## Abstract

For infectious pathogens such as *Staphylococcus aureus* and *Streptococcus pneumoniae*, some hosts may carry the pathogen and transmit it to others, yet display no symptoms themselves. These asymptomatic carriers contribute to the spread of disease but go largely undetected and can therefore undermine efforts to control transmission. Understanding the natural history of carriage and its relationship to disease is important for the design of effective interventions to control transmission. Mathematical models of infectious diseases are frequently used to inform decisions about control and should therefore accurately capture the role played by asymptomatic carriers. In practice, incorporating asymptomatic carriers into models is challenging due to the sparsity of direct evidence. This absence of data leads to uncertainty in estimates of model parameters and, more fundamentally, in the selection of an appropriate model structure. To assess the implications of this uncertainty, we systematically reviewed published models of carriage and propose a new model of disease transmission with asymptomatic carriage. Analysis of our model shows how different assumptions about the role of asymptomatic carriers can lead to different conclusions about the transmission and control of disease. Critically, selecting an inappropriate model structure, even when parameters are correctly estimated, may lead to over- or under-estimates of intervention effectiveness. Our results provide a more complete understanding of the role of asymptomatic carriers in transmission and highlight the importance of accurately incorporating carriers into models used to make decisions about disease control.

## 1. Introduction

For many infectious diseases, an unknown fraction of infected hosts are able to spread disease while remaining symptom-free. We designate these asymptomatic hosts, ‘asymptomatic carriers’ or ‘carriers’ and reserve the term ‘symptomatic infectious’ for those with observable clinical manifestation, who are more readily identified and targeted by disease control efforts.

The elusive nature of asymptomatic carriers has three important implications for understanding and controlling infectious diseases with carriers. First, incidence data typically only reflect symptomatic cases of infection, making the true extent of asymptomatic carriage for particular diseases difficult to assess. For example, estimates of the population-level prevalence of influenza carriage range from 5 to 35% [1]. For Ebola virus, 27–71% of all infections are estimated to be asymptomatic [2,3]. Similarly, estimates of the population-level prevalence of meningococcus carriage can approach 100% in closed and semi-closed populations, but are in the range of 10–35% in young adults [4].

Second, the sparsity of direct evidence of asymptomatic carriers makes interpretation of the epidemiological record difficult, creating uncertainty when defining details of the carrier state and a pathogen’s more complex natural history of disease [5,6]. For example, estimating the duration of asymptomatic carriage is difficult due to limited data on time of infection and of subsequent pathogen clearance [7].

Third, the presence of asymptomatic carriers undermines control interventions that rely on identifying infectious cases, such as border monitoring or isolating and/or treating infectious cases [8]. Asymptomatic carriers can also affect the efficiency of interventions targeting susceptible individuals, such as vaccination or widespread prophylaxis, because susceptible individuals can be difficult to distinguish from asymptomatic carriers [9]. One serious implication of unknowingly treating carriers is that it exerts selective pressure that may contribute to the emergence and transmission of drug-resistant strains [10,11].

Mathematical modelling has helped to unravel some of the complexities of asymptomatic carriage and its implications for control [9,12–14]. For example, modelling studies of influenza [13] and community-associated methicillin-resistant *Staphylococcus aureus* (MRSA) [14] found that control strategies targeting symptomatically infected hosts but not asymptomatic carriers are actually incapable of achieving disease control, even when multiple control types are implemented. Modelling has also revealed the importance of accounting for asymptomatic carriers in epidemiological models. For example, under certain conditions, neglecting pre-symptomatic influenza transmission was shown to overestimate the impact of interventions targeting symptomatically infected hosts [12]. Neglecting asymptomatic carriers in a model of Ebola virus transmission was shown to significantly overestimate the projected cumulative incidence of symptomatic infections, as well as overestimate the population-level vaccination coverage needed to contain epidemics [9]. Accordingly, if carriage is not appropriately represented in mathematical models, there is a risk that they may incorrectly assess the viability of an intervention, potentially leading to missed opportunities for controlling disease or, conversely, the implementation of an ineffective intervention. Therefore, without sufficient knowledge of asymptomatic carriers the parametrization and validation of mathematical models is compromised. More fundamentally, this uncertainty also undermines the choice of model used to represent the transmission and natural history of an infectious disease, as evidenced by the variation in model structure observed across mathematical modelling studies of MRSA [14–22]. This uncertainty in the underlying structure of mathematical models of infectious diseases with asymptomatic carriers casts further doubt on evaluations of interventions from mathematical models, potentially hindering the decision-making process of policy-makers.

There is a need to understand how the structure of a mathematical model, specifically, how a model depicts asymptomatic carriage, affects its behaviour. To address this need, here we review existing mathematical models of infectious disease transmission for the purpose of identifying the types of model structures that are used to describe transmission with asymptomatic carriers. Based on this review, we define a new model of pathogen transmission that enables us to systematically compare the behaviour of alternative model structures and how they influence the estimated impact of control interventions. First, we analyse this model to establish the contribution of asymptomatic carriers to measures of pathogen transmission, the prevalence of symptomatic infections and the total infection (both symptomatic and asymptomatic) prevalence. We then consider how excluding asymptomatic carriers from this model could lead to the misinterpretation of data and the conditions under which this obscures the most important drivers of pathogen transmission. Finally, we evaluate the impact of a range of interventions on transmission and the prevalence of disease, and discuss how the choice of model structure might affect the assessment of these interventions.

Our analysis reveals that interventions that alter the relative incidence of symptomatic infections compared to asymptomatic carriers are particularly vulnerable to being incorrectly assessed by models with inappropriate structure. Examples of this type of intervention are the multivalent *Streptococcus pneumoniae* vaccines. These vaccines only protect against about 10% of this pathogen’s 90 identified serotypes. Those included in the vaccine are selected for their predominance as causes of invasive pneumococcal disease, allowing less virulent (carriage) strains to persist [23], essentially reducing the relative incidence of symptomatic infections versus asymptomatic carriage. Our study provides a better understanding of the dynamic behaviour of models that include asymptomatic carriers, will inform more appropriate design of future models and contribute to a more complete understanding of the impact of asymptomatic carriage on pathogen dynamics.

## 2. Material and methods

### 2.1. Literature review and model categorization

The literature review identified 132 articles presenting models of pathogen transmission accounting for asymptomatic carriers, of which 42 articles (presenting a total of 45 models) met our inclusion criteria (for more details see electronic supplementary material, appendix A). Pathogens modelled in these studies include *S. aureus* (nine studies), hepatitis B virus (4), *Neisseria gonorrhoeae* (3), *Chlamydia trachomatis* (2), *Neisseria meningitidis* (2), hand-foot-mouth disease enterovirus (2), *Mycoplasma mycoides* subsp. mycoides small colony (2), *S. pneumoniae* (1), norovirus (1), *Trichomonas vaginalis* (1), rotavirus (1), *Salmonella enterica* serotype Typhi (1), *Bordetella pertussis* (1), *Acinetobacter baumannii* (1), Middle East respiratory syndrome coronavirus (1), A/H1N1 pandemic influenza virus (1) and HIV (1), while eight studies considered unidentified abstract pathogens.

To make comparisons between these models, we collapsed each model to a generic form where hosts belong to one of four states: susceptible (*S*); asymptomatic carriers (*C*); symptomatically infectious (*I*); and immune (*R*). Models with an exposed state (*E*) were collapsed in a way such that any exposed and adjacent infectious states (either *C* or *I*) were combined into single infectious states. For example, a pathway *R*, from symptomatic infection: *R*). These generic models were sorted according to whether or not hosts can experience infection more than once (because infection leads to either waning or zero immunity to reinfection, or either lifelong immunity or death), and then whether disease progression (

The transmission dynamics of pathogens that can cause infection more than once in individual hosts can differ significantly from those that cannot reinfect hosts following clearance of an infection [24]. This is likely to also hold true when infectious cases can be either asymptomatic or symptomatic. Therefore, we disregard models that do not allow reinfection. This leaves 15 model types in our analysis out of the original 23 found in the literature review.

The remaining models include those with and without the temporary immune class *R* and describe the transmission of either *S. aureus*, *N. gonorrhoeae*, *C. trachomatis*, *N. meningitidis*, hand-foot-mouth disease enterovirus, *S. pneumoniae*, norovirus, *T. vaginalis*, rotavirus, *S. enterica* serotype Typhi, *B. pertussis*, *A. baumannii* or unidentified abstract pathogens. To make comparisons between these models, we collapsed those with the respective flows

We categorized the remaining 10 generic models according to the presence or absence of flows between the asymptomatic carriage state *C* and symptomatic infection state *I* such that each category contains models where either (1) no transitions exist between *C* and *I*; (2)–(3) transitions can occur in one direction only; and (4) transitions can occur in both directions, as shown in figure 1. Each of these broad model categories represent special cases of the model of transmission with asymptomatic carriage shown in figure 2 and described in the next section.

### 2.2. A model of transmission with asymptomatic carriage

We consider the following model of transmission of a non-immunizing infectious disease with asymptomatic carriage:
*S*=*N*−*C*−*I* (shown schematically in figure 2). In this model, there are three host states: susceptible, asymptomatic carrier and symptomatic infected with respective population sizes *S*≥0, *C*≥0 and *I*≥0. The force of infection is given by *λ*(*C*,*I*)=*β*(*ηC*+*I*)/*N*, where *β*>0 is the pathogen’s transmission rate during symptomatic infection, *η*>0 is the relative transmission rate of asymptomatic carriers compared to symptomatically infected hosts, and *N*>0 is the total host population size (which is assumed to be constant).

The model represents the dynamics of a pathogen where the time scale of disease is typically much faster than the time scale of host births and deaths, and where infection typically does not kill hosts, such as norovirus, an influenza virus or hand-foot-mouth disease. Hence, the demographic effects of the population are ignored in the model and there is no input or output from the host population. However, in electronic supplementary material, appendix C, we provide details of an equivalent model accounting for host demography that would better describe pathogens with longer time scales of disease.

New cases of infection are either symptomatic or asymptomatic with complementary proportions *α* and 1−*α*. Hosts may also regress to asymptomatic carriage from symptomatic infection or progress to symptomatic infection from asymptomatic carriage at respective *per capita* rates *ω*≥0 and *τ*≥0. The model allows natural recovery to occur from both symptomatic and asymptomatic infection, resulting in hosts returning to the susceptible class at respective *per capita* rates *γ*≥0 and *ξγ*≥0, where *ξ*≥0 is the relative recovery rate of asymptomatic carriers compared to symptomatic infectives. Hence, neither recovery from infection nor recovery from carriage confers protective immunity. A summary of these parameters and variables is provided in table 1.

In this framework, the four model categories identified in the literature review (figure 1) are defined according to whether hosts can transition between the asymptomatic carriage and symptomatic infection states during one case of infection, i.e. whether their progression and regression rates *τ* and *ω* are non-zero. Specifically, models in category 1 are characterized by *τ*=*ω*=0, category 2 by *τ*>0 and *ω*=0, category 3 by *τ*=0 and *ω*>0, and category 4 by *τ*>0 and *ω*>0.

## 3. Results

### 3.1. Asymptomatic carriers can be key drivers of pathogen transmission

#### 3.1.1. The basic reproduction number

The basic reproduction number *h*_{1}=*τ*/(*τ*+*ξγ*) of asymptomatic carriers progress to symptomatic infection, and a fraction *h*_{2}=*ω*/(*ω*+*γ*) of symptomatic infections regress to asymptomatic carriage. Hence, a fraction *h*_{1} of newly infected asymptomatic carriers progress to symptomatic infection at least once, a fraction *k* times, spending an average of *g*_{1}=1/(*ω*+*γ*) time units with symptoms per instance of symptomatic infection. Similarly, all newly infected asymptomatic carriers experience asymptomatic infection at least once, a fraction *h*_{1}*h*_{2} experience asymptomatic infection at least twice and a fraction *k* times, spending an average of *g*_{2}=1/(*τ*+*ξγ*) time units without symptoms during each instance of asymptomatic carriage. Thus a newly infected asymptomatic carrier spends on average *β* (from symptomatic infections) and *ηβ* (from asymptomatic carriers), and summing gives

If asymptomatic carriers and symptomatically infected hosts experience infection at a proportionally consistent rate so that *ξ*=*η*, then the host-specific reproduction numbers are identical and equal to the population-level basic reproduction number

The basic reproduction number *disease-free equilibrium* (*C*=*I*=0), which is stable when *endemic equilibrium* (

For the analysis that follows, it is convenient to define the *relative reproduction potential of asymptomatic carriers*

The relative reproduction potential of asymptomatic carriers *γ*>0 and *ξ*>0). If recovery is only possible from symptomatic infection (so that *ξ*=0), then *γ*=0 and *γξ*→a constant), then

#### 3.1.2. The prevalence of infection

The total prevalence of infection *P*(*t*) is the proportion of cases in a host population at a given time *t* accounting for both asymptomatic carriers and symptomatically infected cases. In our model *P*(*t*) is given by
*P*_{C}(*t*) is the prevalence of asymptomatic carriers and *P*_{I}(*t*) is the prevalence of symptomatic infection at time *t* such that
*I*→*C* (both directly and indirectly via *S*), and from *C*→*I* (both directly and indirectly via *S*). In electronic supplementary material, appendix B, we show that

### 3.2. Excluding asymptomatic carriers from mathematical models can obscure the key drivers of pathogen transmission

When confronted with a sparsity of direct evidence of asymptomatic carriage, it may be tempting to assume asymptomatic carriers play a minimal role in pathogen transmission and to disregard them in transmission models. Here, we consider the consequences of such an assumption for estimating transmission from symptomatic infections and of a disease’s basic reproduction number and prevalence.

Excluding asymptomatic carriers from our model transforms it into a standard susceptible-infected-susceptible (SIS) model. In the SIS model, the basic reproduction number *β*_{SIS} and *γ*_{SIS}. Similar to our model, in this SIS model the pathogen will become extinct if

If the basic reproduction number can be calculated independently of the model (i.e. from individual-level contact tracing data at the start of an epidemic), then the basic reproduction number will be equivalent in both models so that
*γ*_{SIS} and *γ* of recovery from symptomatic infection in both models should also be equivalent so that
*β*_{SIS} and *β* of transmission from symptomatically infected hosts. Hence, these rates may vary between the two models. By combining equations (3.7) and (3.10)–(3.11), we see that the ratio of these transmission rates satisfies
*β*_{SIS}>*β* when *β*_{SIS}<*β* when *β*_{SIS}=*β* when

Alternatively, if it is not possible to estimate the basic reproduction number independently, it can be estimated using the respective models. In this case, and given empirically observed values for the recovery rate from symptomatic infection and the transmission rate from symptomatically infected hosts, each model may provide different estimates of the basic reproduction number. Combining equations (3.7) and (3.11), we see that the basic reproduction number of our model

Therefore, when the recovery rates and the transmission rates are held fixed between the two models, the SIS model overestimates the basic reproduction number, the total endemic prevalence and the endemic prevalence of disease when asymptomatic carriers have a lower reproduction potential than symptomatically infected hosts. Otherwise, the SIS model underestimates the basic reproduction number, the total endemic prevalence and possibly the endemic prevalence of disease if symptomatic infections appear at a sufficiently fast rate.

To explain this set of results, we note that in our model, asymptomatic carriers arise from the same pool of susceptible hosts as symptomatic infectives so that if an asymptomatic carrier exists, it is at the cost of a symptomatic case. If asymptomatic carriers then have a lower reproduction potential than symptomatic infectives, their presence detracts from the overall transmission and reproduction potential of the pathogen due to lost opportunities for more productive symptomatic cases. When the basic reproduction number is fixed between the two models, the extra contribution to transmission from symptomatically infected hosts required for the overall reproduction number to match that calculated from the epidemiological data is only apparent in our model; the SIS model underestimates transmission from symptomatically infected hosts. On the other hand, if asymptomatic carriers have a higher reproduction potential than symptomatic infectives, they will increase overall transmission and infection reproduction. In this case, and when the basic reproduction number is fixed between the two models, the SIS model overestimates the contribution to transmission by symptomatically infected hosts.

We illustrate these results in figure 3, where the sizes of the endemic subpopulations *a*), and larger if this is not the case (figure 3*b*,*c*). Even when less prevalent, asymptomatic carriers can be responsible for the majority of transmissions when *c*. A necessary but not sufficient condition for this to occur is that asymptomatic carriers are more infectious than symptomatic hosts so that *η*>1 (see electronic supplementary material, appendix D).

### 3.3. Imperfect knowledge of asymptomatic carriage can result in misleading assessments of strategies for control

Now we assess how control interventions that target different aspects of the disease process may affect the basic reproduction number *β*. We also consider interventions that change *η* (e.g. by quarantining symptomatic infectives), *γ* (e.g. through a mass drug administration), *ξ* (e.g. by treating only symptomatic cases), *ω* (e.g. by providing treatment that targets symptoms, not pathogen clearance), *τ* (e.g. by improving sanitation) and *α* (e.g. through a vaccine targeting pathogen strains associated with symptomatic infection, allowing carriage strains to persist).

First, we determine whether the slope of

It is straightforward to show that the condition
*x*∈{*η*,*β*}, and never holds true for the parameters *x*∈{*γ*,*ξ*}, which makes intuitive sense because increasing the infectivity of either infection type (through increasing *β* or *η*) or prolonging their periods of infectiousness (through decreasing *γ* or *ξ*) will always result in more secondary infections on average per infectious case. For parameters controlling the relative frequency of appearance of asymptomatic carriers (*x*∈{*ω*,*τ*,*α*}), the reproduction potential of carriers *x*∈{*ω*}, condition (3.14) only holds true when *x*∈{*τ*,*α*}. In electronic supplementary material, figure S2, we illustrate the dependence of *τ* causes

We also considered the effects of the same set of interventions on the endemic prevalence of disease *x*∈{*β*,*η*}, a similar result is obtained: *x*∈{*β*,*η*}. For parameters controlling the relative frequency of appearance of asymptomatic carriers (*x*∈{*ω*,*τ*,*α*}), the reproduction potential again plays a key role in determining whether *τ* or *α*, or increasing *ω*) will always result in a lower endemic prevalence of disease

Overall, our analyses highlight how the relative reproduction potential of carriers *τ*, *ω* and *α*).

In figure 4, we provide examples of how *ω* (figure 4*a*), the progression rate *τ* (figure 4*b*) or the symptomatic proportion *α* (figure 4*c*). The corresponding endemic prevalences of asymptomatic carriers, symptomatic infections and total prevalence before and after these interventions are shown in electronic supplementary material, figure S3. Accordingly, if estimates of

## 4. Discussion

The presence of asymptomatic carriers can hamper control efforts and make it difficult to uncover the true details of the natural history of an infectious disease and to estimate the total infection prevalence. Here, we have shown that if carriage is not correctly incorporated into mathematical models used to inform control decisions, there is a risk that these models may produce substantially misleading predictions.

### 4.1. The impact of asymptomatic carriers on assessments of pathogen transmission and control

Our analysis revealed the possibility that certain interventions assessed as capable of interrupting transmission may actually lead to the opposite effect, or alternatively the effectiveness of an intervention may be underestimated, making it appear unviable in some circumstances. Specifically, poor estimates of a quantity which we interpret as the relative reproduction potential of carriers *c*, this intervention may actually increase transmission and disease prevalence.

The same quantity also influences whether excluding asymptomatic carriers from our model either overestimates or underestimates the contribution of symptomatic infections to pathogen transmission. If

### 4.2. The impact of model structure on assessments of pathogen transmission and control

From our analysis it is clear that the threshold behaviour determined by the relative reproduction potential of carriers

Therefore, the threshold behaviour due to *b*).

Our results highlight the importance of accurately characterizing the relationship between asymptomatic carriers and symptomatic infections through a purposive process incorporating the clarification of the dominant modes of pathogen transmission, the conduct of carriage studies, seroepidemiology and active contact tracing to inform the correct attribution of infectious states.

### 4.3. Threshold behaviour in other mathematical models that incorporate asymptomatic carriers

The threshold behaviour that characterizes our model has been identified in other models of pathogen transmission incorporating asymptomatic carriage [28,30]. In these studies, it was found that decreasing the proportion of new cases of symptomatic infection will only decrease a pathogen’s basic reproduction number [28] and total prevalence [30] if carrier infectivity is sufficiently low. Here, we have shown that this threshold behaviour also applies to changes in the rate that established infections either become symptomatic or asymptomatic, and that it is the value of the relative reproduction potential of carriers, not just their relative infectivity, which is the critical determinant of this threshold behaviour.

Another threshold behaviour identified in models with asymptomatic carriers and with implications for control is a phenomenon known as *subcritical persistence*, which can allow an infectious disease to persist in a host population post intervention under certain conditions where, otherwise, it would have disappeared. In the models described in [30,31], subcritical persistence can occur if carriers have sufficiently high infectivity. In the model presented here, however, subcritical persistence is not possible [32].

### 4.4. The contribution of asymptomatic carriers to observations of pathogen transmission and control

A key result of our study is that interventions that alter the frequency of symptomatically infected hosts can lead to unintuitive outcomes in the presence of asymptomatic carriers. Vaccines targeted against a multi-strain pathogen almost invariably target only a proportion of strains. This partial coverage has the potential to change the balance of asymptomatic to symptomatic infection episodes, with implications for population-level vaccine impact.

The PCV7 *Streptococcus pneumoniae* vaccine, for example, targeted the seven most-prevalent strains associated with invasive disease when it was first added to routine childhood vaccination schedules in various populations. Numerous studies documented the subsequent declining rates of asymptomatic carriage of PCV7 serotypes and of overall invasive pneumococcal disease (IPD) among young children [23,33], although the overall incidence of asymptomatic carriage remained relatively stable [34]. Overall reductions in the incidence of IPD were also observed in non-vaccinated adults [23,33], although these were accompanied by increases in the overall rates of asymptomatic carriage [34]. The mechanisms behind the extended response to the PCV7 vaccine beyond the vaccinated populations which affected childhood and adult asymptomatic carriage and IPD rates differently are not well understood. However, it has been suggested that both serotype replacement and a decrease in herd immunity from reduced exposure to asymptomatic carriers may play a role in the population-level impact of the PCV7 vaccine [35,36]. Similar mechanisms are also suggested to be behind the extended response beyond the vaccinated populations to the subsequently introduced 13-valent *S. pneumoniae* vaccine [33]. Recently, IPD incidence has begun to increase in populations in the United Kingdom, particularly among persons 5–64 and greater than 65 years of age, and despite the ever-increasing proportion of the population obtaining direct protection through vaccination [37]. Serotype replacement, in conjunction with changes to the invasiveness of the non-vaccine strains, is a possible explanation [37].

In addition to the action of interventions, the frequency of symptomatic infections may also be altered by different host settings. Accordingly, patterns of pathogen transmission may also be expected to vary between host settings. MRSA and *Streptococcus pyogenes*, for example, have mostly commensal interactions with humans and occasionally cause symptomatic skin infections. When such pathogens are transmitted in environments with high rates of host injury (e.g. healthcare settings or regions where other contagious skin diseases like human scabies are endemic [38]), it is conceivable that infected hosts will probably have an increased chance of developing symptoms compared to other environments with lower host-injury rates and lower chances of hosts having breakages in their skin [15]. Our analytical results suggest that different patterns of pathogen transmission may be occurring in these different host settings, and that the effect of controls may also vary across host settings.

For example, asymptomatic carriers are thought to be more important for the transmission of healthcare-associated MRSA (HA-MRSA) than symptomatically infected hosts, reflected in the effectiveness of patient decolonization and improved hand hygiene compliance of healthcare workers for control [39]. Outside healthcare settings, community-associated MRSA (CA-MRSA) has emerged globally and independently of HA-MRSA [40]. One explanation for the sustained transmission of CA-MRSA outside healthcare settings is that it has a higher fitness in the community than HA-MRSA [40]. Our results indicate that the high reproduction potential of asymptomatic carriers is also likely to be playing a role in the transmission of CA-MRSA in the community. Thus carriers should be targets for control, as has been suggested previously [15].

*Streptococcus pyogenes* is characterized by a large difference in the prevalence of infections in high-income settings versus settings of poverty. In fact, children in remote Indigenous communities in northern Australia, New Zealand and the Pacific Islands have the highest prevalence of skin sores caused by *S. pyogenes* in the world [38]. These communities are characterized by higher host-injury rates and poorer host health compared to communities where *S. pyogenes* has a much lower prevalence. From our analytical results, we can deduce that asymptomatic carriers of *S. pyogenes* have a lower reproduction potential than symptomatically infected hosts, because increasing the proportion of symptomatic infections across host settings corresponds to an increase in prevalence.

### 4.5. Limitations and future work

Our proposed model of infectious disease transmission with asymptomatic carriers has a number of limitations. First, it applies to only those pathogens for which asymptomatic carriers are infectious, clearance of infection is possible and does not confer lifelong immunity, and infection does not alter the death rate of hosts. These restrictions mean that our results are not relevant to two of humanity’s leading causes of death attributable to a single infectious agent – *Mycobacterium tuberculosis* and HIV [41]. A natural extension of our work is to consider how relaxing these constraints affects our study’s conclusions to gain insight into the transmission of these important human pathogens. Second, our model assumes the homogeneous mixing of hosts and it does not consider strain diversity. However, for pathogens such as *S. pneumoniae* and *S. pyogenes* age-assortative mixing and strain diversity are likely to be key drivers of observed transmission dynamics [36,42]. Therefore, it is also of interest to explore the implications of asymptomatic carriage in more complex model structures.

## Data accessibility

The code supporting this article has been uploaded as part of the electronic supplementary material.

## Authors' contributions

All the authors made contributions to study design, edited the manuscript and gave their final approval for publication. R.H.C. performed the simulations, analyses and created figures. R.H.C. and N.G. wrote the manuscript.

## Competing interests

We have no competing interests.

## Funding

This work was supported in part by an NHMRC project grant (APP1098319) and NHMRC Centre of Research Excellence (APP1058804). Y.W. is part funded by CANVAS. S.Y.C.T. is supported by an NHMRC Career Development Fellowship (CDF1065736). J.M. is supported by an NHMRC Principal Research Fellowship (PRF1117140). N.G. is supported by an ARC Discovery Early Career Researcher Award (DE130100660).

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3988734.

- Received January 10, 2018.
- Accepted January 11, 2018.

- © 2018 The Authors.

Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.