Searching for Superspreaders: Identifying Epidemic Patterns Associated with Superspreading Events in Stochastic Models
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Abstract
The importance of host transmissibility in disease emergence has been demonstrated in historical and recent pandemics that involve infectious individuals, known as superspreaders, who are capable of transmitting the infection to a large number of susceptible individuals. To investigate the impact of superspreaders on epidemic dynamics, we formulate deterministic and stochastic models that incorporate differences in superspreaders versus nonsuperspreaders. In particular, continuous-time Markov chain models are used to investigate epidemic features associated with the presence of superspreaders in a population. We parameterize the models for two case studies, Middle East respiratory syndrome (MERS) and Ebola. Through mathematical analysis and numerical simulations, we find that the probability of outbreaks increases and time to outbreaks decreases as the prevalence of superspreaders increases in the population. In particular, as disease outbreaks occur more rapidly and more frequently when initiated by superspreaders, our results emphasize the need for expeditious public health interventions.
Keywords
Superspreader Host heterogeneity Middle East respiratory syndrome Ebola Deterministic model Stochastic model1 Background
The prevalence of most infectious diseases is often assumed to emerge from person-to-person interactions among a population of individuals who are considered homogeneous with respect to contact, transmission, and recovery behavior. However, it is more realistic to assume that diseases spread among a heterogeneous population. Host heterogeneity may be due to physiological, behavioral, or immunological differences [33]. Behavioral differences may also be related to the environmental setting [33]. For example, some individuals are at a higher risk for spreading the disease due to increased contact with susceptible persons or longer length of infection. This has been observed in the spread of sexually transmitted and vector-borne diseases, where high-risk individuals are characterized by the “20/80” rule, in which 20% of the infected individuals are responsible for 80% of the disease transmission [21, 41]. The 2002–2003 SARS epidemic highlighted the role of superspreaders (SS), defined as people who infect a large number of individuals, in comparison to nonsuperspreaders (NS) who transmit the disease to few or none [6, 15, 30, 40]. However, the exact characteristics of SS and their impact on disease dynamics are difficult to define. Lloyd et al. studied the effects of heterogeneity in infectiousness and then found that the proportion of SS contributed to high levels of heterogeneity for several infectious diseases (e.g., SARS, measles, influenza, rubella, smallpox, Ebola, and other diseases) [26]. Currently, there are no well-known methods for identifying SS in the population or control efforts to reduce the disease transmission at the individual or population levels based on SS. We consider two infectious diseases, Ebola and Middle East respiratory syndrome (MERS), that are associated with certain cultural and health behaviors for which contact patterns may be traceable. Focusing on these two epidemic cases, we will provide insight into disease patterns associated with superspreading events.
Ebola virus was first discovered in 1976 in Africa, in the country now named the Democratic Republic of the Congo, near the Ebola river. Ebola virus can persist in the environment through animal-to-animal transmission, e.g., bats can transmit the virus to apes, monkeys, antelopes, and other animals. The virus can also be transmitted to humans through contact with infected animals in the environment during hunting, meat preparation, or from an animal bite. Infection can be transmitted to other humans through contact with bodily fluids, such as blood, secretions, and organs of sick or diseased individuals, or with contaminated objects, such as bedding and clothes. According to the World Health Organization (WHO), the 2014–2016 Ebola outbreak in West Africa had the most cases and deaths of any Ebola outbreak to date [38]. This spread might have been increased due to infected health-care workers’ close contact with susceptible individuals. Additionally, burial ceremonies may increase contact with infectious deceased bodies that contain the virus. The incubation period, defined as the time of infection to onset of symptoms, ranges from 2 to 21 days [38]. Individuals can recover from Ebola; however, mortality rates range from 25 to 90%. In 2016, the WHO announced that the first vaccine trial implemented in Guinea was 100% effective [17, 37]. The recent preventive measures announced by the Centers for Disease Control and Prevention (CDC) include: reducing contacts with infected animals or bodily fluids of infected individuals, isolating infected and deceased individuals, early detection of infected individuals, and maintaining a clean environment [8].
MERS was first identified in 2012 from an outbreak that occurred in Saudi Arabia [40]. The source of infection was identified as dromedary camels. However, most cases are not due to camel-to-human infections. MERS outbreaks among humans arise from human-to-human interactions, where many cases occur in healthcare settings with poor health prevention and control practices. In 2015, an outbreak of MERS in South Korea was driven by three SS, initiated with one SS contracting MERS during international travel. The first SS was responsible for 29 secondary infections through various clinical visits. Two subsequently infected individuals were responsible for 106 tertiary infections [39, 40]. Individuals infected with MERS can be asymptomatic, while others may experience the following symptoms: fever, coughs, shortness of breath, diarrhea, and pneumonia. Nearly, 35% of MERS cases resulted in death. While no vaccine or treatments are available, individuals are advised to maintain good hygiene when coming into contact with animals, particularly camels, such as washing hands and avoiding contact with sick animals. Additional prevention strategies include consuming thoroughly cooked and prepared animal products [39].
Mathematical models formulated for recent outbreaks of MERS and Ebola have applied the compartmental setting with various disease stages such as susceptible, exposed, infectious, and recovered (SEIR) or performed statistical analyses to identify important parameters in spread of the disease ([11, 23, 24] MERS and [4, 10, 16, 22] Ebola). Additional classes for asymptomatic, hospitalized, or isolated individuals were also included in MERS models [11, 23]. Time-dependent transmission parameters accounted for superspreading events (e.g., [22, 23, 24]). Superspreading events have also been investigated with multitype branching processes by including individual heterogeneity in offspring generating functions [18, 26]. All of these models have contributed to a better understanding of the role of superspreaders in disease outbreaks. Our models incorporate the compartmental framework and apply stochastic simulations with theory from branching processes to further elucidate the role of superspreaders in disease dynamics.
In this investigation, we develop a mathematical modeling framework that incorporates the heterogeneity of hosts through differences in transmission rates to assess the role of SS in disease spread at the population level. Specifically, we aim to study the disease dynamics in a heterogeneous population consisting of SS and NS individuals, and develop a deterministic model based on ordinary differential equations (ODEs) which is expanded to a stochastic model that is implemented as a continuous-time Markov chain (CTMC) system and approximated by a multitype branching process [1, 2]. We incorporate estimated parameter values from published data of prior MERS and Ebola epidemics into our models. Next, we compute the basic reproduction number for the ODE model, and perform sensitivity analysis using Latin hypercube sampling and partial rank correlation. By varying the initial size of SS and model parameters of the CTMC model, we derive and verify analytical estimates obtained using multitype branching process approximations with model simulations to predict the probability of an epidemic outbreak. In further numerical simulations of the CTMC model, we compute sample paths, probability of outbreak, number of deaths, time to outbreak, time to peak infection, and peak number of infectious individuals. Our analyses and numerical simulations reveal how SS influence the dynamics of epidemic outbreaks, which may provide useful insight for public health interventions.
2 Deterministic Model
Description of the variables used in the ODE model (Fig. 1) and in the CTMC model
Variable | Description |
---|---|
S _{ i} | Number of NS/SS Susceptible individuals |
E _{ i} | Number of NS/SS Exposed individuals |
A _{ i} | Number of NS/SS Asymptomatic individuals |
I _{ i} | Number of NS/SS Infected individuals |
R _{ i} | Number of NS/SS Recovered individuals |
Description of the parameters used in the ODE model (Fig. 1) and in the CTMC model
MERS | Ebola | ||||
---|---|---|---|---|---|
Parameter | Baseline | Range | Baseline | Range | |
β _{1} | Transmission rate NS | 0.06 [31] | (0.04, 0.08) | 0.128 [10] | (0.04, 0.38) |
β _{2} | Transmission rate SS | 0.6931^{∗∗} | (0.4, 0.8) | 1.0150^{∗∗} | (0.5, 1.2) |
\(\alpha _i^{-1}\) | Latent period | 6.3 [12] | (2–8) | 10 [10] | (7, 14) |
\(\delta _i^{-1}\) | Duration of asymp. stage | 0.4 [12] | (0.1, 2) | 10^{−4} [10] | (9.9, 100) × 10^{−5} |
μ _{ ji} | Disease-induced death rate | 0.08 [7] | (0.02, 0.14) | 0.09 [9] | (0.075, 0.125) |
γ _{ i} | Recovery rate | 0.075 [19] | (0.05, 0.1) | (0.04, 0.1) |
2.1 Basic Reproduction Number
3 Markov Chain Model
If the number of hosts/pathogens is sufficiently small, an ODE model is not appropriate. To that end, we utilize a continuous-time Markov chain (CTMC) model, which is continuous in time and discrete in the state space, to study the variability at the initiation of an outbreak, in time to outbreak, and in the peak level of infection. For simplicity, we use the same notation for the state variables as in the ODE model. In particular, time t ∈ [0, ∞) and the states are discrete random variables, e.g., S_{i}, E_{i}, A_{i}, I_{i}, R_{i} ∈{0, 1, 2, …}. The Markov property implies that the future states of the stochastic process only depend on the current states. In particular, there is an exponential waiting time between events.
State transitions and rates for the CTMC model with Poisson rates r_{i} Δt + o( Δt)
Event | Description (i = 1, 2) | Transition | Rate, r_{i} |
---|---|---|---|
1, 2 | Infection of S_{i} | S_{i} → S_{i} − 1 | \(\frac {S_i}{N}\sum _{k=1}^2\beta _k(A_k+I_k)\) |
E_{i} → E_{i} + 1 | |||
3, 4 | Transition to A_{i} | E_{i} → E_{i} − 1 | α _{ i} E _{ i} |
A_{i} → A_{i} + 1 | |||
5, 6 | Transition to I_{i} | A_{i} → A_{i} − 1 | δ _{ i} A _{ i} |
I_{i} → I_{i} + 1 | |||
7, 8 | Death of A_{i} | A_{i} → A_{i} − 1 | \(\mu _{A_i}A_i\) |
9, 10 | Death of I_{i} | I_{i} → I_{i} − 1 | \(\mu _{I_i}I_i\) |
11, 12 | Recovery of I_{i} | I_{i} → I_{i} − 1 | γ _{ i} I _{ i} |
R_{i} → R_{i} + 1 |
3.1 Branching Process Approximation
The theory of multitype (Galton–Watson) branching processes has a long history (e.g., [2, 13, 36] and references therein). It has been used to approximate the dynamics of the CTMC model near the DFE and the stochastic threshold for a disease outbreak [1, 2, 3, 36]. In fact, the stochastic threshold (i.e., probability of a disease outbreak) is directly related to the basic reproduction number as defined in the corresponding deterministic model (2.1) (see [3, 36]). More specifically, if the basic reproduction is less than unity, then disease extinction occurs with probability one. In this case, the branching process is called subcritical. However, if the basic reproduction number is greater than unity, the probability of disease extinction is less than one (probability of outbreak is greater than zero) and the process is referred to as supercritical.
The expectation matrix M = (m_{ij}) can be shown to be directly related to the basic reproduction number [3] with \(m_{ij}=\frac {\partial f_j}{\partial x_i}|{ }_{\boldsymbol {X}=\boldsymbol {1}}\). We include this calculation in Appendix 2. It is known that the spectral radius of M, denoted as ρ(M), determines whether the disease extinction probability is equal to or less than the unity [2, 3, 13]. Specifically, if ρ(M) < 1, q_{1} = ⋯ = q_{6} = 1, then the extinction probability is one; if ρ(M) > 1, then there exists a unique fixed point (q_{1}, ⋯ , q_{6}) ∈ (0, 1)^{6}, and hence the extinction probability is strictly less than one. By the Threshold Theorem of reference [3], it follows that the spectral radius of the matrix M is strictly less than one if and only if the basic reproduction number is strictly less than one. Analogous statements hold whenever the spectral radius of M is equal to one or is strictly greater than one.
4 Parameter Sensitivity Analysis
We perform a sensitivity analysis on the parameters ranges given in Table 2 for the ODE models for MERS and Ebola using a uniform distribution for the values. Latin hypercube sampling (LHS), first developed by McKay et al. [29], with the statistical sensitivity measure partial rank correlation coefficient (PRCC), performs a sensitivity analysis that explores a defined parameter space of the model. The parameter space considered is defined by the parameter intervals depicted in Table 2. Rather than simply exploring one parameter at a time with other parameters held fixed at baseline values, the LHS/PRCC sensitivity analysis method globally explores multidimensional parameter space. LHS is a stratified Monte Carlo sampling without replacement technique that allows an unbiased estimate of the average model output with limited samples. The PRCC sensitivity analysis technique works well for parameters that have a nonlinear and monotonic relationship with the output measure. PRCC shows how the output measure is influenced by changes in a specific parameter value when the linear effects of other parameter values are removed. The PRCC values were calculated as Spearman (rank) partial correlations using the partialcorr function in MATLAB 2016. Their significances, uncorrelated p-values, were also determined. The PRCC values vary between −1 and 1, where negative values indicate that the parameter is inversely proportional to the output measure. Following Marino et al. [27], we performed a z-test on transformed PRCC values to rank significant model parameters in terms of relative sensitivity. According to the z-test, parameters with larger magnitude PRCC values had a stronger effect on the output measures.
5 CTMC Analysis
For the CTMC model, we numerically simulate sample paths to compute the probability of an outbreak, number of deaths, time to outbreak, time to peak infection, and peak number of infectious individuals. For sample paths and probability of outbreak, we compare our results with the deterministic model. In the remainder of this analysis, we assume that the initial total population size is N(0) = 2000. Reference to infected individuals will imply the variables I_{1} and I_{2}, unless stated otherwise. For example, peak number of infectious individuals refers to the maximum value of I_{1} + I_{2} and the time to peak infection refers to the time t at which this maximum occurs. However, an outbreak means that the total number in classes E_{i}, A_{i}, and I_{i} for both NS and SS has reached at least 50, i.e., \(\sum (E_i+A_i+I_i)\geq 50\). In addition, we note that for the CTMC model, an outcome measure (e.g., peak values, time to peak, and number of deaths) is defined by a corresponding probability distribution and a sample path yields one outcome from the distribution.
5.1 Sample Paths
5.2 Probability of Outbreak
We note a negative correlation between the proportion of SS in the S class and the probability of outbreak (Fig. 4) and attribute this to the fact that the value of β_{2} is varied in order to maintain a constant value of \(\mathcal {R}_0\) (MERS, \(\mathcal R_0=2.5\) and Ebola, \(\mathcal R_0=2.39\)). In other words, as the fraction of SS susceptible individuals is increased, the value of β_{2} decreases and results in a reduction in the probability of outbreak (1 − q_{6}). Results in Fig. 4 are shown only for q_{3} and q_{6} since these outputs are similar to q_{1} and q_{4}, respectively. We also note that q_{1} = q_{2} and q_{4} = q_{5} and therefore exclude those plots as well.
As expected, the probability of an outbreak is dependent on the initial fraction of the population that is infected, with an increasing chance of an outbreak as the number of initially infected individuals increases (Fig. 4). Furthermore, the probability of outbreak is significantly enhanced when the initially infected population is composed of SS rather than NS individuals. We also find a strong agreement between the probability of outbreak predicted by stochastic simulations of the CTMC model and the associated branching process approximations for all of these analyses (Fig. 4a–f).
5.3 Number of Deaths
Utilizing our stochastic model of MERS and Ebola dynamics within a population of individuals, we next sought to investigate whether the presence of SS individuals within the population could be reflected in key metrics that capture the severity of disease outbreak: the number of deaths, time to disease outbreak, probability of outbreak, time to peak number of infections, and the peak number of infectious individuals.
We first assess the impact of SS individuals on the number of deaths that accumulate over a 150-day time frame following disease initiation. We observe a modest increase in the frequency of deaths as the size of the susceptible SS class of individuals is increased from 5 to 50% of the total population for both MERS and Ebola disease simulations (not shown). We note a higher frequency of epidemics with lower numbers of deaths when the fraction of SS individuals in the susceptible fraction is lower (not shown).
5.4 Time to Outbreak
We also find a clear separation between the time to outbreak of an epidemic initiated by a fraction of SS versus NS infected individuals. Mean differences were significantly distinct for each percentage in (Fig. 8a–b), p < 0.001. In fact, if a MERS or Ebola outbreak is initiated by 1.5% or more of the initial population size and these individuals are SS, then the time to outbreak is predicted to be no more than 20 days where an outbreak is defined as 2.5% of the population becoming infected (Fig. 8c–f).
5.5 Time to Peak Infection and Peak Number of Infectious Individuals
We repeated the same analysis to assess mean differences in the peak number of infections. Surprisingly, we did not find a significant difference between the peak number of infections for epidemics initiated from a single infected NS versus SS individual for Ebola (Fig. 9d). However, significant differences were observed when 5% or 50% of the susceptible population was SS for MERS.
6 Discussion
In this investigation, we capture the dynamics of MERS and Ebola epidemics by applying both deterministic and stochastic modeling strategies. To investigate the role of SS on the epidemic dynamics and to compare our results, we keep the \(\mathcal {R}_0\) constant for both MERS and Ebola while varying β_{2}, the transmission rate of SS. Parameter sensitivity analysis, using Latin hypercube sampling and partial rank correlation coefficient, shows that β_{2} has a significant effect on all the output measures (Fig. 2).
From Fig. 4, we can conclude that the stochastic model simulations agree with the branching process analytical results. As the value of \(\mathcal {R}_{0}\) increases, we observe that the probability of an outbreak increases for both diseases. This result is expected since more individuals in the population are infected. The probability of an outbreak is greater for Ebola than MERS, which is due to the transmission parameters for Ebola being larger than MERS. Furthermore, these results show that if the outbreak is initiated by an SS, then the probability of an outbreak is significantly higher. Additionally, fewer SS individuals than NS individuals are sufficient to cause an outbreak irrespective of the disease (MERS or Ebola).
As an outbreak initiated by SS has a greater probability of occurrence and peaks earlier than with NS, the accumulated number of deaths is more severe in an epidemic initiated with the same proportion of SS than NS (Figs. 5 and 6). Disease severity (number of deaths) for both MERS and Ebola occurs earlier with SS than NS. Our findings agree with prior epidemiological studies on superspreading events [15, 32, 40]. For example, the 2003 outbreak of the respiratory infection SARS in Beijing found that SS had higher mortality rates, higher attack rates, and greater number of contacts in comparison to NS [40]. From a public health perspective, as SS events will be observed more frequently, intervention/prevention methods must have rapid response to reduce disease severity. For example, Wong et al. [40] suggested that several community-based efforts could have been made to reduce the number of MERS and Ebola cases in Guinea and Sierra Leone, such as tracking contacts, earlier diagnosis, treatment strategies, and community education. Effective responses to control superspreading events and reduce disease transmission in MERS and Ebola outbreaks included: “early discovery, diagnosis, intervention, and quarantine of confirmed cases.” [40]. Other epidemics that are more likely to occur in hospital settings, e.g., SARS, could be controlled through hospital administrative strategies, such as reducing contact between the infected patient and healthcare workers, visitors, or other patients whose immune system may be comprised due to other infections [32]. Thus, a rapid response is needed to reduce disease severity of SS events.
Evident in Figs. 7 and 8, when an outbreak is initiated by an SS rather than an NS, the time to outbreak is shorter and has less variability. Therefore, if the number of disease cases rises rapidly, there may be SS in the community. In this scenario, healthcare managers should search for potential SS. Similar results apply for time to peak infection, Fig. 9. If peak infection occurs quickly, it is more likely that there is an SS in the population.
Interestingly, varying the percentage of SS in the population has little influence on the peak number of infections (Fig. 9c, d). This is likely due to the fact that the \(\mathcal {R}_0\) values are held constant.
7 Future Work
We have formulated, analyzed, and numerically simulated deterministic and stochastic epidemic models that include heterogeneity in transmission for NS and SS. We applied our models to emerging and re-emerging infectious diseases, MERS and Ebola, where the models were parameterized with data from the literature but with a fixed initial population size of 2000. There are a number of extensions and generalizations that we will consider in the future work. We assumed homogeneous mixing and only two types of classifications of individuals (NS/SS) for the entire population. Generalizing this model to include heterogeneous mixing and spatial components are key features that can provide insight on how a superspreaders can be classified. In our model, we considered inter-host variability, which naturally leads to constructing a model with intra-host variability utilizing stochastic differential equations or other types of models. In addition, variability of the pathogen on epidemic dynamics can be explored. Additionally, we will validate our models’ findings against time series data, test our models’ abilities to detect the presence of SS, and interpret the results for public health implementation. Finding answers to these problems will lead to our ultimate goal of constructing novel ways to quantify, characterize, and identify an SS during the initiation of an outbreak.
Notes
Acknowledgements
The work described in this chapter was initiated during the Association for Women in Mathematics collaborative workshop Women Advancing Mathematical Biology (WAMB) hosted by the Mathematical Biosciences Institute (MBI) at Ohio State University in April 2017. Funding for the workshop was provided by MBI, NSF ADVANCE “Career Advancement for Women Through Research-Focused Networks” (NSF-HRD 1500481), Society for Mathematical Biology, and Microsoft Research. We give special thanks to the WAMB organizers: Ami Radunskaya, Rebecca Segal, and Blerta Shtylla. Anarina L. Murillo acknowledges that this work has been supported in part by the grant T32DK062710 from the National Institute of Diabetes and Digestive and Kidney Diseases and grant T32HL072757 from the National Heart, Lung, and Blood Institute. Omar Saucedo acknowledges that this research has been supported in part by the MBI and the grant NSF-DMS 1440386. Nika Shakiba is the recipient of the NSERC Vanier Canada Graduate Scholarship. This work was partially supported by a grant from the Simons Foundation (#317047 to Xueying Wang). In addition, we thank Texas Tech University for hosting our second WAMB group meeting and the Paul Whitfield Horn Professorship of Linda JS Allen for providing financial support. We thank the two anonymous reviewers for their helpful suggestions on the original manuscript.
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