Emergence of microbial host dormancy during a persistent virus epidemic

TL;DR

This work develops a minimal stochastic model for a microbial host population under a persistent lytic virus epidemic and analyzes the emergence of a costly dormancy trait introduced by a single mutant. By coupling a stochastic invasion phase (via multi-type branching processes) with a large-population deterministic phase (six-dimensional ODE dynamics), the authors derive explicit conditions for dormancy emergence and quantify the invasion probability and the time to macroscopic establishment in terms of the system’s parameters, notably the dormancy initiation probability $q$, the host birth rates $ ext{rate}( ext{1})= ext{something}$ and $ ext{rate}( ext{2})$, the virus-escape rates, and the carrying capacity $K$. They show that dormancy can invade only within a nontrivial parameter range (the Inv conditions) where the reduced reproduction of the dormancy-bearing host is compensated by the ability to escape infection, and they derive the corresponding largest eigenvalues $ ext{eigenvalues of }J^*$ and $ ilde{J}$ that govern growth. After invasion, the model suggests either a stable six-type coexistence (all host types and the virus persist) or fixation of one host type with the virus continuing to persist, with precise asymptotics and conjectures supported by simulations. The results illuminate how contact-mediated dormancy can arise and persist in microbial communities facing ongoing viral pressure, highlighting the trade-offs between reproduction costs and infection-avoidance benefits, and they lay groundwork for future rigorous proofs of global stability and extensions to other dormancy forms.

Abstract

We study a minimal stochastic individual-based model for a microbial population challenged by a persistent (lytic) virus epidemic. We focus on the situation in which the resident microbial host population and the virus population are in stable coexistence upon arrival of a single new ``mutant'' host individual. We assume that this mutant is capable of switching to a reversible state of dormancy upon contact with virions as a means of avoiding infection by the virus. At the same time, we assume that this new dormancy trait comes with a cost, namely a reduced individual reproduction rate. We prove that there is a non-trivial range of parameters where the mutants can nevertheless invade the resident population with strictly positive probability (bounded away from 0) in the large population limit. Given the reduced reproductive rate, such an invasion would be impossible in the absence of either the dormancy trait or the virus epidemic. We explicitly characterize the parameter regime where this emergence of a (costly) host dormancy trait is possible, determine the success probability of a single invader and the typical amount of time it takes the successful mutants to reach a macroscopic population size. We conclude this study by an investigation of the fate of the population after the successful emergence of a dormancy trait. Heuristic arguments and simulations suggest that after successful invasion, either both host types and the virus will reach coexistence, or the mutants will drive the resident hosts to extinction while the virus will stay in the system.

Figures (2)

• Figure 1: Conjectured invasion outcome depending on $\lambda_2\in(1.2\color{black},4)$ and $q\in(0.01,0.99\color{black})$ for fixed $\lambda_1=3.15,\mu_1=1, C=1, D=0.5, r=1, v=1, \kappa=0.1, \sigma=2, m=10, \mu_3=0.5$. $\color{red1}$ Red (left): fixation of type 1 (coex. with 3), $\color{lightgreen}$ light green (top mid/left): stable 6-dim. coexistence (type 2 is not able to coexist with type 3 in absence of type 1), $\color{darkgreen}$ dark green (mid/left): stable 6-dim. coexistence (type 2 is able to coexist with type 3), $\color{orange1}$ orange (top right): fixation of type 2a (without 3), $\color{purple1}$ purple (bottom mid/left): fixation of type 1 (coex. with 3), $\color{blue1}$ blue (bottom right): fixation of type 2 (coex. with 3). The curve separating red from purple, light green from dark green, and orange from blue corresponds to $\bar{n}_{2a}=\widetilde{n}_{2a}$. Type 2 only coexists with type 3 below this curve. The light green and the orange regime are separated by the line $\lambda_2=\lambda_1=3.15$; below this value of $\lambda_2$, fixation of type 2a without 3 not possible. The dark green area reaches this line at 0, with vanishing width.
• Figure 7: Here, the parameters are the same as in Figure \ref{['figure-regimes']}, apart from $r$, which is increased to $3$, and $\kappa$, which is increased to $1$, so that $r\kappa\mu_1 > v \sigma$. Compared to Figure \ref{['figure-regimes']}, only the following colours have a new meaning. $\color{lightgreen}$ Light green (top mid/right): founder control (type 2 is not able to coexist with type 3 in absence of type 1), $\color{darkgreen}$ dark green (bottom mid/right): founder control (type 2 is able to coexist with type 3). The meaning of the following colours is unchanged. $\color{red1}$ Red (left): fixation of type 1 (coex. with 3), $\color{orange1}$ orange (top right): fixation of type 2a (without 3), $\color{purple1}$ purple (bottom mid): fixation of type 1 (and 3), $\color{blue1}$ blue (bottom right): fixation of type 2 (and 3). The dark green regime reaches the black line $\lambda_2=\lambda_1$ at the $\lambda_2$ axis with a vanishing width.

Theorems & Definitions (34)

• Remark 1.1
• Theorem 1.2: Invasion of a dormancy trait -- informal version
• Remark 1.3: On the necessity of a persistent (stable) virus epidemic
• Proposition 2.1: Stability of equilibria inherited from sub-systems
• Remark 2.2: Link to classical invasion analysis in adaptive dynamics
• Proposition 2.3
• Remark 2.4: Heuristics for the emergence of a full coexistence equilibrium
• Proposition 2.5
• Remark 2.6: Conjectured stability of the coexistence equilibrium for non-positive $\widetilde{n}_3$
• proof : Proof of Proposition \ref{['prop-fixation']}