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Transmission of multiple pathogens across species

Clotilde Djuikem, Julien Arino

TL;DR

The paper develops a multi-host SLIR framework to study the cross-species spread of multiple pathogens across populations. It combines deterministic ODEs, a continuous-time Markov chain representation, and multitype branching-process approximations to quantify outbreak probabilities near the disease-free equilibrium. Through targeted cases with two species and one or two pathogens, it derives explicit expressions for basic reproduction numbers, extinction probabilities, and case-specific dynamics, highlighting how demographic and incubation parameters often dominate outbreak risk. The approach provides a structured way to assess spillover and pathogen establishment in aquatic ecosystems, with potential extensions to more complex interactions and spatial structures.

Abstract

We analyse a model that describes the propagation of many pathogens within and between many species. A branching process approximation is used to compute the probability of disease outbreaks. Special cases of aquatic environments with two host species and one or two pathogens are considered both analytically and computationally.

Transmission of multiple pathogens across species

TL;DR

The paper develops a multi-host SLIR framework to study the cross-species spread of multiple pathogens across populations. It combines deterministic ODEs, a continuous-time Markov chain representation, and multitype branching-process approximations to quantify outbreak probabilities near the disease-free equilibrium. Through targeted cases with two species and one or two pathogens, it derives explicit expressions for basic reproduction numbers, extinction probabilities, and case-specific dynamics, highlighting how demographic and incubation parameters often dominate outbreak risk. The approach provides a structured way to assess spillover and pathogen establishment in aquatic ecosystems, with potential extensions to more complex interactions and spatial structures.

Abstract

We analyse a model that describes the propagation of many pathogens within and between many species. A branching process approximation is used to compute the probability of disease outbreaks. Special cases of aquatic environments with two host species and one or two pathogens are considered both analytically and computationally.
Paper Structure (26 sections, 4 theorems, 102 equations, 9 figures, 6 tables)

This paper contains 26 sections, 4 theorems, 102 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The general model
  3. Formulation of the deterministic model
  4. Notation
  5. Basic analysis of the deterministic model
  6. The continuous time Markov chain model
  7. Branching process approximation of the CTMC
  8. Case of one pathogen and two species
  9. The models and their basic analysis
  10. The ODE model when $P=2$ and $V=1$
  11. The CTMC model when $P=2$ and $V=1$
  12. Branching process approximation
  13. Infectious Hematopoietic Necrosis ($P_1\leftrightarrow P_2$)
  14. Transmission from wild to farmed fish ($P_1\to P_2,P_2\not\to P_1$)
  15. Viral Hemorrhagic Septicaemia ($P_1\to P_2,P_2\not\to$)
  16. ...and 11 more sections

Key Result

Lemma 1

The disease-free equilibrium $\bm{E}_0^{sys:ODE-multi-species}$ of sys:ODE-multi-species is locally asymptotically stable if $\mathcal{R}_0^{sys:ODE-multi-species} < 1$ and unstable if $\mathcal{R}_0^{sys:ODE-multi-species} > 1$.

Figures (9)

  • Figure 1: Partial rank correlation coefficient (PRCC) of the probability \ref{['eq:pro-inf12']} of IHN outbreak for different initial conditions $y_0=(l_{10},l_{20}, i_{10}, i_{20}) \in (e_1,e_2,e_3,e_4)$. The range of parameter values remaining in Table \ref{['tab:param-and-range-2p1v']}.
  • Figure 2: PRCC of the probability \ref{['eq:P_2nonfec1']} of disease outbreak $\mathbb{P}_{\text{outbreak}}^{W\to F}$ for four different initial conditions $y_0=(\ell_{10},\ell_{20}, i_{10}, i_{20}) \in (e_1,e_2,e_3,e_4)$. Parameter values ranges in Table \ref{['tab:param_ranges_2noninfec1']}.
  • Figure 3: Probability \ref{['eq:P_2nonfec1']} of disease outbreak as a function of mortality rates of wild and farmed fish. For two initial conditions $y_0=(\ell_{10},\ell_{20}, i_{10},i_{20})\in (e_3,e_4)$. Parameter values in Table \ref{['tab:param-and-range-2p1v']}, with $\beta_{11}=\beta_{22}=10^{-5}$, $\beta_{21}=5\beta_{11}$.
  • Figure 4: PRCC of the probability of disease outbreak \ref{['eq:P_2noinfec']} for different initial conditions $y_0=(\ell_{10},\ell_{20}, i_{10}, i_{20}) \in (e_1,e_2,e_3,e_4)$. Parameter values ranges in Table \ref{['tab:param-and-range-2p1v-2noinf']}.
  • Figure 5: Equilibrium prevalence of infection $I_2^\star$ in population 2 as a function of the reproduction number $\mathcal{R}_{02}$ for the pathogen in species 2. The different curves correspond to different values of prevalence $I_1^\star$ in population 1.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • Conjecture 5