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Multiscale feedback drives viral evolution and epidemic dynamics

Juan C. Muñoz-Sánchez, J. Tomás Lázaro, Josep Sardanyés, Santiago F. Elena

TL;DR

The paper presents a minimal multiscale framework that links fast within-host quasispecies dynamics of two viral variants to slow population-level SIRS transmission via explicit bidirectional coupling. Transmission rates depend on instantaneous virion loads, while within-host fitness is modulated by the population prevalence, enabling a slow-fast reduction that yields a unified expression for $\mathcal{R}_0$ and context-dependent, epidemic-state–driven error thresholds $\mu_c(\nu)$. The model predicts damped oscillations and two contrasting regimes: a vaccine-like avirulent strain that benignly replaces the master sequence, and a burnout-like hypervirulent strain that self-limits through host depletion, with macroscopic dynamics tracing signatures in incidence and within-host composition. These insights provide a principled link between viral load, variant composition, and epidemic outcomes, offering testable predictions for how micro-scale fidelity interacts with macro-scale transmission in shaping epidemics.

Abstract

We introduce a minimal multiscale framework that links within-host virus dynamics to population-level SIRS epidemiology through explicit, bidirectional coupling. At the microscopic layer, a two variant quasispecies (master and mutant genomes with packaged virions) evolves on a fast timescale. At the macroscopic layer, two infectious classes (master- and mutant-infected), susceptible, recovered, and deceased individuals evolve slowly. The two scales are connected through transmission rates that depend on instantaneous virion abundance and through prevalence-weighted effective replication rates. Exploiting the timescale separation, we formalize a coarse-grained slow-fast closure: the genome-virion subsystem rapidly relaxes to quasi-steady states that parameterize time-varying transmission in the slow epidemiological system. This yields an integrated expression for the basic reproduction number and sharp inequalities that delineate coexistence versus exclusion. A key prediction is a context-dependent error threshold that shifts with the prevalence ratio, enabling transient pseudo-error catastrophes driven by epidemic composition rather than intrinsic fidelity. Linearization reveals parameter regions with damped oscillations arising solely from the microscopic-macroscopic feedback. Two illustrative extremes bracket the model's behavior: an avirulent strongly immunizing strain that benignly replaces the master, and a hypervirulent weakly immunizing that self-limits via host depletion and collapses transmission. This framework yields testable signatures linking viral load, incidence, and within-host composition.

Multiscale feedback drives viral evolution and epidemic dynamics

TL;DR

The paper presents a minimal multiscale framework that links fast within-host quasispecies dynamics of two viral variants to slow population-level SIRS transmission via explicit bidirectional coupling. Transmission rates depend on instantaneous virion loads, while within-host fitness is modulated by the population prevalence, enabling a slow-fast reduction that yields a unified expression for and context-dependent, epidemic-state–driven error thresholds . The model predicts damped oscillations and two contrasting regimes: a vaccine-like avirulent strain that benignly replaces the master sequence, and a burnout-like hypervirulent strain that self-limits through host depletion, with macroscopic dynamics tracing signatures in incidence and within-host composition. These insights provide a principled link between viral load, variant composition, and epidemic outcomes, offering testable predictions for how micro-scale fidelity interacts with macro-scale transmission in shaping epidemics.

Abstract

We introduce a minimal multiscale framework that links within-host virus dynamics to population-level SIRS epidemiology through explicit, bidirectional coupling. At the microscopic layer, a two variant quasispecies (master and mutant genomes with packaged virions) evolves on a fast timescale. At the macroscopic layer, two infectious classes (master- and mutant-infected), susceptible, recovered, and deceased individuals evolve slowly. The two scales are connected through transmission rates that depend on instantaneous virion abundance and through prevalence-weighted effective replication rates. Exploiting the timescale separation, we formalize a coarse-grained slow-fast closure: the genome-virion subsystem rapidly relaxes to quasi-steady states that parameterize time-varying transmission in the slow epidemiological system. This yields an integrated expression for the basic reproduction number and sharp inequalities that delineate coexistence versus exclusion. A key prediction is a context-dependent error threshold that shifts with the prevalence ratio, enabling transient pseudo-error catastrophes driven by epidemic composition rather than intrinsic fidelity. Linearization reveals parameter regions with damped oscillations arising solely from the microscopic-macroscopic feedback. Two illustrative extremes bracket the model's behavior: an avirulent strongly immunizing strain that benignly replaces the master, and a hypervirulent weakly immunizing that self-limits via host depletion and collapses transmission. This framework yields testable signatures linking viral load, incidence, and within-host composition.
Paper Structure (21 sections, 5 theorems, 71 equations, 9 figures)

This paper contains 21 sections, 5 theorems, 71 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Multiscale mathematical model
  3. Multiscale coupling between microscopic and macroscopic dynamics
  4. Results and discussion
  5. Equilibrium points
  6. Computation of the basic reproduction number $\mathcal{R}_0$
  7. On the condition for pandemic growth
  8. Case of interest 1: a vaccine-like viral strain.
  9. Equilibrium points
  10. Equilibria stability
  11. The limit case $\mathbf{\pi_1 \rightarrow +\infty}$
  12. Case of interest 2: the burnout viral strain.
  13. Discussion
  14. Concluding remarks
  15. Acknowledgements
  16. ...and 6 more sections

Key Result

Lemma 1

Let us assume that $(I_0^*,I_1^*)$, both positive, are the $I$-variables of an equilibrium point for the macroscopic system eq:sirs:S-eq:sirs:D, and denote $\nu^*=\nu_0^*= \nu(I_0^*,I_1^*)\in (0,1)$, as defined in def:nu, the relative prevalence of $I_0^*$ with respect to $I_0^*+I_1^*$. Assume that for $\nu^*>0$. Then, the equilibrium points for the genome-virion system associated to the $I$-equi

Figures (9)

  • Figure 1: Schematic representation of the multiscale model (left) and summary of the associated parameters (right). Microscopic variables include the master and mutated genomes ($g_0$, $g_1$) and their corresponding virions ($v_0$, $v_1$), while macroscopic variables comprise susceptible ($S$), infected ($I_0$, $I_1$), recovered ($R$), and deceased ($D$) individuals.
  • Figure 2: Schematic representation of the four different equilibrium points (DFE, NME, NmutE, and CSE) in terms of the parameters $\beta_{00}, \beta_{01}, \beta_{11}$, and $\chi$. Dashed lines represent open faces of the prism not included in the corresponding domain. The black dot indicates the origin.
  • Figure 3: (A) $\omega$-limits of the principal trajectory with i.c. \ref{['case:interest:2:ic']} and fixed parameters \ref{['caseinterest1:CSE:parameters']} for three different combinations of parameters. Each point of the plot is colored according to the type of equilibrium reached: NME, DFE, and CSE. The first and second panels correspond to $a_1 = 6$, while the third one fixes $\mu = 0.675$. These, together with $f_1 = 0.2$, will be the nominal values employed for the case study. They are highlighted in the plots using solid white lines to facilitate visual reference. (B) Time evolution of the macroscopic and microscopic variables for three scenrios, highlighted in the upper panel. Each simulation illustrates convergence toward a different equilibrium.
  • Figure 5: (A) Evolution of the family of CSE$_1$ points, varying with $\pi_1$ (see Figure \ref{['fig:CEEpoints']} (left)), for the $(S,I_0,I_1)$ variables (left) and $\beta_{00}^*$, $\beta_{01}^*$ (right). (B) Mechanism leading the CSE$_2$ point (see Figure \ref{['fig:CEEpoints']}(right)) towards its collision with the DSE point $(1,0,0,0)$ as $\pi_1$ tends to $\pi_1^{\diamond}$. Left: $(S,I_0^*,I_1^*)$ variables. Right: transmission rates $\beta_{00}^*$ and $\beta_{01}^*$.
  • Figure 6: Eigenvalues spectrum of the jacobian matrix at the NME point. The background color stands for the corresponding numerical $\omega$-limit of the principal trajectory \ref{['case:interest:2:ic']}, see Figure \ref{['fig:CoI3_highIRmut']}, which are NME, DFE and CSE. The two vertical green lines indicate the region in which the system exhibits complex eigenvalues, i.e., where the jacobian has eigenvalues with non-zero imaginary parts, corresponding to the interval $\pi_1 \in (0.357.., 5.129..)$.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Lemma 1: macroscopic to microscopic equilibrium points
  • Proposition 1: disease-free equilibrium (DFE) points
  • Proposition 2: no master equilibrium (NME) points
  • Proposition 3: no mutant equilibrium (NmutE) points
  • Proposition 4: Co-circulating strains equilibrium (CSE) points
  • Remark 1
  • Remark 2