When might host heterogeneity drive the evolution of asymptomatic, pandemic coronaviruses?

Abstract

Controlling many infectious diseases, including SARS-Coronavirus-2 (SARS-CoV-2), requires surveillance followed by isolation, contact-tracing and quarantining. These interventions often begin by identifying symptomatic individuals. However, actively removing pathogen strains causing symptomatic infections may inadvertently select for strains less likely to cause symptomatic infections. Moreover, a pathogen’s fitness landscape is structured around a heterogeneous host pool; uneven surveillance efforts and distinct transmission risks across host classes can meaningfully alter selection pressures. Here, we explore this interplay between evolution caused by disease control efforts and the evolutionary consequences of host heterogeneity. Using an evolutionary epidemiology model parameterized for coronaviruses, we show that intense symptoms-driven disease control selects for asymptomatic strains, particularly when these efforts are applied unevenly across host groups. Under these conditions, increasing quarantine efforts have diverging effects. If isolation alone cannot eradicate, intensive quarantine efforts combined with uneven detections of asymptomatic infections (e.g., via neglect of some host classes) can favor the evolution of asymptomatic strains. We further show how, when intervention intensity depends on the prevalence of symptomatic infections, higher removal efforts (and isolating symptomatic cases in particular) more readily select for asymptomatic strains than when these efforts do not depend on prevalence. The selection pressures on pathogens caused by isolation and quarantining likely lie between the extremes of no intervention and thoroughly successful eradication. Thus, analyzing how different public health responses can select for asymptomatic pathogen strains is critical for identifying disease suppression efforts that can effectively manage emerging infectious diseases.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11071-022-07548-7.

Introduction

The SARS-Coronavirus-2 (SARS-CoV-2) pandemic reinforced the centrality of non-pharmaceutical interventions to suppress and mitigate outbreaks of emerging pathogens [14, 21, 27]. In particular, it showed how testing, isolation, and contact-tracing followed by quarantining can be an effective and targeted disease control strategy [24, 64, 78]. However, testing regimes are subject to financial and capacity constraints (e.g., [38]). Hence, in the absence of wide-spread, randomized testing, these interventions are typically initiated by identifying symptomatic individuals.

Under these conditions, intervention measures that fall short of disease eradication could allow strains that are able to circulate with minimal host removal to readily outcompete strains that cause their hosts to be detected. This is because when asymptomatic carriers go undetected, the pathogens they harbor can spread with less friction through the host population. By contrast, viral lineages more likely to cause symptomatic infections are subject to detection and quarantine. Moreover, concern about such lineages can lead to the detection of asymptomatic infections. Thus, the effort to isolate infected individuals, independently of their symptoms, can prune viral lineages prone to causing symptomatic infections. To the extent that testing, isolation, and contact-tracing followed by quarantining rely on initially identifying symptomatic individuals, these non-pharmaceutical interventions may inadvertently select for strains less likely to result in symptomatic infections.

The potential evolutionary consequences of targeting symptomatic individuals for intervention present major challenges for controlling pathogens such as SARS-CoV-2. Epidemiologically, the 2020 SARS-CoV-2 pandemic differs markedly from the 2003 SARS-Coronavirus-1 (SARS-CoV-1) outbreak. While the underlying pathogens and human host populations in which the outbreak occurred differ along several important biological and social axes (e.g., [20, 77, 86, 96, 98, 119, 144]), one of the major differences has been the degree to which individuals infected with SARS-CoV-2 are unlikely to present symptoms ([65, 83, 89, 91, 101, 108, 139]; but see [66]). This results in fewer cases being detected, and a higher proportion of unwittingly infectious carriers circulating in the population. Strikingly, it seems plausible that many cases were likely at most only mildly symptomatic, with infection prevalence as high as 31% of the population in the USA alone, far below the value implied by officially reported case counts [110]. Nevertheless, improved diagnostic criteria for SARS-CoV-2 have been critical in facilitating case detection since the first cases in 2019 [3, 74, 80, 102, 114].

Elucidating how different symptoms-based control measures govern evolution is therefore critical for controlling pathogens such as SARS-CoV-2. Analyzing these evolutionary consequences requires accounting for the reality that selection on widespread, novel pathogens operates within a fitness landscape characterized by host heterogeneity. By definition, emerging zoonotic pathogens navigate a heterogenous landscape of hosts (e.g., [48, 60]). Even after evolving to circulate in humans, the intensity of non-pharmaceutical intervention efforts such as symptoms-based control likely differs across host classes. Economic disparities are especially likely to result not only in distinct transmission and infection risks among hosts, but also in variable public health attention and responses [9, 17, 34, 38, 57, 88, 122]. The lack of pharmaceutical interventions, and the heavy reliance on underfunded, or structurally adjusted, public institutions for disease control may exacerbate existing disparities in vulnerability to infection among host populations [4, 22, 79, 120, 129]. This is because non-pharmaceutical interventions, ranging from highly targeted case-detection and expansive contact-tracing to population-wide guidelines on physical distancing to lockdowns and economic shutdowns, require both a robust public health infrastructure and high social and economic development to implement successfully [9, 33, 40, 46, 48, 53, 54, 71, 87, 137, 141]. Under these conditions, the adequacy of any broad-scale non-pharmaceutical intervention (and hence the selection pressures imposed on pathogens) is likely to be compromised by differential attention and resources aimed at different host groups (e.g., [69, 73, 130]).

Here, we develop and analyze an evolutionary epidemiology model to asses how symptoms-based control measures drive viral evolution. We explore how multiple viral strains, some more symptomatic than others, circulate in a heterogenous host population. The nonlinearities inherent to the evolution and circulation of novel viral strains among heterogenous hosts necessitate a mathematical framework that can elucidate the conditions under which non-pharmaceutical interventions select for strains less likely to cause symptomatic infections. Superficially, selection for a largely asymptomatic strain may seem like a tolerable epidemiological outcome. However, we argue that high prevalence of undetected infections illustrates precisely why host heterogeneity is important. As the SARS-CoV-2 pandemic illustrates, widespread circulation of a generally asymptomatic pathogen raises risks for host subpopulations in which the infection is unlikely to be mild [13, 19, 70, 72, 143], and there is growing evidence of disparities in vulnerability to infection among host populations driving the efficacy of public health interventions and transmission risks [1, 17, 120, 125]. Hence, our analyses focus on two host types: hosts vulnerable to infections but experiencing limited public health attention, and hosts less likely to be infected and experiencing greater public health attention. To our knowledge, our study is the first that seeks to elucidate how such heterogeneity interacts with the selective effects of non-pharmaceutical intervention strategies on pathogen evolution, particularly as it concerns selection for disease severity. Below, we present the structure of our model and our analyses.

Materials and methods

Model development

We consider a pathogen circulating within a heterogeneous host population. The host population consists of two main host classes: vulnerable hosts, which experience potentially higher disease transmission but are also potentially less likely to be subject to disease surveillance, and resilient hosts, who have reduced infection risk and are potentially more likely to be the target of disease surveillance. We focus our analyses on the case where the individual transition of hosts between the two host classes over the course of their lifetime is negligible (as may occur, for instance, in demographically or socio-economically segregated settings) or nonexistent (as in the case of sylvatic and human hosts).

We adopt SARS-coronaviruses, and SARS-CoV-2 in particular, as a case study. We apply our analysis to this system for several reasons. First, transmission from asymptomatic infections is highly likely [44, 70, 108, 145], suggesting that strains less likely to cause symptomatic infections still replicate and spread. Second, mitigation strategies against SARS-CoV-2 had initially been fairly representative of many emerging pathogens. In particular, while pharmaceutical interventions for controlling SARS-CoV-2 are, as of autumn 2021, available for some populations [6, 11, 75, 97], for much of the pandemic these interventions remained unable for a sizeable share of the world’s population [14, 50, 56]. As is the case with most emerging infectious diseases, pandemics are highly challenging to predict, resulting in a lack well-tested, widely and immediately available pharmaceutical interventions [8, 85, 100]. Thus, the selection pressures these pathogens experience from public health interventions, and non-pharmaceutical interventions in particular, have the potential to be very different from the selection pressures operating on endemic pathogens. Third, several novel variants of SARS-CoV-2 with distinct epidemiological profiles have readily emerged [49, 93, 113, 115]. This speaks to the considerable evolvability of the virus, making it an attractive case study for evolutionary epidemiology. Finally, while there has been quite a bit of work on molecular evolution in SARS-CoV-2 [61, 81, 94, 99, 114, 123, 124, 131, 133, 148], given the strong selection potentially imposed by non-pharmaceutical interventions, there is growing interest in understanding how different selective pressures can drive phenotypic evolution in this system [2, 25, 28, 47].

A baseline model without interventions

We assume that transmission to be frequency-dependent. Thus, if a susceptible host of type h encounters infectious hosts of type $h^{\prime }$, pathogen transmission occurs at a per-capita rate $\beta r_{h,h^{\prime }}/N$, where N is the density of all susceptible and infectious hosts, $\beta$ is the baseline per-capita infection rate, and $r_{h,h^{\prime }}$ describes the extent of mixing between hosts h and $h^{\prime }$ (for instance, as could be caused by heterogeneous contact networks—e.g., [7, 26, 58, 127, 128, 147]). Following infection, we assume individuals of host type h infected with strain j become asymptomatic with strain-specific probability ${\pi }_{j,h}$. During infection, strain i may emerge via mutation, and subsequently out-compete strain j within a host at a per-infected individual rate ${\mu }_{j\to i}$ (e.g., [12, 103, 109]). For brevity, we refer to this process by which an infected individual of strain j undergoes a within-host replacement by strain i as “mutation”. Furthermore, because of the relatively low case fatality rate of SARS-CoV-2 (e.g., [82, 107]), we consider infection-induced mortality to have minimal epidemiological effects over the time scales of interest. Similarly, the incubation period is treated as being sufficiently short to not affect evolutionary trajectories.

Finally, infectious individuals recover from an infection at rate $\gamma$. In the absence of evidence of antagonistic pleiotropy, we treat this rate as consistent across strains. For SARS-coronaviruses, there is limited evidence of lifelong immunity, although some degree of at least temporary immunity appears likely [37, 45, 59]. We recognize that over epidemiological timescales, SARS-CoV-2 outbreak data can be accurately described by susceptible-exposed-infectious-resistant-susceptible (SEIRS)-type models. However, for purposes of our analysis, we consider the period of complete immunity over evolutionary timescales to be relatively short, in part because (i) host demographic turnover provides a supply of newly susceptible individuals, (ii) over the relevant evolutionary time-scales, coronaviruses are subject to repeated and regular zoonotic emergences and re-emergences (e.g., [67]) for which there is unlikely to be pre-existing immunity and (iii) the duration of immunity relative to changes in the antigenic profiles of SARS-coronaviruses, particularly in relation to the mutation rates of SARS-coronaviruses, remains an open question (e.g., [15, 29]). Thus, for purposes of our analyses, a more tractable susceptible-infectious-susceptible (SIS) model appropriately captures the essential dynamic components that can drive the evolution of more transmissible, asymptomatic respiratory coronaviruses. We further highlight that, from the standpoint of nonlinear dynamical systems, SIS models with distributed time lags also recreate the trajectories of SARS-coronavirus outbreak data on epidemiological timescales [126].

Taking all of the above into consideration, in the absence of any public-health interventions (pharmaceutical or otherwise), the resulting evolutionary epidemiology can be summarized as:

$$\begin{array}{cc}\hfill \frac{\text{d}{S}_h}{\text{d}t}& =-\frac{{\sum }_{j=1}^J\beta {\sum }_{h^{\prime }=1}^H{r}_{h,h^{\prime }}(I_{h^{\prime },\sigma ,j}+a{I}_{h^{\prime },a,j})}{N}{S}_h\hfill \\ \hfill & +\gamma \sum _j^J(I_{h,j,\sigma }+I_{h,j,A})\hfill \\ \hfill \frac{\text{d}{I}_{h,j,A}}{\text{d}t}& ={\pi }_{h,j}\frac{\beta {\sum }_{h^{\prime }=1}^H{r}_{h,h^{\prime }}(I_{h^{\prime },\sigma ,j}+a{I}_{h^{\prime },a,j})}{N}{S}_h\hfill \\ \hfill & -\sum _{i\ne j}^J{\mu }_{j\to i}{I}_{h,j,A}\hfill \\ \hfill & +\sum _{i\ne j}^J{\pi }_{h,j}{\mu }_{i\to j}(I_{h,i,A}+I_{h,i,\sigma })-\gamma I_{h,j,A}\hfill \\ \hfill \frac{\text{d}{I}_{h,j,\sigma }}{\mathit{dt}}& =(1-{\pi }_{h,j})\frac{\beta {\sum }_{h^{\prime }=1}^H{r}_{h,h^{\prime }}(I_{h^{\prime },\sigma ,j}+a{I}_{h^{\prime },a,j})}{N}{S}_h\hfill \\ \hfill & -\sum _{i\ne j}^J{\mu }_{j\to i}{I}_{h,j,\sigma }\hfill \\ \hfill & +\sum _{i\ne j}^J(1-{\pi }_{h,j}){\mu }_{i\to j}(I_{h,i,A}+I_{h,i,\sigma })\hfill \\ \hfill & -\gamma I_{h,j,\sigma }.\hfill \end{array}$$ 1

Table 1 summarizes the parameter values of model (1). Although several of the parameter values are drawn from the literature, many parameters remain context and strain-specific. For instance, the extent to which a new strain causes less symptomatic infections depends, among other things, on the specific point mutations at the genomic level and how these mutations cascade to alter the intra-host viral replication cycle (e.g., [28, 47]). Therefore, in order to elucidate how the phenotypic consequences of the mutation (in this case, the probability of causing asymptomatic infections), we aim to explore a range of possible mutations that cover qualitatively distinct scenarios: from mutations causing very few asymptomatic infections, to mutations resembling the ancestral strain, to mutations causing almost all infections to be asymptomatic. Hence, for these parameters, we considered numerical ranges that aim to cover a large set of qualitatively distinct scenarios and characterize in Table 1 the epidemiological relevance of the ranges considered.

Table 1.

Variables for the baseline model without any interventions

Variable	Interpretation	Value or range	Notes	Epidemiological implications of ranges
$S_h$	Density of hosts of type h	–	–	–
$I_{h,j,A}$	Density of hosts of type h infectious, asymptomatic with viral strain j	–	–	–
$I_{h,j,\sigma }$	Density of hosts of type h infectious, symptomatic with viral strain j	–	–	–
N	Total density of susceptible and infectious hosts	–	–	–
$\beta$	Baseline per-capita infection risk	0.085 ${\text{host}}^{-1}{\text{time}}^{-1}$	[18]	–
$r_{h,h^{\prime }}$	Constant of proportionality describing interactions leading to pathogen transmission to hosts of type h from hosts of type $h^{\prime }$	0.01–100	–	The range was specified to span four orders of magnitude, characterizing qualitatively distinct extremes ranging from when hosts of type h drive only $1\%$ of the force of infection among hosts of type $h^{\prime }$, to when $h^{\prime }$ drive only $1\%$ of the force of infection among hosts of the same type
a	Relative infectiousness of asymptomatic, infectious hosts	0.1	[42, 84]	–
${\pi }_{h,j}$	Probability that an infection of host type h by viral strain j becomes asymptomatic	0.075–0.975	For lower bound, [138]	The upper bound of the range was specified to characterize the opposite extreme whereby all but a tiny fraction (2.5$\%$) of infections are asymptomatic
${\mu }_{j\to i}$	Within-host replacement rate of strain j to strain i	${10}^{-6}{\text{time}}^{-1}$	[90]	–
$\gamma$	Per-capita recovery rate	$0.033{\text{time}}^{-1}$	[18]	–

The effect of public health interventions

To model the effect of non-pharmaceutical public health interventions as well as their potential unequal application across host classes on the evolution of viral phenotypes, we focus on two types of interventions commonly used at the beginning of a pandemic. First, we assume that symptomatic individuals can be successfully identified and isolated at a potentially time-varying, per-capita rate ${\theta }_{h,j}(t)$ that depends on the host type h and the strain type j with which they are infected. Symptomatic individuals that are isolated are removed from the population of infectious hosts. Because isolated hosts eventually recirculate in the population after recovery, the density of isolated hosts is denoted by $K_{h,j}$ for isolated hosts of type h infectious with strain j.

Second, we model the effect of subsequent contact tracing and testing aimed at identifying asymptomatic carriers. We assume that this contact tracing and testing can identify infected, asymptomatic hosts of type h infected with strain j at a potentially time-varying, per-capita rate $q_{h,j}(t)$. Once asymptomatic carriers are identified through contact tracing and testing, they are also removed from the population of infectious hosts. As with isolated, symptomatic hosts, these asymptomatic cases are ultimately returned to the population after the infection clears, so we track the density $Q_{h,j}$ of successfully removed, asymptomatic hosts of type h carrying strain j. To distinguish the removal of asymptomatic and symptomatic hosts, we refer to the removal of asymptomatic hosts following contact tracing and testing as “quarantining”, and the removal of symptomatic hosts as “isolation”. We use $v_q,v_{\theta }$ to describe the relative difference in quarantining and isolation efficacy between the two strains ($q_{h,j}(t)/q_{h,j^{\prime }}(t)$ and ${\theta }_{h,j}(t)/{\theta }_{h,j^{\prime }}(t)$, respectively). [3, 74, 80, 102, 114]. Successfully quarantined or isolated hosts do not come into contact with susceptible and infectious individuals; thus, their densities do not contribute to N, which, as noted above, we define as the sum of infectious and susceptible hosts. We note that both the isolation rate ${\theta }_{h,j}(t)$ and the quarantining rate $q_{h,j}(t)$ are composite parameters that depend on both the accuracy of identifying infectious individuals (asymptomatic or otherwise) and the efficacy of removing those individuals from the population. Thus, the magnitude of these parameters are governed, in part, by the reliability, efficiency and availability of diagnostic tools (e.g., [114]). We further treat the testing regime during the decision to isolate or quarantine as sufficiently effective (e.g., [3, 32, 136, 146]) that the accidental removal of uninfected hosts (i.e., false positives) is negligible.

Successfully isolated or quarantined hosts recover at an accelerated rate $\delta$ that represents, for instance, the effects of administering antiviral drugs [11, 132] or the successful management of symptoms [51]. Because asymptomatic individuals may, potentially, be subject to ongoing monitoring, we treat the enhanced per-capita recovery rate of quarantined individuals as comparable to the enhanced per-capita recovery rate of isolated individuals.

The combined eco-evolutionary dynamics of the interacting hosts and pathogens in the presence of a public health response are given by:

$$\begin{array}{cc}\hfill \frac{\text{d}{S}_h}{\text{d}t}& =-\frac{\beta {\sum }_{h^{\prime }=1}^H{r}_{h,h^{\prime }}{\sum }_{j=1}^J{I}_{h^{\prime },j}}{N}{S}_h\hfill \\ \hfill & +\gamma \sum _j^J(I_{h,j,\sigma }+I_{h,j,A}+\delta \sum _j^J(Q_{h,j}+K_{h,j}))\hfill \\ \hfill \frac{\text{d}{I}_{h,j,A}}{\text{d}t}& ={\pi }_{h,j}\frac{\beta {\sum }_{h^{\prime }=1}^H{r}_{h,h^{\prime }}{I}_{h^{\prime },j}}{N}{S}_h\hfill \\ \hfill & -\sum _{j\ne i}^J{\mu }_{j\to i}{I}_{h,j,A}-{\mathit{q}}_{\mathit{h}\mathbf{,}\mathit{j}}\mathbf{(}\mathit{t}\mathbf{)}{\mathit{I}}_{\mathit{h}\mathbf{,}\mathit{j}\mathbf{,}\mathit{A}}-\gamma I_{h,j,A}\hfill \\ \hfill & +\sum _{j\ne i}^J{\pi }_{h,j}{\mu }_{i\to j}(I_{h,i,A}+I_{h,i,\sigma })\hfill \\ \hfill \frac{d{I}_{h,j,\sigma }}{\mathit{dt}}& =(1-{\pi }_{h,j})\frac{\beta {\sum }_{h^{\prime }=1}^H{r}_{h,h^{\prime }}{I}_{h^{\prime },j}}{N}{S}_h\hfill \\ \hfill & -\sum _{i\ne j}^J{\mu }_{j\to i}{I}_{h,j,\sigma }-{\mathit{\theta }}_{\mathit{h}\mathbf{,}\mathit{j}}\mathbf{(}\mathit{t}\mathbf{)}{\mathit{I}}_{\mathit{h}\mathbf{,}\mathit{j}\mathbf{,}\mathit{\sigma }}-\gamma I_{h,j,\sigma }\hfill \\ \hfill & +\sum _{i\ne j}^J(1-{\pi }_{h,j}){\mu }_{i\to j}(I_{h,i,A}+I_{h,i,\sigma })\hfill \\ \hfill \frac{\text{d}{Q}_{h,j}}{\text{d}t}& =q_{h,j}(t)I_{h,j,A}-\delta \gamma Q_{h,j}\hfill \\ \hfill \frac{\text{d}{K}_{h,j}}{\text{d}t}& ={\theta }_{h,j}(t)I_{h,j,\sigma }-\delta \gamma K_{h,j}\hfill \end{array}$$ 2

for pathogen strains $j=1,\dots ,J$ and host types $h=1,\dots ,H$.

We highlight two further points regarding model (2). First, because successfully quarantined and isolated hosts do not contribute to onward pathogen transmission, we ignore mutation dynamics within successfully isolated or quarantined hosts as those viral mutants cannot spread further. Second, we do not model the effects of public health policies that aim to reduce the supply of susceptible hosts (e.g., mass lock-downs, culling, etc...). This is because we seek to model how interventions based on surveillance in the pre-pandemic stage select for different pathogen strains. Thus, here we concern ourselves with evolution that occurs before large-scale public health interventions reducing the density of susceptibles become necessary. Nevertheless, we hasten to add that our model does allow for public health measures aimed at transmission reduction (e.g., by providing access to personal protective equipment or physical distancing): The composite term ${\beta }_{j,h}{r}_{h,h^{\prime }}$ governing transmission is able to account for such effects. Table 2 summarizes the key parameters added to model (1) in model (2). As was the case for Table 1, we characterize the qualitatively distinct epidemiological scenarios captured by the ranges of parameter values we considered.

Table 2.

Additional variables used to model public health interventions

Variable	Interpretation	Range	Epidemiological Implications of ranges
$Q_{h,j}$	Density of quarantined hosts of type h infected with viral strain j	–
$K_{h,j}$	Density of isolated hosts of type h infected with viral strain j	–
$q_{h,j}(t)$	The efficacy of quarantining asymptomatic infections of hosts of type h by viruses of type j	0.0001–1	At the lower extreme, only one in 10,000 asymptomatic infections are successfully identified and removed per unit time; at the upper extreme, a high fraction of hosts can be quarantined effectively immediately
${\theta }_{h,j}(t)$	The efficacy of isolating asymptomatic infections of hosts of type h by viruses of type j	0.0001–1	At the lower extreme, only one in 10,000 asymptomatic infections are successfully isolated per unit time; at the upper extreme, a high fraction of hosts can be quarantined effectively immediately
$v_{·}$	The relative ability to successfully identify and remove symptomatic and asymptomatic infections involving the novel virus	0.0001–1	At the lower extreme, the mutant strain is 10,000$\times$ more effective at evading public health interventions than the ancestral strain; at the upper extreme, the mutant strain is just as readily removed as the ancestral strain
$\delta$	The accelerated recovery of quarantined or isolated hosts	0.05–0.85	At the lower extreme, isolated or quarantined hosts experience similar recovery times (1.05$\times$) as hosts that continue to circulate in the general population; at the upper extreme, isolation and quarantining approximately doubles (1.85$\times$) the recovery time of removed hosts

Model analyses

We explore how public health interventions and differences between host classes drive the evolutionary emergence of asymptomatic strains of SARS-coronaviruses. We focus on two scenarios. We begin by examining the scenario where an ongoing, background surveillance effort leads to generally constant, host-class and virus-strain specific quarantining and isolation efforts over time (i.e., $q_{h,j}(t)\approx q_{h,j}$ and ${\theta }_h(t)\approx {\theta }_{h,j}$). We then assess the effects of reactive surveillance efforts, in which the quarantining and isolation efforts depend on the prevalence of symptomatic infections $P_{\sigma }(t)=\frac{1}{N(t)}{\sum }_{j=1}^J{\sum }_{h=1}^H{I}_{h,j,\sigma }(t)$. Because infection prevalence is time-varying, under this scenario, $q_{h,j}(t)=q_{h,j}P(t)$ and ${\theta }_{h,j}(t)={\theta }_{h,j}P(t)$. Under ideal conditions, integrating real-time data about epidemiological dynamics allows public health stakeholders to fine-tune their intervention efforts, thereby improving disease management (e.g., [111, 112]). Moreover, the efficacy of disease surveillance and intervention efforts are often integrally tied to the dissemination of accurate information on disease control among individuals affected (e.g., [134, 135]). Thus, our characterization of reactive control efforts in Table 2 aims to capture a gradient in the robustness and accuracy of epidemiological surveillance, as well as in the ability to have individuals comply with the intervention regimes from marginal to highly reliable. Finally, we assume that, over the evolutionary timescales we consider, the ability of public health interventions to respond to prevalence is effectively instantaneous.

Because of the high-dimensionality of model (2), we conduct our analyses in two steps: First, we explore the basic question of when isolation of symptomatic individuals and contact-tracing, followed by quarantining, favor the emergence of asymptomatic strains in the limiting case where there is an absence of host heterogeneity (i.e., the limiting case where for model (2) whereby ${\pi }_{h,j}={\pi }_{h^{\prime },j},r_{h,h^{\prime }}=1,q_{h,j}=q_{h\prime ,j}$ and ${\theta }_{h,j}={\theta }_{h\prime ,j}$). Second, we explore how host heterogeneity modifies these conclusions about when public health interventions promote the evolution of an asymptomatic strain. When there is host heterogeneity, even the relatively simpler baseline model (1) did not lend itself to ready algebraic or analytic characterization of the equilibria (Supplementary File S1), so the long-term dynamics of models (1 and 2) had to be analyzed numerically. To facilitate biological interpretation, we present results for two pathogen strains - an ancestral strain and a novel, less-symptomatic strain, systematically varying the extent to which infection by the novel strain is likely to be asymptomatic. All numerical analyses are conducted using the function $\mathtt{lsoda}$ from the package deSolve in R [116], with absolute error tolerance set at ${10}^{-50}$. We initialized our analyses with a total host population density of ${10}^6$ individuals per unit area, with a ten-to-one ratio of vulnerable to resilient hosts. Epidemics in all analyses are seeded with a single infectious individual with the ancestral, symptomatic strain. For parameters which did not have fixed numerical values, we generated all possible parameter combinations from evenly distributed points within each parameter’s range (summarized in Tables 1, 2). For further implementation details, we refer the reader to the underlying source code used in the analysis, which is accessible on github (https://anonymous.4open.science/r/9a717a70-7812-4c1e-a686-c018cc3664d4/) and is released under the GNU Public License v3 [118].

Results

We highlight results showing how the efficacy of public health interventions and, ultimately, their heterogeneous application to distinct host classes drives the evolution and spread of asymptomatic coronaviruses.

The evolutionary effects of public health interventions without host heterogeneity

To establish a baseline set of expectations, we first begin with a limiting case of model (2) that assesses whether symptoms-driven isolation and contact tracing, followed by quarantining, can select for the emergence of a viral strain more likely to be asymptomatic. Our limiting case explores the scenario where host classes are equally likely to be infected, the propensity to exhibit symptomatic infections does not depend on host class, and both host classes are equally likely to be isolated or quarantined. Under these conditions, a more asymptomatic strain can emerge when the resident strain is at a disease-endemic equilibrium in model (2) when

$$\begin{array}{cc}& \frac{{\beta }_m({\theta }_{·,R}+\gamma )(q_{·,R}+\gamma )({\theta }_{·,M}{\pi }_{·,M}+q_{·,M}(1-{\pi }_{·,M})+\gamma )}{({\theta }_{·,M}+\gamma )(q_{·,M}+\gamma )({\theta }_{·,R}{\pi }_{·,R}+q_{·,R}(1-{\pi }_{·,R})+\gamma )}\hfill \\ \hfill & >1,\hfill \end{array}$$ 3

where R, M denote the resident and less symptomatic mutant strain, respectively (e.g., [52]) and ${\beta }_m$ denotes the differential infectivity of the more asymptomatic strain. Briefly, condition (3) is obtained by constructing a next-generation matrix (e.g., [31, 41]) for the mutant strain M in model (2). When the dominant eigenvalue of this matrix (evaluated when strain M in model (2) is exceedingly rare) exceeds unity, the mutant strain M is able to increase when rare [30]. For a more detailed derivation of condition (3), we refer the reader to the Mathematica code [140] and outputs in Supplementary File S2.

Condition (3) illustrates how the interplay between each virus strain’s propensity to be asymptomatic (${\pi }_{·,·}$) and the prevailing efficacy of the public health responses drive the emergence of a more asymptomatic strain. In particular, when the mutant strain is likely to be asymptomatic (${\pi }_{·,M}>>{\pi }_{·,R}$) then the marginal efficacy of isolating individuals infected with this strain relative to identifying and quarantining asymptomatic individuals infected with this strain (${\theta }_{·,M}-q_{·,M}$) becomes critical in suppressing the emergence of a more asymptomatic strain. By contrast, this difference between isolation and quarantine efficacy matters less when the novel strain is not particularly asymptomatic.

A key consequence of this result is that, at least in relatively homogenous host populations, reducing the discrepancy between isolation efficacy and the efficiency of contact-tracing followed by quarantining can help prevent the emergence of asymptomatic variants. This is particularly the case when it concerns the novel strain; if quarantine efficacy is less than unity, reducing this discrepancy between isolation and quarantine efficacy could improve the ability of the mutant, more asymptomatic strain to invade if the ability ($q_{·,M}$) to identify and remove asymptomatic infections is low (condition 3).

Heterogeneous host populations

Next, we highlight results showing how the efficacy of public health interventions and their heterogeneous application to distinct host classes drives the evolution and spread of asymptomatic coronaviruses. Unlike the previous analysis where host heterogeneity was limited, in the presence of host heterogeneity, only numerical analyses of models (1 and 2) proved tractable. To facilitate a comparison of different scenarios, we summarize the results of our numerical analyses using long-term dynamics.

In the context of public health, the prevalence of infection by the novel strain is of greater concern than the frequency of the asymptomatic virus in the viral population per se. Thus, we illustrate our results using the long-term prevalence of hosts infected with the evolved, asymptomatic strain to characterize the joint evolutionary and epidemiological predictions. In the comparisons that follow, we distinguish between a baseline isolation efficacy and a baseline efficacy of contact-tracing followed by quarantining for the ancestral strain, and the effectiveness of isolation and contact tracing followed by quarantining for the derived, more asymptomatic strain. We also compare the effects of increased infection risk and reduced intervention efficacy for the vulnerable host class.

Scenario 1: constant effort for public health interventions

When isolation and contact-tracing followed by quarantine (hereafter “quarantining”) are carried out at a constant rate ($q_{h,j}(t)=q_{h,j}$ and ${\theta }_{h,j}(t)={\theta }_{h,j}$), a wide range of qualitatively distinct evolutionary outcomes result (Fig. 1). Generally, increasing isolation effort selects for a novel strain to spread in the host population, until the isolation efficacy is sufficiently high that disease control occurs before an asymptomatic mutant can evolve. Exceptions are Fig. 1J and scenarios resulting in complete suppression or prevalences of 100% across the entire parameter space (Supplementary Material S3 and S4).

Fig. 1.

The range of possible qualitative behavior of long-term prevalence of all hosts infected with the novel, more asymptomatic virus in model (2) as a function of quarantine and isolation effort. In addition to the results above, the model also produced outcomes where the novel virus could not successfully spread or would infect all hosts irrespective of the isolation and quarantine efforts (results not shown). A $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. B $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. C $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. D $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. E $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. F $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. G $r_{h,h\prime }=0.1,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. H $r_{h,h\prime }=0.1,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. I $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=0.001,v_{\theta }=1$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. J $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. K $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. L $r_{h,h\prime }=0.1,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. M $r_{h,h\prime }=0.1,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. N $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. O $r_{h,h\prime }=0.1,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. P $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. Here, and for subsequent figures, we note that varying the rate $\delta$ at which isolated or quarantined hosts recovered had very little effect on long-term prevalence (Supplementary Material S3 and S4)

Across the parameter space we explore, a critical distinguishing feature among the range of dynamical behaviors is whether the interaction between isolation and quarantine effort promotes the evolution and spread of the novel, more asymptomatic virus. The evolutionary consequences of isolation can, in some cases, occur largely irrespective of the quarantine effort (e.g., Fig. 1A, D, G, H, O). When quarantining also affects viral evolution and spread, we find increasing quarantine efforts can have divergent effects. At times, increased removal of symptomatic hosts selects for an asymptomatic strain, but this can be mitigated by more effective quarantining. For instance, Fig. 1B–C, J–L, P show how even at levels of isolation effort selecting for an asymptomatic strain, effective quarantining can mitigate the spread of the evolved virus (cooler colors in upper regions of those panels). By contrast, Fig. 1E, F, I, M, N illustrate the opposite effect: higher quarantining efforts interact with levels of isolation that select for asymptomatic viruses to drive high prevalence. Even within the latter scenarios, the joint effects of isolation and quarantining are not always consistent. For instance, Fig. 1E, F, N show how the interaction between high quarantine levels and isolation on selecting for the more asymptomatic strain diminishes as isolation efficacy increases.

Supplementary Tables S5 and S6 systematically group all the scenarios we analyzed into each of these qualitative long-term patterns based on direct visual inspection of all parameter combinations. Supplementary Table S5 categorizes the effects for the case where the two host classes receive disparate public health intervention efforts; Supplementary Table S6 evaluates the conditions when the two host classes receive comparable public health intervention efforts.

A comparison of scenarios in Supplementary Table S5 reveals how the ability to detect asymptomatic cases can be a major driver of the distinctions highlighted above. Indeed, the differing effects of increasing quarantine effort arise because, like isolation effort, it is ultimately intermediate levels of quarantining that select for the emergence and spread of an asymptomatic variant. In our model, quarantine efficacy is a composite of the ability to detect asymptomatic hosts (through contact tracing, testing or diagnosis—[3, 114]) and the successful removal of such hosts. Thus, whether increasing quarantining selects for an asymptomatic virus critically depends upon the ability to detect asymptomatic infections. Comparing Fig. 1C–F illustrates how an ability to detect asymptomatic infections shifts the evolutionary effects of quarantining. When the ability to detect asymptomatic infections is limited (Fig. 1F), higher quarantining efforts promote the evolution and spread of an asymptomatic strain. Yet as the ability to detect asymptomatic infections improves (Fig. 1C), increasing quarantining efforts can successfully suppress the asymptomatic strain.

A further driver of the qualitative differences is the transmission risk for the resilient host (Table S3). Because these hosts are more likely to be the target of strong public health interventions, increased transmission in this host class can result in greater selection on the asymptomatic strain, until sufficiently high isolation efforts in this host class facilitate successful disease suppression. Supplementary Table S5 again illustrates how this outcome depends on whether there is a reasonable prospect of identifying asymptomatic mutant infections.

Finally, in addition to public health interventions and host heterogeneities, the probability (${\pi }_{h,j}$) that infections from the evolved strain are asymptomatic also mediates the nature of how quarantine and isolation effort interact to drive the evolution and spread of the asymptomatic virus. For instance, when the ability to detect asymptomatic cases is low, as the evolved virus becomes increasingly asymptomatic (particularly towards the resilient host), the effect of increasing quarantine efforts changes from successful suppression to facilitating the evolution of the asymptomatic strain at intermediate isolation efforts (Supplementary Table S5). This shift occurs because strains that cause more asymptomatic infections are harder to suppress even when quarantine efforts are high, whereas viruses that are more readily detected are easier to control by increasing quarantine efforts. Once more, we see how these effects are magnified when the probability of being asymptomatic is high for resilient hosts, because the resilient host is also more likely to be subject to increased isolation and quarantine efforts.

These results thus far characterize the evolutionary and epidemiological consequences of host heterogeneities in public health responses. When there is a more even application of isolation and quarantining across host classes (so that $q_{h,j}=q_{h^{\prime },j},h\ne h^{\prime }$), a somewhat different pattern emerges. In general, we find that more uniformity in transmission risk and isolation and quarantining efforts between vulnerable and resilient hosts usually reduces the prevalence of the more asymptomatic virus (Fig. 2A–F). An exception is when isolation efforts are very low but disease suppression otherwise results (Fig 2G–I). In this scenario, very modest isolation efforts prevent the evolutionary emergence of the asymptomatic strain when the public health response is unequal (Fig. 2G), but can result in selective pressure for the more asymptomatic strain when public health responses and transmission risk are more even across host classes (Fig. 2H, I).

Fig. 2.

The effect of increasing isolation and quarantine efforts or reducing transmission risk for the neglected host, in terms of long-term prevalence of among hosts infected with the novel, more asymptomatic virus in model (2) as a function of quarantine and isolation effort. In this, and in subsequent figures, the color scheme follows Fig. 1. A $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. B $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. C $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. D $r_{h,h\prime }=100,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.11$. E $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. F $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=13,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. G $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. H $r_{h,h\prime }=0.1,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. I $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$

Supplementary Table S6 summarizes the conditions under which the distinct evolutionary outcomes identified in Fig. 1 result when both host types are subject to the same extent of isolation and quarantining. A key result is that as a generality, even modest levels of isolation can prevent the evolution and spread of the novel virus. Improved detection of symptomatic infections by the evolved virus readily causes suppression (Supplementary Table S6). Even when the ability to detect such symptomatic infections is more limited, modest isolation efforts can help limit the spread of the more asymptomatic, novel strain, with high isolation facilitating disease suppression. Still, if the ability to detect asymptomatic infections is low, there is more potential for quarantining and isolation to drive infections by the novel virus. This is especially the case when there is a high transmission risk to the resilient host class, as well as a lower probability that infections by the evolved strain are asymptomatic (Supplementary Table S6 and Fig. 2A–C).

Scenario 2: intervention efforts reflect prevalence of symptomatic infections

As was the case for when isolation and quarantine efforts are constant through time, model (2) predicts several qualitatively distinct evolutionary outcomes when these efforts respond to the prevalence of symptomatic hosts (Fig. 3). First, above a certain isolation efficacy, most infections are with the novel strain, with the prevalence of such infections potentially depending on the quarantine efficacy (Fig. 3A–D). In contrast to the case with constant control, when control efforts track symptomatic infections, we find increasing public health responses can prove counterproductive; higher removal efforts ultimately select for the asymptomatic strain, which subsequently spreads to high prevalence.

Fig. 3.

The range of possible qualitative behavior of long-term prevalence of all hosts infected with the novel, more asymptomatic virus in model (2) as a function of time-varying quarantine and isolation efforts. In addition to the results above, the model also produced outcomes where the novel virus could not successfully spread or would infect all hosts irrespective of the isolation and quarantine efforts (results not shown). A $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=v_{\theta }=1$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. B $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=0.001,v_{\theta }=1$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. C $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. D $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. E $r_{h,h\prime }=0.1,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=13,v_q=0.001,v_{\theta }=1$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. F $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=0.001,v_{\theta }=1$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. G $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=1.1,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.1$. H $r_{h,h\prime }=0.1,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=1,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$

A second class of possible dynamical behavior results when higher isolation efforts either facilitate suppressing the evolution and spread of the asymptomatic strain, particularly as quarantine effort decreases (Fig. 3E, F). Finally, although very subtle, in some cases modest to intermediate levels of isolation coupled with low quarantining slightly reduced the prevalence of the novel strain (Fig. 3G), while in others increasing isolation efforts are what reduced the strain’s prevalence (Fig. 3H).

Supplementary Tables S57-S8 characterize the processes driving these qualitatively distinct eco-evolutionary predictions. In comparison to the case where intervention efforts are constant through time, the qualitative differences in dynamical behaviors are driven much more by the detection ability of the novel strain rather than by different likelihoods of the novel virus causing asymptomatic infections.

One important result is that unlike in the case where intervention efforts are constant, high isolation efforts coupled with low quarantine efficacy most readily promote the emergence of the novel strain when the detection of asymptomatic and symptomatic infections are both high. When intervention intensity tracks the prevalence of symptomatic infections, the more symptomatic strain is quite readily removed as isolation efforts increase, and as quarantine efforts are relaxed following a reduction in the prevalence of symptomatic infections, the novel virus can then escape suppression. This outcome stands in contrast to what we found when intervention efforts were constant. There, pathogen suppression often occurred when the ability to detect asymptomatic and symptomatic infections by the evolved virus were high (Supplementary Tables S5-S6).

A somewhat different pattern emerges when most asymptomatic infections by the novel viral strain go undetected. Under this scenario, even if the detection of symptomatic infections is high, greater quarantine effort can still promote the evolution and spread of the more asymptomatic strain (Supplementary Table S7). Thus, the qualitative trend is somewhat akin to the case when intervention efforts were constant. Because quarantining efficacy is a composite of quarantining effort and the ability to detect asymptomatic cases, these results highlight how at intermediate levels of removing asymptomatic hosts via quarantining the novel virus spreads to high prevalence. If asymptomatic infections are readily detected and removed, then high isolation efforts can facilitate disease eradication. But if the ability to quarantine asymptomatic infections is very low, this reduces the selective advantage for more asymptomatic viruses.

We further find conditions under which greater equality between host classes in public health responses diffuse selection pressure for the asymptomatic strain (Fig. 4A, B vs. E and Supplementary Table S8). This contrasts to the case when the intensity of public health interventions is constant through time. The discrepancy arises because when intervention efforts track symptomatic cases, selection for being asymptomatic is low when infection rates are low. When infection rates are low, even symptomatic infections are not removed efficiently from the system. This relaxes selection pressure against the symptomatic strain. This relaxed selection pressure, in turn, prevents the asymptomatic strain from increasing to high frequency.

Fig. 4.

The effect of increasing isolation and quarantine efforts for the neglected host or equalizing transmission risk, in terms of long-term prevalence of among hosts infected with the novel, more asymptomatic virus in model (2) as a function of time-varying quarantine and isolation efforts. A $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=v_{\theta }=1$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. B $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=0.001,v_{\theta }=1$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. C $r_{h,h\prime }=1,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=0.001,v_{\theta }=1$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. D $r_{h,h\prime }=10,{\pi }_{1,2}/{\pi }_{1,1}=5,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=1$. E $r_{h,h\prime }=100,{\pi }_{1,2}/{\pi }_{1,1}=1.1,{\pi }_{2,2}/{\pi }_{2,1}=5,v_q=0.001,v_{\theta }=0.001$ and $q_{h,·}/q_{h\prime ,·}={\theta }_{h,·}/{\theta }_{h\prime ,·}=0.11$

At the same time, in some cases even quite low isolation efforts can induce the evolution and spread of the more asymptomatic virus, with quarantine effort having little effect (Fig. 4C, D). This outcome mirrors the results for the constant control effort case characterized in, e.g., Fig. (2I). We find this to especially be the case if most infections are of the more common, vulnerable host class that experiences lower isolation and quarantine efficacy. When the intensity of public health intervention is even, a higher infection risk for the vulnerable host class can amplify selection pressure for the more asymptomatic strain. Indeed, increasing the ability to detect infections of the vulnerable host class selects for the novel virus when there is enhanced infection risk among the vulnerable host class (Supplementary Table S8). When infection risks are more even across host classes, improving isolation and quarantine efforts for the vulnerable host class reduces selection for the more asymptomatic strain (Supplementary Table S8).

Discussion

When epidemiological monitoring fails to detect asymptomatic carriers, the pathogens they harbor are able to spread with less friction through the host population. By contrast, potentially symptomatic lineages are subject to detection and, through contact-tracing even asymptomatic infections are identified quarantined, thereby pruning that particular viral lineage. Modeling provides a uniquely systematic approach linking quantifiable public health interventions such as isolation efficacy to potential evolutionary epidemiological outcomes (i.e., the infection prevalence of a mutant, less symptomatic virus strain). These models can therefore allow us to consider the evolutionary implications of public health interventions in a rapidly evolving viral lineage.

Our analyses using SARS-coronaviruses as a case study illustrate how reliance on symptoms-driven reporting and control, however well-meaning, can ultimately shift the pathogen’s fitness landscape to select for highly-transmissible, pandemic strains. We show that such selection for an asymptomatic pathogen is often most acute when isolation and quarantine efforts are intense, but fall short of complete disease suppression. Our analyses further indicate that when host removal depends on the prevalence of symptomatic infections, even very high levels of isolation effort can facilitate the emergence and extensive spread of more asymptomatic strains. The rapid spread in late 2021 of the potentially less severe omicron (B.1.1.529) variant of SARS-CoV-2 [10, 23] suggests how a viral genotype less likely to cause symptomatic infections can potentially remain undetected, thereby evading surveillance efforts to increase in frequency.

Our results further highlight the critical role host heterogeneity plays in driving the evolutionary consequences of isolation and quarantining symptomatic individuals. We find in general that implementing isolation and quarantine evenly across host classes can facilitate disease suppression and thereby reduce selection for more asymptomatic viruses. If reducing disease burden is a long-term objective, localized containment, particularly if it is haphazardly implemented across host classes, may prove to be evolutionarily unsustainable. However, under some conditions, reducing disparities among host classes can also increase overall selection pressure against more symptomatic strains. For instance, when isolation and quarantine efficacy are just low enough to permit the suppression of a more asymptomatic virus by a less asymptomatic virus, evenly carrying out isolation or quarantining efforts across host classes can occasionally shift the selective balance to promote the spread of the more asymptomatic virus.

One dimension of host heterogeneity that had a slightly counter-intuitive effect was when transmission risk was more even across host classes, this in fact promoted suppression. We found that at least in some cases, increasing transmission for the resilient host operates, to some degree, in a manner analogous to the dilution effect, in which viral infections of hosts that cannot transmit the pathogen slow the epidemic (e.g., [104]). When resilient hosts are more likely to be successfully identified and removed from the population of infectious hosts, the ability of isolation and quarantine to suppress the disease becomes more apparent when these hosts are more likely to get infected. One consequence of this is that when resilient hosts, who are more likely to be successfully identified and removed if infected, are less likely to be exposed to the pathogen, then this can create conditions conducive the evolution and spread of a more asymptomatic strain. Thus, evening transmission risk across host classes may be one strategy for disease control when infections in some hosts are more likely to be removed from the population. Taken together, these results further suggest the importance of broad-based information dissemination aimed at reducing transmission risk not only for managing disease prevalence [134, 135], but also for minimizing the risk of novel strains emerging.

We wish to be clear. Our goal here is not to argue against the value of case detection at the point of care and subsequent, vigorous contact tracing. Because many emerging zoonotic pathogens lack effective prophylactics or treatment, identifying and removing infectious individuals remains the primary control strategy [8, 35]. At its best, disease control relying on isolating and tracking severe cases can ensure that limited public health resources are optimally allocated to reducing transmission [5, 35, 62, 95, 106]. These policies proved tentatively effective in suppressing SARS-CoV-2 in some cases when carried out with precision, multi-sector societal collaboration and adequate financial and operational support [18, 21, 76, 92, 136, 142]. By contrast, several jurisdictions also struggled to contain the outbreak well into the pandemic [16, 117, 142].

Rather, our claim is that symptoms-based surveillance is a necessary, but, from an evolutionary perspective, potentially insufficient component of a robust public health strategy aimed at preventing disease emergence. One implication is that although most public health responses to emerging zoonotic pathogens are necessarily reactive, an anticipatory approach to case identification may be necessary for highly infectious pathogens such as coronaviruses known to cause asymptomatic infections. For instance, routine, widespread randomized metagenomic surveys of known and suspected reservoirs would more closely approximate the “constant intervention efforts” scenario we model. Our results indicate, however, that such efforts need to be effective and broadly applied across hosts to prevent the emergence of asymptomatic viruses. Whether anticipatory control measures such as these are realistic given constraints on public health budgets remains an open question [129].

By design, but also due to limitations in the data, our numerical analyses surveyed a wide segment of parameter space. Figures 1, 2, 3 and 4 illustrate the diverse dynamical outcomes predicted by our model. Given the frequency with which a more asymptomatic coronavirus emerged and spread in our simulations, as well as the distinct qualitative outcomes of the role of isolation and quarantine efforts, we highlight the need for better measurements of key quantities. A case in point is the degree to which mutations enable viruses to cause asymptomatic infections. Because selection against symptomatic strains involves a reciprocal interplay between epidemiological and evolutionary processes, the evolution and spread of the novel strain at times depended on the probability that infection with the novel virus is asymptomatic. Thus, any intervention measure aimed at preventing the emergence of an asymptomatic strain must take into account the viral genotype-phenotype map. To be sure, whether a virus causes symptoms also depends heavily on the host’s biology; a strain that is initially less asymptomatic may eventually be selected for greater virulence as a result of within-host competition [36, 43, 68, 105]. Indeed, the emergence and rapid spread of the omicron (B.1.1.529) variant of SARS-CoV-2 in late 2021 became most apparent following increased hospitalizations caused by the strain [15, 39, 55]. Thus, in light of our results, given the growing capacity to characterize large amounts of viral sequence variation, explaining how this variability drives the propensity of viral infections to be symptomatic (such as [63]) seems particularly warranted.

Concluding remarks

Our analyses show that intense symptoms-driven disease control selects for asymptomatic strains, particularly when these efforts fail at complete suppression. Moreover, when interventions reflect the prevalence of symptomatic infections, higher removal efforts more readily select for asymptomatic strains. These inadvertent evolutionary outcomes are exacerbated when disease control efforts are unevenly applied across host classes. These phenomena are especially likely to be observed where a wide gap exists between resilient and vulnerable host classes. For instance, underestimating the severity of population spread may have amplified the morbidity caused by the delta (B.1.617.2) variant of SARS-CoV-2 in India in 2021 [29]. Thus, the evolution and spread of asymptomatic viruses is driven by a reciprocal interplay between public health intervention measures and prevailing host population structure, on the one hand, and the nature of the host-pathogen interaction at the level of individual hosts (particularly as it concerns the likelihood of symptomatic infections), on the other. Our approach integrates two critical themes at the heart of the SARS-CoV-2 pandemic: the ability of a rapidly evolving, often asymptomatic pathogen to circulate widely (e.g., [82, 91, 114, 121, 131, 139]) and considerable evidence of systematic disparities in public health intervention efforts (e.g., [1, 17]). Understanding the interplay between these two components is critical to elucidating the evolutionary dynamics of viruses such as SARS-CoV-2. To our knowledge, our study is the first to explore the relationship between epidemiological dynamics, uneven allocation of public health resources and viral evolution, particularly as it concerns the emergence of strains more likely to be less symptomatic.

These results also have implications for other pathogens. At the time of writing, several other major human pathogens (dengue, Methicillin-resistant Staphylococcus aureus) similarly lack effective and widespread prophylactics or treatment. In light of our conclusions, we urge epidemiological monitoring efforts to seriously consider undertaking randomized testing to avoid inadvertently creating a surveillance regime that selects for more asymptomatic strains, and to do so in a way that is consistent across host classes. Our investigation provides a critical first step towards providing quantitative guidance for determining evolutionarily appropriate levels of interventions for coronaviruses and other highly transmissible pathogens.

Supplementary Information

Below is the link to the electronic supplementary material.

Author Contributions

Conception and design: KWO, RW, RW, LFC. Coding and creation of new software: KWO, VO. Analysis and interpretation: KWO, VO, LFC. Drafted: KWO, LFC critical revision: all authors. Approved final version: all authors.

Funding

V.O. was supported by funding from the College of Arts and Sciences at the University of St. Thomas. R. W. was supported with a grant from the Robert Wood Johnson Foundation.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Code availability

All code used in the analysis is accessible on github (https://anonymous.4open.science/r/9a717a70-7812-4c1e-a686-c018cc3664d4/) and is released under the GNU Public License v3 [118].