociometric network measures using Time Directed Partner Graphs (TDPG). Relationships between individuals rather than individuals themselves constitute the nodes of TDPG. Directed links between these partnership nodes are defined when one individual is common to two nodes (partnerships) and the timing of formation and dissolution of those two relationships meet specified criteria chosen to reflect the potential for infection transmission. We were inspired to treat partnerships as network nodes by the work of Morris and Kretzschmar.²⁶ Our approach differs in that time relationships are used to establish directed connections, while their connections are always bidirectional. Our approach reduces to the Morris and Kretzschmar approach under specified timing rules.

The sociometric data for a TDPG include when a relationship began and ended as well as the members of the relationship. Such data are presented in Figure 2 where the timing data are presented graphically in the third column. In columns 4 and 5 of Figure 2 we have constructed TDPG using rules that are presented after the graph. More refined rules are possible that would make the TDPG correspond even more closely to the chances for chains of transmission. Columns 4 and 5 use a different “δ” parameter corresponding to time between onset of infection and the end of contagiousness in the graph construction.

Fig. 2: Data structure and construction of Time Directed Partner Graphs.

From TDPG it is possible to construct a “source” count for each partnership-the number of previous partnerships that could have originated a chain of infection that could reach the partnership under concern. The source count for an individual is the sum of source counts for each of that individual's past partnerships that could be a source of their infection if they were currently infected. Our plan is to determine the statistics relevant to source counts with the most predictive power.

Network Measure Design Using Egocentric Data

The κ₃ measure from Kretzschmar and Morris is a measure that can be derived from egocentric data consisting of partnership concurrency counts.²⁶ We will explore analogous measures that use source counts rather than concurrency counts. To develop source counts, we will have to use inferences from data on the counts of partners with specific interaction characteristics. These will include place of partnership formation, condom use, and courtship interval. Theory provides an insufficient basis for deciding what data to use or how to use them. There are hundreds of decisions to be made on how to derive source counts from data and how to develop statistics from source counts that have maximal predictive power.

Because network patterns are unique at every instant and because there is insufficient theory to define an optimal classification of network patterns, the development of useful network statistics is highly dependent upon the use of simulations. The GERMS simulator provides an empirical environment where insights can be gained through experience in a way that insights could never be gained in the real world. We will use models in this task the way models should be used for all policy decision-making or design purposes. We will define a set of decisions to be made. The decisions will include things like whether the source counts should be dynamic or cumulative, and whether it is better to use a few precise interaction variables in deriving counts or many different but less precisely measurable variables. Then we will define explicit outcome criteria for making those decisions; for example, we might define measures of the accuracy of predictions across a range of conditions based on the use of different network measures as outcome criteria. Then we will explore the best way to meet these criteria using simple models that include only those elements that we have reason to believe a priori will affect both the network measures and population infection levels. Next a range of model assumptions that might not be realistic and that should be relaxed in a sensitivity analysis will be defined and the appropriate sensitivity analyses performed. Note that we will explore the sensitivity of the design decision, not the sensitivity of model behavior.

Concluding Comments

Stochastic discrete individual models of transmission have three advantages:

• They provide a means to model data collection that continuous models do not.
• They make it possible to model situations that deterministic differential equation models cannot address, such as unlimited potential for concurrent contacts.
• They can address stochastic issues like the chances that infection will die out of a population.

Major disadvantages of discrete individual models have been that they are hard to analyze and validate. While differential equation models may be more tractable in certain situations, GERMS actually has an advantage with regard to mathematical analysis of model structure in some situations. We have demonstrated that the continuous time aspect of GERMS allows for mathematical analysis of equilibrium values that cannot be mathematically analyzed using differential equations.³

GERMS also facilitates two types of validation. First, it helps validate decisions based on model analysis by providing a framework for assessing the sensitivity of those decisions to assumptions intrinsic to the form of the model rather than to just the parameter values of the model. Second, GERMS facilitates validation of simulation code by providing differential equation model outcomes with which mean GERMS model behavior must be consistent.

GERMS facilitates the first validation process by providing a framework for relaxing the following assumptions of differential equation models:

• large population size
• deterministic behavior
• lack of duration of contact in models of population segments
• concurrency in models of populations of pairs of individuals
• patterns of contact or network structure
• distribution of individual characteristics in a population
• natural history of immunity and infection

Another aspect of GERMS that facilitates realistic relaxation of restrictive assumptions is its model formulation within geographic and social structures. Data or intuition about such structures are more likely to be available than are data about network conformation.