- Research
- Open Access
Effects of pathogen dependency in a multi-pathogen infectious disease system including population level heterogeneity – a simulation study
- Abhishek Bakuli^{1, 2}View ORCID ID profile,
- Frank Klawonn^{1, 3},
- André Karch^{2, 4} and
- Rafael Mikolajczyk^{4, 5, 6}Email author
https://doi.org/10.1186/s12976-017-0072-7
© The Author(s). 2017
- Received: 22 September 2017
- Accepted: 22 November 2017
- Published: 13 December 2017
Abstract
Background
Increased computational resources have made individual based models popular for modelling epidemics. They have the advantage of incorporating heterogeneous features, including realistic population structures (like e.g. households). Existing stochastic simulation studies of epidemics, however, have been developed mainly for incorporating single pathogen scenarios although the effect of different pathogens might directly or indirectly (e.g. via contact reductions) effect the spread of each pathogen. The goal of this work was to simulate a stochastic agent based system incorporating the effect of multiple pathogens, accounting for the household based transmission process and the dependency among pathogens.
Methods
With the help of simulations from such a system, we observed the behaviour of the epidemics in different scenarios. The scenarios included different household size distributions, dependency versus independency of pathogens, and also the degree of dependency expressed through household isolation during symptomatic phase of individuals. Generalized additive models were used to model the association between the epidemiological parameters of interest on the variation in the parameter values from the simulation data. All the simulations and statistical analyses were performed using R 3.4.0.
Results
We demonstrated the importance of considering pathogen dependency using two pathogens, and showing the difference when considered independent versus dependent. Additionally for the general scenario with more pathogens, the assumption of dependency among pathogens and the household size distribution in the population cohort was found to be effective in containing the epidemic process. Additionally, populations with larger household sizes reached the epidemic peak faster than societies with smaller household sizes but dependencies among pathogens did not affect this outcome significantly. Larger households had more infections in all population cohort examples considered in our simulations. Increase in household isolation coefficient for pathogen dependency also could control the epidemic process.
Conclusion
Presence of multiple pathogens and their interaction can impact the behaviour of an epidemic across cohorts with different household size distributions. Future household cohort studies identifying multiple pathogens will provide useful data to verify the interaction processes in such an infectious disease system.
Keywords
- Agent based model
- Epidemic
- Household size
- Pathogen dependency
- Multi-pathogen
Background
Respiratory infections are the most common type of infections that contribute to loss of productive time due to acute conditions [1]. Households play an important role for the transmission process of respiratory infective agents, since they serve as confined structures due to the proximity of contacts among individuals that belong to such a confinement [2]. Approximately a third of the influenza like infection transmissions occur within households [3–5]. Studies on modelling epidemics spread in populations distributed into household clusters of varying sizes have been conducted to investigate possible control measures against epidemic outbreaks where larger households were associated with more infection transmissions [6–10].
Individual level stochastic models, also known as agent based models are highly flexible constructs to study complex phenomena by simulating the behaviour of multiple agents (individuals or grouped entities) simultaneously. FluTE [11] and FRED [12] are examples of such agent based models that have been built incorporating the community structure to study the progression of influenza like infections in the population [13–15].
Epidemic studies, till date, have mostly focused on the effect of a single pathogen in determining the population behaviour and spread of infections. Seasonal epidemics of respiratory infections are a common phenomenon during the winter months annually with several emergent and dominant pathogens circulating in the society. Additionally, there is always the possibility for antigenic drifts which are due to mutations of viruses impacting the protective effect of immunity from further infections [16]. Thus there is a need to study the epidemic reality of several pathogens co-existing in the community, with differential seasonality patterns, as well as differential severity and transmissibility characteristics. The idea of dynamic interaction between pathogens or ecological interference has been studied for diseases with differential seasonality in case of measles and whopping cough [17] and for the impact of vaccination for pandemic influenza [18].
The study of the infection process with multiple interacting pathogens has been lacking in the agent based models that have been developed in the past. Infection from one pathogen along with an intervention strategy, like household isolation, can not only have an impact on the individual’s exposure to the specific pathogen but also to other pathogens which can eventually impact parallel epidemic processes from other co-existing pathogens. This involves cross immunity caused by an infectious pathogen, and changes in the contact structure among individuals within and between households. In addition to this, if there are two pathogens with exactly the same characteristics, they create a competition within the scope of the epidemic process. Additional factors like household structure and presence of an immunized proportion of individuals can impact the course of the epidemic since they can potentially accelerate or decelerate the transmission of infections in the population [8–10]. Moreover they are also directly related to the household isolation strategy since they impact the within household transmission. The aim of our study is to investigate how multi-pathogen interaction impacts the epidemic process when compared to scenarios with only a single pathogen if different household structures and the proportion of already immune individuals are taken into account.
Methods
Agent-based modelling of disease transmission
We use an agent-based approach with the basic structure of an SEIR (Susceptible, Exposed, Infectious, and Recovered) model. During the exposed state we assume that individuals are asymptomatic and do not impact the transmission process. After a period of being asymptomatic the individuals enter the infectious phase where they are symptomatic and can transmit infections. The assumption that during the infectious phase, there is household isolation making an individual nullify the risk of external infection from other pathogens, causes the interaction between pathogens in the multi-pathogen setting. The degree of this reduction of the external transmissibility depends on pathogen characteristics.
The transition probability matrix for a single pathogen with the SEIR states
Time = t + 1 | |||||
Susceptible | Exposed | Infectious | Recovered | ||
Susceptible | 1 − TP _{1}(t) | TP _{1}(t) | 0 | 0 | |
Time = t | Exposed | 0 | 1 − TP _{2} | TP _{2} | 0 |
Infectious | 0 | 0 | 1 − TP _{3} | TP _{3} | |
Recovered | 0 | 0 | 0 | 1 |
\( {P}_{external}\left(p,t\right)=v\ s(t)\ z\ \left(\frac{I\left(t-1\right)}{N}+{P}_0\right), \)with \( N=S(t)+E(t)+I(t)+R(t)\ and\ \frac{I\left(t-1\right)}{N}+{P}_0\le 1 \).
Description of the symbols used in the mathematical formulation of the transition probabilities for describing the agent based model
Symbols | Description |
---|---|
N | Total number of individuals in the cohort (10,000 individuals considered as a population cohort) |
S(t); E(t); I(t); R(t) | Number of individuals in the Susceptible, Exposed, Infectious, and Recovered states at time point t |
P _{ external }(t) | Probability of a susceptible individual acquiring infections from contacts in society |
P _{ family }(t) | Probability of a susceptible individual acquiring infections from contacts within household |
v | Baseline infectivity of a given pathogen. Always present in determining the probability of acquiring an infection by a susceptible individual (Fixed at 0.025 for each day) |
(I(t))/N | Proportion of infectious individuals in society at time t. Impacts the probability of susceptible individuals acquiring infections from society at time t + 1 |
P _{0} | Influx of infection from outside of the studied population to avoid permanent extinction of the epidemics (fixed at 0.0001 for a single day) |
z | Pathogen specific reduction factor; Expression of severity of symptoms thus extent of isolation from the society; Multiplicative factor on the sum of the proportion of infectious individuals and the influx of infections from outside the population (Range: 0.3–0.9) |
s(t) | Seasonality parameter at time point t; expression for the seasonal variability in the transmission probability of the infection from contacts at the society level for a specific pathogen |
A | Amplitude for the seasonality characteristics of the pathogen; indicates the extent of seasonal variation of transmissibility of a given pathogen (Range: 0.5–5, lower values indicate lack of seasonality whereas higher values are indicative of seasonality) |
c | Factor for increased closeness of contacts within household as differentiated from the society contacts; Multiplicative factor on the baseline infectivity for determining the within household transmission probability for a specific pathogen (Fixed at 9 for all pathogens) |
Λ | Coefficient for the degree of household isolation. In case of complete pathogen dependency with full household isolation of 100%, risk of acquiring infections from outside household when already infectious is zero. For the independent pathogens scenario, the household isolation is 0% which means that there is a complete risk of acquiring a co-infection from outside household. |
t _{0} | Coefficient for the phase shift. It helps in varying the temporal trend of the pathogen. It is set to zero for most cases. Except for pathogen 10, we examine the case when the value is +/− 45 days and remains zero for the other pathogens |
I _{ h }(t) | Number of infectious persons in the same household at time t; impacts the probability of acquiring an infection from household contacts at time t + 1; exponential factor on the product of baseline infectivity and closeness of contacts |
LP | Length of asymptomatic infection(latency period);Average time spent from being exposed to becoming infectious for a specific pathogen (Range: 1–6) |
IP | Length of symptomatic infections(infectious period); Average time spent in-between becoming infectious and acquiring immunity (Range: 2–9) |
Transmission in the family depends on pathogen characteristics - baseline infectivity v, factor for within family closeness of contacts c and number of infectious persons in the same household I _{ h }(t).
\( {P}_{family}\left(p,t\right)=1-{\left(1-v\ c\right)}^{I_h\left(t-1\right)} \) (Description provided in Table 2.)
Z(t) ∈ {Susceptible, Exposed, Infectious, Recovered} ∀ t ≥ 1, where Z is an individual in the study.
TP _{1} = Probability(Z(t + 1) = Exposed | Z(t) = Susceptible) (Table 1).
=1 − (1 − P _{ external }(p, t)) ∗ (1 − P _{ family }(p, t)) (Table 2).
The probability TP _{1} describes the transition probability from being Susceptible to becoming Exposed. The above formulation includes the specific scenario, when there is no possibility of a family based transmission, which is always the case for a single member household.
Let LP (>0) and IP (>0) (description in Table 2) be the average latency period and infectious period, respectively for a given pathogen p. TP _{ 2 } describes the transition probability of an Exposed individual becoming Infectious for the pathogen it is already exposed to. TP _{ 3 } describes the transition probability for an Infectious individual to obtain immunity or become Recovered for that pathogen for the remaining time in the study period. In this paper, we assume that LP and IP are independent constructs.
X(i, p)~Geometric(q)− After time X(i, p), that is, at time X(i, p) + 1, the i ^{ th } individual becomes Infectious, since the time it became Exposed for pathogen p. (Description in Table 2).
Y(i, p)~Geometric(r)− After time Y(i, p), that is, at time Y(i, p) + 1 the i ^{ th } individual acquires immunity, since the time it became Infectious from the pathogen p, for the remaining study period (Description in Table 2).We also assume that X(i, p) and Y(i, p) are independently distributed as a geometric distribution.
\( Probability\left(\boldsymbol{Z}\left(\boldsymbol{t}+1\right)={Susceptible}^{+}\kern0.1em |\kern0.1em \boldsymbol{Z}\left(\boldsymbol{t}\right)= Susceptible\right)=1\kern0.1em \mathrm{for}\kern0.17em \mathrm{pathogen}\kern0.1em {p}^{\hbox{'}} \) (Fig. 1).
When Z(t) = Infectious for pathogen p and p ≠ p ^{'}.
=(1 − (1 − P _{ family }(p ^{'}, t)) ∗ (1 − (1 − λ)P _{ external }(p ^{'}, t))) ∗ (1 − TP _{3}( p)) (Fig. 2).
Simulation
Population structure
The household size distributions for the different populations considered to describe the epidemic outcomes from simulations using the agent based model
Pathogen characteristics
Pathogen characteristics. This table with the input parameters for the simulation of the agent based model with ten pathogens. I indicates influenza type while C indicates common cold type of pathogen
Pathogen Characteristics | Pathogen Number | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Interpretation | Symbol | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Seasonality | A | 0.5 | 0.5 | 3.0 | 1.5 | 2.0 | 4.0 | 1.0 | 3.0 | 2.5 | 5.0 |
Baseline infectivity | v | 0.025 | 0.025 | 0.025 | 0.025 | 0.025 | 0.025 | 0.025 | 0.025 | 0.025 | 0.025 |
Closeness family to society | c | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
Reduction of contacts with society | z | 0.6 | 0.9 | 0.3 | 0.4 | 0.4 | 0.8 | 0.7 | 0.3 | 0.6 | 0.9 |
Duration of latent period(days) | X | 1.5 | 3.0 | 6.0 | 5.0 | 4.0 | 4.0 | 2.0 | 4.0 | 1.5 | 1.5 |
Duration of infectious period(days) | Y | 4 | 4 | 7 | 3 | 5 | 9 | 3 | 6 | 3 | 4 |
Proportion immune at start | R(1) | 0.50 | 0.20 | 0.20 | 0.25 | 0.50 | 0.15 | 0.30 | 0.20 | 0.20 | 0.15 |
Number infectious at start(per 10,000) | I(1) | 17 | 18 | 71 | 62 | 66 | 52 | 45 | 58 | 06 | 52 |
Number of infections from outside (per 10,000) | P _{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Pathogen type | I | C | C | C | C | I | I | C | I | C |
Computation
The simulation proceeded in discrete time steps. Each step denoted a day in the follow up period. Based on the initial number of Infectious individuals, the epidemic process began its course of action. It followed the seasonal trend of the pathogen, the relation to other household members, and the prevalence of the infection for the specific pathogen in society at a given time point. We started initially with two pathogens from Table 4 (Pathogen 6 and Pathogen 10). Pathogen 6 would be in accordance with the characteristics of a pandemic influenza strain whereas pathogen 10 would correspond to the characteristics of human respiratory syncytial virus (HRSV). Then we observed the scenario where both the pathogens jointly interact. Finally we looked at the general scenario with 10 pathogens jointly which would be a more appropriate representation of the reality during the winter season [18] (https://grippeweb.rki.de).
The comparisons were done for the scenarios of pathogen dependencies, 1) assuming all pathogens existed independently (λ = 0%), and 2) assuming the pathogens worked together and influenced each other (λ = 100%) (indicated by Pathogen Dependency- Yes or No); household size distribution based on different household size distributions in different countries (Country – Germany, India or Hypothetical). At the start of the simulation few people were infectious for every pathogen, which was denoted as I(1) to kick start the infection process, while the number of people already immune at the start were represented as R(1).
In addition to the above, we assumed that, for every pathogen there would be a small chance that an individual could acquire an infection from outside the system. This has been described as the external influx of infection. We had set this value to one in a ten thousand, at each observational time point (day) in the epidemic process. Besides this, the maximum number of days spent as infectious had been censored to 55 days. Each of the scenario combinations were replicated 100 times for a study period of 150 days in the peak season for respiratory illness.
In our base case scenario with the German population and pathogen dependency (λ = 100%), we look at the same temporal tend (seasonality) for all the pathogens considered. However to present the effect of differential seasonality, we introduce a different temporal trend for pathogen 10, by modifying the value of t _{0} as +45 days and −45 days. This results in shifting the peak of the epidemic for these pathogens and impacts the overall epidemic process when multiple pathogens are present. Also for this scenario we evaluated the effect of change in λ from 0% to 100% in steps of 10% which would allow us to infer on the importance of the pathogen dependency assumption through the introduction of household isolation.
Statistical methods
We measure the epidemiological parameters of interest which are 1) height of the epidemic peak (peak prevalence), 2) time taken to reach the peak of the epidemic, 3) incidence proportion (attack rate) of infections in the study period, and 4) incidence proportion stratified by household size for the different populations in consideration, through our simulations as described above. Summary statistics are presented for all the outcomes described above.
We observe the peak prevalence and the incidence proportion for the pathogens 6 and 10, both individually and jointly. We are interested in the hypothesis that jointly modelling pathogens creates a competition, and hence we would observe lower values of the peak prevalence and incidence proportion, compared to observing them individually. The observations are compared using the non-parametric Mann–Whitney–Wilcoxon test for evaluating the difference when observing joint epidemics. The parametric version with the paired t-test also gives us similar results, however due to no necessity of normal distribution assumptions the Mann–Whitney–Wilcoxon test values are reported [24].
We also use a simple linear regression model [25] on the outcomes described above and show the confidence intervals for the slope across different outcomes to indicate the impact of pathogen dependency on the country variable (used to describe the different household size distribution) in the scenario with 10 pathogens. The covariate used is the coefficient for the degree of household isolation (0 to describe the independent scenario and 1 to describe complete household isolation in case of the dependency). The confidence intervals show the variability in the slopes across different country variables indicating different household size distributions. For studying the degree of household isolation, we use the Generalized Additive Model (GAM). GAM’s are an extension of the generalized linear model (GLM) allowing for some kind of smoothing of the predictor variables. The advantage of GAMs is that it allows us to deal with highly non-linear and non-monotonic relationships between the response and the predictor variables often driven by the observed data at hand [26, 27]. GAMs are also used in this work to model the dependency of incidence proportion of infections stratified by household size where a non-linear relationship is observed.
Results
Summary and comparison of two pathogen system (S2) vs. one pathogen system (S1). The pathogen is indicated in the parenthesis. S1(P6 + P10) indicates the sum of the individual values from the pathogen independently whereas S2(P6 + P10) indicates the system where the household isolation introduces pathogen dependency and the pathogens function jointly. The outcomes of peak prevalence and incidence proportion (during the 150 day period) along with their 95% confidence intervals (based on Monte-Carlo simulations) are shown in the summary section. The comparison section displays the non-parametric p values (based on the Mann-Whitney-Wilcoxon test) obtained when comparing the pathogen systems over the simulation runs
Peak prevalence | Incidence proportion | |||||
---|---|---|---|---|---|---|
Hypothetical | India | Germany | Hypothetical | India | Germany | |
S1 (P10) | 0.0006 (0.0004,0.001) | 0.078 (0.072,0.084) | 0.019 (0.015,0.025) | 0.005 (0.002,0.009) | 0.617 (0.586,0.640) | 0.318 (0.276,0.357) |
S1 (P6) | 0.006 (003,0.011) | 0.179 (0.170,0.187) | 0.096 (0.089,0.104) | 0.061 (0.021,0.102) | 0.763 (0.753,0.771) | 0.654 (0.640,0.669) |
S2 (P10) | 0.0006 (0.0003,0.001) | 0.070 (0.065,0.079) | 0.014 (0.009,0.021) | 0.004 (0.002,0.009) | 0.589 (0.564,0.612) | 0.279 (0.198,0.334) |
S2 (P6) | 0.006 (0.003,0.010) | 0.169 (0.162,0.177) | 0.094 (0.087,0.101) | 0.055 (0.026,0.099) | 0.761 (0.750,0.771) | 0.652 (0.636,0.667) |
S2 (P6 + P10) | 0.006 (0.003,0.011) | 0.211 (0.198,0.225) | 0.105 (0.098,0.113) | 0.061 (0.030,0.102) | 1.351 (1.318,1.378) | 0.931 (0.857,0.983) |
S1 (P6 + P10) | 0.007 (0.003,0.012) | 0.258 (0.246,0.267) | 0.116 (0.104,0.125) | 0.068 (0.026,0.107) | 1.383 (1.341,1.410) | 0.970 (0.933,1.015) |
Comparison | ||||||
S1(P10) vs. S2(P10) | 0.590 | <0.001 | <0.001 | 0.543 | <0.001 | <0.001 |
S1(P6) vs. S2(P6) | 0.471 | <0.001 | 0.029 | 0.350 | 0.029 | 0.134 |
S1(P6 + P10) vs. S2(P6 + P10) | 0.175 | <0.001 | <0.001 | 0.672 | <0.001 | <0.001 |
Comparison of slopes across the different country locations. This indicates the observed difference in the outcomes from the epidemics due to the differences in the coefficient of household isolation (the extreme scenarios of complete dependency versus pathogens functioning independently) and the household size distribution in the country location used as shown in Fig. 3 (3.1, 3.2)
Hypothetical | India | Germany | |
---|---|---|---|
Peak Prevalence | (−0.0006,0.0008) | (−0.015,-0.011) | (−0.006,-0.003) |
Time to reach peak prevalence | (−22.542, 4.102) | (1.252, 2.427) | (−0.932,1.732) |
Incidence proportion | (−0.008,0.006) | (−0.079, −0.060) | (−0.081,-0.056) |
Discussion
We have proposed an agent based model to study the behaviour of epidemics under the influence of multiple pathogens working simultaneously in the population. With the presence of two pathogens in such a system without the influence of any other effect, we could demonstrate how the interference of the pathogens in the infection process played a role in controlling the epidemic process (lower number of infected individuals as well as lower daily incidence proportion). The interference among pathogens was introduced through the assumption of household isolation during the period of being symptomatically infectious, where the individual was immune to the risk of acquiring infections from outside the household. To our knowledge this was the first time for studying the behaviour of an epidemic process incorporating the influence of multiple pathogens using an agent based model. We further went on to present a more general scenario where there are 10 pathogens, and also the impact from recovered individuals being present in the population at the start of the epidemic process.
Our simulations were performed to study the impact of the dependency among pathogens as opposed to pathogens functioning independently (2 extreme levels for the coefficient of household isolation during the infectious period), and the household size distributions of different populations (three different populations with varying household size distributions) The population system reached a stable state at the end of simulation period, confirming that the epidemic had almost died out in 150 days (approximately 5 months during winter season). The dependencies among pathogens were important determinants in controlling the epidemic process. Additionally, the household size distributions did produce significant differences in the peak of the epidemic (peak prevalence) and the incidence proportion in the study period of interest. For common respiratory infections like influenza and common cold, household size can be an important factor determining their spread as seen for influenza or influenza like illnesses based hospitalizations: the population structure difference has accounted for a third of the observed variation [28]. In our simulations we observed that household size distribution influences the speed of the epidemic. Population with larger household sizes reached the peak of the epidemic much faster than those with smaller household sizes. Looking at the incidence of infections across household sizes, we could see that larger households were associated with more infections due to the intra household infection spread, consistently with assumed random mixing within the household.
Looking across the different pathogens, we observed that the infectious period also is important in shaping the severity of the epidemic. Pathogen 6 and 10 as considered in the simulation have almost similar characteristics except for the duration of the infectious phase, but this resulted in different severity of the epidemic. Also in the multipathogen scenario, the epidemic characteristics are dominated by pathogen 10. Shifting of the temporality to introduce a peak 45 days before for pathogen 10 allows for more infections in the multipathogen system as opposed to a delay in the peak.
Our simulation study does come with limitations. There are common challenges associated with agent based models, especially in statistical methods for hypothesis testing in combination with determining the number of appropriate simulation runs [29]. In addition to the standard challenges, our assumptions are largely simplistic in nature, assuming for random mixing within the household and the population is looked upon as an assortment of homogenous agents. The increase in contacts with the increasing household size may not necessarily take place. Secondly, we induce a sort of isolation for the transmission process of infections, but we do not account for the specific severity of the infections, except for the duration of being symptomatic and infectious. The severity of the pathogen can directly influence the duration of isolation. Even for our sensitivity analysis, we assume this parameter to be same for all the pathogens. Additionally, we also assume same transmissibility characteristics for all the pathogens. These are strong assumptions that have been made for the realization of the system in a simplistic way. However, this model can be extended easily to observe more complex realizations of a realistic system.
Conclusion
Through our agent based model formulation, we could demonstrate the importance of considering the multi-pathogen interactions in controlling the spread of infections during an epidemic process. Household size and dependency among pathogens are important factors in determining the outcome of the epidemic. Future prospective studies in household cohorts looking at pathogen identification and coinfections can provide quantitative measures for specific characteristics of the multi-pathogen system. This kind of data can also be used to test the validity of the assumptions made in simulation models.
Declarations
Acknowledgments
We thank all the group members of the Epidemiological and Statistical Methods research group at HZI, Braunschweig for their valuable feedback and discussions.
Funding
Internal funding of the Helmholtz Centre for Infection Research.
Availability of data and materials
The R code for the simulations supporting the conclusions of this article is available from the corresponding author upon request.
Authors’ contributions
RM and FK conceived the idea of the simulation study. All authors contributed to the theoretical development of the model. AB programmed the simulations and statistical analyses. AB, FK made contributions to the statistical analyses and simulations. AB drafted the manuscript. All authors contributed to the interpretation of the data, writing, and revising of the manuscript and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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