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Contact process with viral load

Marco Seiler

TL;DR

A duality relationship is derived between the two variant, which is used to uncover a phase transition regarding invariant distributions in the second variant, and for both variants a Poisson construction is presented.

Abstract

In this article, we present two novel variants of the contact process. In the first variant individuals carry a viral load. An individual with viral load zero is classified as healthy and otherwise infected. If an individual becomes infected it begins with a viral load of one, which then evolves according to a Birth-Death process. In this model, viral load indicates severity of the infection such that individuals with a higher load can be more infectious. Moreover, the recovery times of individual is not necessarily exponentially distributed and can even be chosen to follow a power-law distribution. In the second variant individuals are permanently infected albeit in two states: actively infected or dormant. The dynamics of these individual states are again governed by a Birth-Death process. Dormant infections do not interact with neighbouring individuals but may reactivate spontaneously. Active infections reactivate dormant neighbours at a constant rate and may become dormant themselves. We present for both variants a Poisson construction. For the first model, we study the phase transition of survival and discuss existence of a non-trivial upper invariant law. Additionally, we derive a duality relationship between the two variant, which we use to uncover a phase transition regarding invariant distributions in the second variant.

Contact process with viral load

TL;DR

A duality relationship is derived between the two variant, which is used to uncover a phase transition regarding invariant distributions in the second variant, and for both variants a Poisson construction is presented.

Abstract

In this article, we present two novel variants of the contact process. In the first variant individuals carry a viral load. An individual with viral load zero is classified as healthy and otherwise infected. If an individual becomes infected it begins with a viral load of one, which then evolves according to a Birth-Death process. In this model, viral load indicates severity of the infection such that individuals with a higher load can be more infectious. Moreover, the recovery times of individual is not necessarily exponentially distributed and can even be chosen to follow a power-law distribution. In the second variant individuals are permanently infected albeit in two states: actively infected or dormant. The dynamics of these individual states are again governed by a Birth-Death process. Dormant infections do not interact with neighbouring individuals but may reactivate spontaneously. Active infections reactivate dormant neighbours at a constant rate and may become dormant themselves. We present for both variants a Poisson construction. For the first model, we study the phase transition of survival and discuss existence of a non-trivial upper invariant law. Additionally, we derive a duality relationship between the two variant, which we use to uncover a phase transition regarding invariant distributions in the second variant.

Paper Structure

This paper contains 10 sections, 21 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Models and main results
  3. The contact process with viral load
  4. Contact process with lingering infections
  5. Discussion
  6. Construction of the processes
  7. Some consequences of the Poisson construction for the CPVL
  8. Duality of CPVL and CPLI
  9. Survival of the CPVL and its upper invariant law
  10. Phase transition of the CPLI

Key Result

Theorem 2.2

Let $\Lambda,b,d:\mathds{N}_0\to [0,\infty)$ satisfy Assumption ass:RateAssumption. Then, there exists a Feller process $\boldsymbol{\eta}=(\boldsymbol{\eta}_t)_{t\geq 0}$ with values in $\mathds{N}_0^V$, whose generator agrees with the operator associated to the transitions CPViral on $\mathcal{C}_

Figures (2)

  • Figure 1: Visualisation of the Poisson construction of the viral load on vertex $x$ with rate function $b(n)=n$ and $d(n)=n+\tfrac{1}{2}$ for all $n\geq 1$. The squares and circles signify points from $\mathbb{U}$ and $\mathbb{D}$, where the coloured ones are part of the construction. Furthermore, red cross indicates a recovery of $x$ and the red dot an infection from a neighbour.
  • Figure 2: Visualisation of the on site dynamic for both process with $b(n)=n$ and $d(n)=n+\tfrac{1}{2}$ for all $n\geq 1$, where the symbols are the same as in Figure \ref{['fig:PoissonConstruction']}. Note that the meaning of the squares and circles are reversed in the right picture compared to the left.

Theorems & Definitions (25)

  • Theorem 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Corollary 2.10
  • Proposition 2.11
  • ...and 15 more