Effect of stochasticity on coinfection dynamics of respiratory viruses
Respiratory viral infections are a leading cause of mortality worldwide. As many as 40% of patients hospitalized with influenza-like illness are reported to be infected with more than one type of virus. However, it is not clear whether these infections are more severe than single viral infections. Mathematical models can be used to help us understand the dynamics of respiratory viral coinfections and their impact on the severity of the illness. Most models of viral infections use ordinary differential equations (ODE) that reproduce the average behavior of the infection, however, they might be inaccurate in predicting certain events because of the stochastic nature of viral replication cycle. Stochastic simulations of single virus infections have shown that there is an extinction probability that depends on the size of the initial viral inoculum and parameters that describe virus-cell interactions. Thus the coinfection dynamics predicted by the ODE might be difficult to observe in reality.
In this work, a continuous-time Markov chain (CTMC) model is formulated to investigate probabilistic outcomes of coinfections. This CTMC model is based on our previous coinfection model, expressed in terms of a system of ordinary differential equations. Using the Gillespie method for stochastic simulation, we examine whether stochastic effects early in the infection can alter which virus dominates the infection.
We derive extinction probabilities for each virus individually as well as for the infection as a whole. We find that unlike the prediction of the ODE model, for similar initial growth rates stochasticity allows for a slower growing virus to out-compete a faster growing virus.
KeywordsViral coinfection Respiratory virus Within-host model Continuous-time Markov chain Multi-type branching process Extinction probability
Continuous time Markov chain
Defective interfering particle
Egg infectious dose
Human immunodeficiency virus
Influenza A virus
Influenza B virus
Ordinary differential equation
Plaque forming unit
Respiratory syncytial virus
Tissue culture infectious dose
With the advent of molecular diagnostic techniques, respiratory tract specimens from patients with influenza-like illness (ILI) are now being recognized as having multiple viruses [1, 2, 3, 4]. Around 40% of the hospitalized patients with ILI have coinfections with influenza A virus (IAV), influenza B virus (IBV), respiratory syncytial virus (RSV), human rhinovirus (hRV), adenovirus (AdV), human enterovirus (hEV), human metapneumovirus (hMPV), coronavirus (CoV), parainfluenza virus (PIV), human bocavirus (hBoV) and many others [5, 6, 7, 8, 9]. These patients are reported to suffer from heterogeneous disease outcomes such as enhanced [10, 11, 12], reduced [13, 14] and unaltered [14, 15, 16] severity compared to patients with single virus infections. However, it is not clear how the virus-virus and virus-host interactions influence disease severity and lead to these varied outcomes. Two or more virus agents can interact in diverse ways which may arise from the consequences of their inoculation order, inter exposure time, initial inoculums, different combinations of viruses, number of coinfecting viruses and host immune state [17, 18]. Thus, coinfections pose a combinatorial problem which can be challenging to study in a laboratory set up alone.
Coinfection can be better understood using mathematical modeling. While mathematical modeling of single virus infections at the cellular level has proven crucial for finding answers where laboratory experiments are impossible, impractical or expensive [19, 20, 21, 22, 23], little has been done in viral coinfection modeling. A few studies [24, 25, 26] have used within host models considering interactions of two different strains of the same virus. Among them, Pinilla et al.  and Petrie et al.  used their models to study competitive mixed-infection experiments of pandemic A/H1N1 influenza with its H275Y mutant strain and Simeonov et al.  considered a spatio-temporal model to explain in vitro cellular susceptibility due to the simultaneous presence of RSV A2 and RSV B. Pinky and Dobrovolny  proposed a two virus coinfection model to investigate viral interference observed in an experimental study of IAV-RSV coinfection (Shinjoh et al. ) where they concluded that distinct viruses interact through resource competition. In further investigations [29, 30], they used the model to quantify the impact of resource availability, finding the possibility of chronic single infection if constant cellular regeneration was considered and chronic coinfection if both cellular regeneration and superinfection were considered. However, the majority of the two virus models studied so far have focused on the deterministic approach that reproduces the average behavior of infection kinetics. The exceptions are Dobrovolny et al.  and Deecke et al.  who investigated two strains of the same virus (wild-type and drug resistant mutant) using a stochastic model to determine mechanisms driving the emergence of drug-resistant mutants during the course of a single infection. Since in real life viral infections are stochastic and discrete events, stochastic simulations of infection models will provide further insight into coinfection dynamics.
For example, stochastic simulations of single virus infections have shown that there is an extinction probability that depends on the size of the initial viral inoculum and parameters that describe virus-cell interactions . Similarly, experimental studies of viral infections in animals have shown that viruses do not always establish infection in every animal under study . Although the causative phenomenon is still unidentified, there are some possible factors suggested by researchers such as host defense mechanisms, spatial heterogeneity in the target cell population, and the stochastic nature of the virus life cycle . Moreover, evaluation of this quantity can be useful in many situations where the viral dynamics cannot be explained with a simple deterministic model. Numerous stochastic models have been developed to study various aspects of the single viral infection process such as virus release strategies (i.e. budding and bursting) for HIV [33, 34], impact of initial viral dose , length of eclipse and infectious phases [33, 34], impact of the immune response [34, 35], and how ongoing proliferation of immune cells acts to decrease the emergence probability of mutated strains . These models have been studied using Monte Carlo simulations of the multi-type branching process [37, 38], or by simulating solutions to stochastic differential equations where processes involved in the virus life cycle are diffusion processes (stochasticity is represented by noise terms in the equations) [35, 39, 40].
Of particular interest for stochastic models is the probability of extinction, a feature that ODE models cannot capture. In stochastic models, analytic expression of extinction probability is formulated by keeping track of the number of infected cells , the number of virions  and both  in single virus models, mostly for HIV infection. Yan et al.  used a similar method to calculate the extinction probability that includes time dependent immune responses in a single influenza virus model. Stochastic extinction could be a factor in coinfection dynamics since one virus could have a higher extinction probability, even if the two viruses have the same initial viral inoculum or initial growth rate, making it possible for one virus to go extinct while the other viral infection grows. Thus the coinfection outcomes predicted by the ODE model might be difficult to observe in reality.
In this work, we implement a stochastic counterpart of our previously published ODE coinfection model , in the form of a continuous-time Markov chain (CTMC) model. Trajectories for the CTMC model are simulated using Gillespie’s tau-leap algorithm. In order to investigate how stochastic effects early in the infection impact coinfection, we vary the initial growth rate and compare to predictions from the ODE model. We also derive the extinction coefficient analytically for the model using multi-type branching method. While the ODE model found that the virus with a higher growth rate consumes more target cells and produces higher peak viral load compared to the slower growing virus, we find that stochasticity can allow slower growing viruses to consume more target cells and produce more virus than the faster growing virus.
Derivation of extinction coefficient
Stochastic dynamics of identical viruses
While the probability of virus extinction is an important feature of stochastic models, we are also interested in understanding if stochasticity affects the predicted dynamics of coinfections that survive. Previously in our ODE model , we found that the virus with the higher growth rate always out-competes the slower growing virus. While ODEs can give us the average behaviors of the coinfection process, in real systems the biological processes are stochastic. The randomness associated with births and deaths during the initial infection process may lead to virus extinction even in an exponentially growing virus population . Yan et al.  reported that the invasion of viral infection is dependent on the initial viral dose and growth rate of each virus. Here, we are interested in knowing how the coinfection dynamics change with change in growth rates of each virus. First, we will observe the dynamics of coinfection with identical viruses.
Stochastic dynamics for different viruses
The time of viral peak also shows some differences between the ODE and CTMC models. For the ODE model, the time of viral peak is similar for both viruses when the relative viral production rate is greater than 100, although the time of peak decreases as the relative viral production rate increases. This is because the viral production rate of virus 1 is increased over its baseline value causing an earlier time of peak. This drives the earlier time of peak of virus 2, which is the weaker competitor in this case. The decline in time of viral peak is not as sharp in the CTMC model since stochasticity can temper the effect of the increased production rate of virus 2 by allowing virus 1 to still have an opportunity to infect some cells.
Finally, we compare the predicted duration of coinfection variation for ODE and stochastic models (Fig. 5 (bottom row)). Viruses do not coexist for more than about a week in either model. The longest coinfection durations are seen, for both models, when the two viruses have the same growth rates. This is because the faster growing virus out-competes the slower growing virus leading to short infections for the slower growing virus.
The dynamics of coinfection were previously modeled deterministically in several studies [24, 25, 29]. However, ODE models do not capture the very earliest dynamics of infection where stochastic effects may play an important role. The stochastic model presented here indicates that stochastic effects can dramatically alter the time course of the infection. Our previous ODE coinfection model  could not distinguish between two identical/similar viruses, as the predicted time courses are identical. Simulations of the stochastic model, however, indicate that for a particular realization of the model, two identical viruses can have very different time courses, with ∼12% of infections initiated with two viruses resulting in infections with only one detectable virus. When viruses have different growth rates, the ODE model predicts that the virus with the higher growth rate will have a higher peak viral titer. This is not the case for the CTMC where early stochastic effects can allow a slower growing virus to infect more target cells than the faster growing virus, giving the slower virus a competitive advantage that continues over the course of the infection.
The ODE coinfection model resulted in a simple rule for determining which virus would be dominant in a coinfection — the virus with a higher growth rate. Replication of the slower-growing virus is suppressed due to lack of accessibility to target cells. This simple rule suggests that we can easily determine which viruses will be suppressed in coinfections. For example, application of the ODE model to several respiratory viruses indicated that parainfluenza virus (PIV) replication is substantially reduced during coinfection with other respiratory viruses , suggesting that it should be difficult to detect PIV in coinfections. However, PIV is detected in coinfections from 30–80% of the time [15, 47, 48, 49, 50]. Some of this unexpectedly high detection rate might be due to stochasticity. PIV detection in coinfection is, however, lower than what is observed for two identical viruses as described in the previous paragraph. The slow growth rate of PIV means that most viruses will out-compete PIV more often than viruses with identical growth rates.
Stochasticity also impacts our ability to use viral interference as a possible mechanism for treating or preventing more serious infections. If we cannot guarantee that a fast-growing virus will suppress growth of a slow-growing virus, then this strategy might be risky. For example, some have suggested using defective interfering particles (DIPs) as a possible method for blocking infections [51, 52, 53, 54, 55]. DIPs cannot replicate on their own, but have a high growth rate when fully-functioning virus is present. Our results indicate that even when there is a large difference in the viral growth rate, there is a non-zero probability that the slower-growing virus (in this case the fully-functional virus) will rise to a higher peak than the faster growing virus, suggesting that use of DIPs for treatment will not be completely effective.
While our extension of the simple coinfection model has provided insight into how stochasticity might affect coinfections, this simple model does not capture all biological processes during the infection. More complex ODE models that include cell regeneration  and superinfection  have been proposed and reproduce a broader range of behaviors observed during viral coinfections. Stochastic versions of these models can also be developed in the future to examine how stochasticity affects behaviors such as chronic coinfections. Other limitations include the lack of an explicit immune response, which will likely increase the probability of extinction of the coinfection , and the inclusion of realistic delays to account for intracellular replication . Despite these shortcomings, this stochastic implementation of a viral coinfection model has shown the extent of variability in the time course of coinfections when stochasticity is introduced.
While ODE models are useful for giving a broad picture of possible dynamical behaviors of infection, in reality each infection is distinct with disease outcome dependent on early stochastic events. This is particular important when considering interactions between viruses during coinfection since stochasticity can drive one or both viruses to extinction before the infection has time to take hold. Our models show that for viral coinfections, this sometimes leads to a less fit virus out-competing a more fit virus.
Continuous-time Markov chain model
State transitions and propensities for the CTMC coinfection model
Infection by V1
β 1 T V 1
Infection by V2
β 2 T V 2
Infection to infectious
k 1 E 1
Infection to infectious
k 2 E 2
Death of I1
δ 1 I 1
Death of I2
δ 2 I 2
Production of V1
p 1 I 1
Production of V2
p 2 I 2
Decay of V1
c 1 V 1
Decay of V1
c 2 V 2
It has been shown that, the stochastic representations of chemical reactions converge to the differential equations as the number of particles goes to infinity when we can assume that the probability of a reaction depends on the density of the reactants [57, 58, 59]. We make a similar assumption for the “reactions” involved in viral replication where infection of a cell, for example, depends on the density of both cells and virus. Since biological processes, particularly at the microscopic level, are really a series of chemical reactions, there is an inherent stochasticity to the system that is not simply averaged out because we are not specifically considering the detailed chemical reactions in the model. For instance, the infection of a cell in this model includes binding of the virus to the cell receptor, fusion of the virus with the cell membrane, and opening of the virus membrane to release the contents, among other steps. These are all chemical reactions that can be assumed to occur with probability proportional to density of the reacting chemicals. It seems reasonable then to assume that the overall infection process is also dependent on the density of the larger entities (viruses and cells) that contain these chemicals and we can expect a similar convergence of the Markov chain to the differential equation when there are large numbers of viruses and cells.
Stochastic simulation algorithm
Parameter values for the CTMC coinfection model
cell −1 (TCID50/mL) −1 d −1
TCID50/mL d −1
Availability of data and materials
All data is contained within the manuscript. All code is available at https://doi.org/http://github.com/hdobrovo/stochastic_coinfection.
LP, GGP and HMD conceived the experiment(s), LP conducted the experiment(s), LP analyzed the results, LP and HMD wrote the manuscript. All authors reviewed the manuscript.
Ethics approval and consent to participate
Consent for publication
The authors declare that they have no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
- 3.Jartti T, Soderlund-Venermo M, Hedman K, Ruuskanen O, Makela MJ. New molecular virus detection methods and their clinical value in lower respiratory tract infections in children. Paediatr Resp Rev. 2013; 14:38–45.Google Scholar
- 4.Anderson TP, Werno AM, Barratt K, Mahagamasekera P, Murdoch DR, Jennings LC. Comparison of four multiplex PCR assays for the detection of viral pathogens in respiratory specimens. J Virol Meth. 2013; 191(2):118–21.Google Scholar
- 5.Kenmoe S, Tchendjou P, Vernet MA, Tetang SM, Mossus T, Ripa MN, et al.Viral etiology of severe acute respiratory infections in hospitalized children in Cameroon, 2011–2013. Influenza and Other Res Vir. 2016; 10(5):386–393.Google Scholar
- 7.Asner SA, Rose W, Petrich A, Richardson S, Tran DJ. Is virus coinfection a predictor of severity in children with viral respiratory infections?Clin Microbiol Infect. 2015; 21(3):264.e1–6.Google Scholar
- 8.Pretorius MA, Madhi SA, Cohen C, Naidoo D, Groome M, Moyes J, et al.Respiratory Viral Coinfections Identified by a 10-Plex Real-Time Reverse-Transcription Polymerase Chain Reaction Assay in Patients Hospitalized With Severe Acute Respiratory Illness-South Africa, 2009-2010. J Infect Dis. 2012; 206(S1):S159–65.PubMedGoogle Scholar
- 23.González-Parra G, Dobrovolny HM. Assessing Uncertainty in A2 Respiratory Syncytial Virus Viral Dynamics. Comput Math Meth Med. 2015; 2015:567589.Google Scholar
- 24.Pinilla LT, Holder BP, Abed Y, Boivin G, Beauchemin CAA. The H275Y Neuraminidase Mutation of the Pandemic A/H1N1 Influenza Virus Lengthens the Eclipse Phase and Reduces Viral Output of Infected Cells, Potentially Compromising Fitness in Ferrets. J Virol. 2012 October; 86(19):10651–60.PubMedPubMedCentralGoogle Scholar
- 25.Petrie SM, Butler J, Barr IG, McVernon J, Hurt AC, McCaw JM. Quantifying relative within-host replication fitness in influenza virus competition experiments. J Theor Biol. 2015; 382(7):256–71.Google Scholar
- 26.Simeonov I, Gong X, Kim O, Poss M, Chiaromonte F, Fricks J. Exploratory Spatial Analysis of in vitro Respiratory Syncytial Virus Co-infections. Virus. 2010; 2:2782–802.Google Scholar
- 33.Pearson JE, Krapivsky P, Perelson AS. Stochastic Theory of Early Viral Infection: Continuous versus Burst Production of Virions. PLoS Comp Biol. 2011; 7(2):e1001058.Google Scholar
- 37.Heffernan JM, Wahl LM. Monte Carlo estimates of natural variation in HIV infection. J Theo Bio. 2005; 236(2):137–53.Google Scholar
- 38.Allen LJ, Lahodny GE. Extinction thresholds in deterministic and stochastic epidemic models. J Bio Dyna. 2012; 6(2):590–611.Google Scholar
- 39.Kirupaharan N, Allen LJ. Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality. Bull Math Bio. 2004; 66(4):841–64.Google Scholar
- 40.Allen LJ. An introduction to stochastic processes with applications to biology. In: CRC Press. Boca Raton: Chapman and Hall/CRC: 2010.Google Scholar
- 41.Merrill SJ. The stochastic dance of early HIV infection. J Com App Math. 2005; 184(1):242–57.Google Scholar
- 43.Pinky L. Viral coinfections of the respiratory tract. UMI Thesis. Fort Worth: TCU Press; 2018.Google Scholar
- 44.Allen LJ, Jang SR, Roeger LI. Predicting population extinction or disease outbreaks with stochastic models. Lett Biomath. 2017; 4(1):1–22.Google Scholar
- 48.Dierig A, Heron LG, Lambert SB, Yin JK, Leask J, Chow MYK, et al.Epidemiology of respiratory viral infections in children enrolled in a study of influenza vaccine effectiveness. Influenza Other Resp Viruses. 2014; 8(3):293–301.Google Scholar
- 49.Martin ET, Kuypers J, Wald A, Englund JA. Multiple versus single virus respiratory infections: viral load and clinical disease severity in hospitalized children. Influenza Other Resp Viruses. 2011; 6(1):71–7.Google Scholar
- 51.Dimmock NJ, Easton AJ. Cloned Defective Interfering Influenza RNA and a Possible Pan-Specific Treatment of Respiratory Virus Diseases. Viruses. 2015; 2015(7):3768–88.Google Scholar
- 52.Zhao H, To KK, Chu H, Ding Q, Zhao X, Li C, et al.Dual-functional peptide with defective interfering genes effectively protects mice against avian and seasonal influenza. Nature Communications. 2018; 15(9):2358.Google Scholar
- 53.Wasik MA, Eichwald L, Genzel Y, Reichl U. Cell culture-based production of defective interfering particles for influenza antiviral therapy. Appl Microbiol Biotech. 2018; 102(3):1167–77.Google Scholar
- 57.Kurtz T. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Prob. 1971; 8(2):344–56.Google Scholar
- 58.Kurtz T. Reationship between stochastic and deterministic models for chemical reactions. J Chem Phys. 1972; 57(7):2976–8.Google Scholar
- 59.Kurtz T. Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Prob. 1970; 7(1):49–58.Google Scholar
- 60.Gillespie DT. Exact stochastic simulation of coupled chemical reactions. J Phys Chem. 1977; 81(25):2340–61.Google Scholar
- 61.Bartlett MS. Stochastic processes or the statistics of change. J R Stat Soc: Ser C: Appl Stat. 1953; 2(1):44–64.Google Scholar
- 63.Perelson AS, Rong L, Hayden FG. Combination antiviral therapy for influenza: predictions from modeling of human infections. Journal Infect Dis. 2012; 205(11):1642–5.Google Scholar
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