A stochastic model for immune response with mutations and evolution: the non-spatial setting

TL;DR

The paper analyzes a non-spatial stochastic pathogen model with ancestral-order immune killing and antigenic mutations. It first establishes a sharp survival phase transition for the beneficial-mutation model with threshold $\lambda_c=(1+\sqrt{r})^{-2}$ and derives explicit finite-mean total progeny in the subcritical regime, using branching-process arguments and a recursive distributional equation. It then extends the model to allow deleterious mutations and obtains a closed-form survival criterion in terms of $(\lambda,r,p)$, revealing regimes with intermediate mutation windows. The proofs combine thinning techniques, recursive distributional equations, and connections to the birth-and-assassination process, yielding exact expressions for survival and moment quantities. Overall, the work provides a rigorous minimal framework for understanding immune escape vs mutational load in a non-spatial setting and sets the stage for spatial extensions.

Abstract

We consider a stochastic model for a pathogen population in the presence of an immune response, in which pathogen types are partially ordered by ancestry and the immune system must eliminate ancestor types before it can eliminate their descendants. In this model, pathogens reproduce independently at rate $λ>0$ and, at each birth, a mutation occurs with probability $r\in(0,1]$, producing a novel type that is antigenically distinct and whose elimination by the immune system is delayed relative to its ancestors. We provide an explicit characterization of the survival--extinction phase transition and compute the expected total progeny in the subcritical regime. We then extend the model by allowing mutations to be deleterious: conditional on mutation, with probability $p\in(0,1]$ the mutation is beneficial and with probability $1-p$ it is deleterious, producing a sterile offspring. For this extension, we obtain an explicit survival criterion in terms of $(λ,r,p)$ and identify parameter regimes in which survival is possible only for an intermediate range of mutation probabilities, reflecting the balance between immune escape and mutational load.

Figures (3)

• Figure 2.1: Illustration of the non-spatial process $\mathcal{B}(\lambda, r)$ with $0<r<1$. Pathogens of the same type share the same number and shape. They are numbered according to the order their type first entered into the system. Pathogens alive are in white and pathogens already dead are in black. Pathogen types at risk are encircled. Pathogens of the same type die all together. For instance, if the next event to occur is the death of all pathogens of type 5. All pathogens of type 5 would change color, and pathogens of types 9 and 10 would become encircled.
• Figure 2.2: The process $\mathcal{B}(\lambda,r)$ survives with positive probability if and only if $(\lambda,r)$ lies above the critical curve $r=(1-\sqrt{\lambda})^{2}/\lambda$.
• Figure 3.1: Critical curves $r\mapsto \lambda_c(r,p)$ for different values of $p$. For fixed $p$, the region above the curve corresponds to survival ($\lambda>\lambda_c(r,p)$), while the region on and below corresponds to almost sure extinction ($\lambda\le \lambda_c(r,p)$).

Theorems & Definitions (18)

• Definition 2.1
• Theorem 2.2
• Proposition 2.3
• Definition 3.1
• Theorem 3.2
• Remark 3.3: Phase diagram for fixed $p$
• Proposition 3.4
• Lemma 4.1
• proof
• proof : Proof Theorem \ref{['TNS']} (Necessary condition).