An offer you cannot refuse: down-regulation of immunity in response to a pathogen's retaliation threat

According to the Red Queen hypothesis, hosts and pathogens are engaged in an escalating coevolutionary arms race between resistance and virulence. However, the vast majority of symbionts colonize their hosts' mucosal compartments without triggering any immune response, resulting in durable commensal associations. Here, I propose a simple extension of previous mathematical models for antagonistic coevolution in which the host can mount a delayed immune response; in response, the symbiont can change its virulence following this activation. Even though the levels of virulence in both phases are assumed to be genetically determined, this simple form of plasticity can select for commensal associations. In particular, coevolution can result in hosts that do not activate their immune response, thus preventing phenotypically plastic pathogens from switching to a higher virulence level. I argue that, from the host's point of view, this state is analogous to the mafia behaviour previously described in avian brood parasites. More importantly, this study provides a new hypothesis for the maintenance of a commensal relationship through antagonistic coevolution.


Model definitions
The baseline population dynamic model as presented in the main text ( Fig. 1) is described by the following set of differential equations: An extended model describing acquired immunity with memory ( Fig. S1) is given by the following set of differential equations: All symbols are defined in Table S1. In the following I refer to model (S1) as the SIAS model, and (s2) as the SIARA model.

Pathogen basic reproductive ratio
In order to determine the pathogen's basic reproductive ratio, I used next generation matrices as described by Hurford et al. [1]. While there are simpler heuristics to derive the expression of R0, the use of the next-generation theorem is more systematic and can also be used to determine invasion criteria for host and pathogen evolution. In brief, I re-write the system of differential equations in vectorial form dx/dt = Ax, where x(t) is the vector of state variables of the system and A is a matrix of constant coefficients. The next-generation theorem states that, if A can be written as F-V where F≥0, V -1 ≥0 and if all the eigenvalues of -V have negative real parts, then all the eigenvalues of A have negative real parts if and only if all the eigenvalues of FV -1 lie within the unit circle [1]. In practice, "F is a matrix which gives the rate at which new individuals appear in class j, per individual of type i. The matrix V describes the movement of existing individuals among the different classes, as well as the loss of these individuals.
[…] Hence, FV -1 is sometimes referred to as the next-generation matrix. Moreover, ρ(FV -1 ) = R0, which has an interpretation as the expected lifetime reproductive output of a newborn individual." [1] To calculate R0 for the pathogen in either model, I assume that the host population is fully susceptible and at the carrying capacity K. The state vector for the pathogen is x = (I A) T , the reproduction matrix is and the transition matrix is The dominant eigenvalue of FV -1 is then Note that the same expression could be obtained by reasoning from first principles, writing R0 as the sum of reproductive ratios during the two phases of infection: where the factor between brackets is the probability of reaching the second phase of infection.
Even though the full expressions of the equilibrium points of either system cannot be obtained analytically, it can be shown by solving system (S1) that, the endemic equilibrium (S * , I * , A * ) must satisfy the following relations:

Pathogen evolution
To study pathogen evolution, I extend equations (S1) and (S2) to two strains of pathogens that compete for infection of susceptible hosts (S and R), assuming that currently infected hosts (I or A) cannot be reinfected. In addition, the two pathogen strains only differ by the values of virulence (αi and αa) and infectivity (δi and δa) and are antigenically identical, so that hosts are equally susceptible to both strains.
Labelling the two strains with subscripts 1 and 2, the SIAS model can be written as: Assuming that both strains have basic reproductive ratios R0,1 and R0,2 greater than unity, I consider the scenario where strain 1 is initially present and has reached endemic equilibrium (S * , I1 * , A1 * ) and strain 2 is introduced at a very low prevalence.
Using the next generation theorem as before, the mutant's fitness can be written as: Hence the mutant will be able to invade if and only if its basic reproductive ratio is larger than that of the resident strain. The same analysis can be done easily with the SIARA model and leads to the same conclusion.
As explained in the main text, I chose to impose a 'classical' constraint on pathogen evolution by assuming that virulence α and infectivity δ are positively linked. In a single-stage infection (here when µ = 0), if infectivity increases less than linearly with virulence, there is a single phenotype that maximises R0 and it is therefore an ESS [2,3]. Here I consider two cases, depending on whether the pathogen has a plastic response or not.

a. Non-plastic virulence
First, if the pathogen cannot change its virulence, let αi = αa = α and δi = δa = δ(α). Then the first order condition for α to be an ESS is: Using the function δ α ( ) = δ 0 α α + ε , equation (S4) becomes a quartic polynomial. While it cannot be solved analytically, it is possible to prove that equation (S4) has exactly one positive root α * with can then be re-written as: The second-order condition R 0 ′′ α * ( ) < 0 boils down (after a few lines of tedious algebra) to γ a > γ i which is the working assumption. I therefore conclude that a non-plastic pathogen has a single ESS

b. Plastic virulence
If the pathogen can change its virulence during the second phase of infection, so that the values of αi and αa are evolving independently, then any candidate ESS has to be a solution of the following system of two equations: Using the function δ α ( ) = δ 0 α α + ε for both δi and δa, (S5) has a unique solution: There remains to calculate the Hessian matrix of R 0 α i ,α a ( ) : Since δ is a concave function, the two terms on the diagonal are negative, hence α i * ,α a * ( ) is a maximum of R0, so it is an ESS.

c. Numerical results
Pathogen's ES levels of virulence plotted against the host's activation rate (µ), host mortality (m), and recovery rate before (γi) or after immune activation (γa). The dashed black line shows the ES virulence α * of non-plastic pathogens, whereas the amber and red lines show the respective ES levels αi * and αa * for plastic pathogens. Same numerical values as on Fig. 2.

Host evolution a. SIAS model
In this section I consider two competing host genotypes (labelled with subscripts 1 and 2) that can differ in three traits: immune activation rate µ, fecundity b and mortality m. There is a single strain of pathogen which does not evolve. The SIAS model can be written as: As with the pathogen, I follow Hurford et al.'s [1] next-generation matrix method to determine the conditions under which the mutant genotype 2 can invade the resident genotype 1. I assume that the system has reached its stable equilibrium S 1 * , I 1 * , A 1 * ( ) in the absence of genotype 2 and that genotype 1 is such that the pathogen's basic reproductive ratio is greater than unity. From system (S6), I define the hosts' reproduction matrix F as: and the transition matrix V as: where Λ 1 * = β δ i I 1 * + δ a A 1 * ⎡ ⎣ ⎤ ⎦ is the force of infection at equilibrium. It follows that the nextgeneration matrix FV -1 has the same zero elements as F, and therefore has two non-zero eigenvalues ω1 and ω2 which represent the respective relative fitnesses of genotypes 1 and 2:  Table S1 and m0=0.1, ν=10.
These graphs show a single ESS which is both evolutionarily stable (it cannot be invaded by any mutant) and convergent-stable (it can evolve through a series of small mutations from any other genotype). The ESS drops to zero when the effective benefit of mounting an immune defence is too low, for example if virulence is too high during the second phase of infection: Here only a strong benefit of mounting an immune response (middle frame) gives rise to a non-zero ES activation rate. Even then, a closer look reveals that µ = 0 is also an ESS (the two ESS are separated by an evolutionary repeller): (iv) Linear cost on fecundity: b = b 0 1 − µ / ν ( ) . Numerical values as in Table S1 and b0=1, ν=20. Here a non-zero ESS exists only if the cost on fecundity is very low (with a large value of ν) and the benefit of the immune response very high.

b. Supplementary result
ES activation rate (µ) plotted against the two levels of pathogen virulence. Same as figure 4, except that infectivity parameters are kept constant: δ i = δ a = 1 .

c. SIARA model
The same analyses as above can be done for the SIARA model, which is just slightly more complicated because of the extra variable. With two host genotypes we have 8 equations: As before, I assume that genotype 1 is initially on its own in the population with the pathogen and reaches its stable equilibrium S 1 * , I 1 * , A 1 * , R 1 * ( ) before genotype 2 appears by mutation. This leads to define the hosts' reproduction matrix F as: and the transition matrix V as: The eigenvalues of the next-generation matrix FV -1 are: As with the SIAS model, I use ω2 as a measure of the mutant's fitness in order to produce Pairwise Invasibility Plots and calculate ESS, with a set of alternative cost functions (see next page).

Coevolution
As explained in the main text, I have chosen to model coevolution by modifying the host evolutionary algorithm under the assumption that the pathogen's phenotype is at the ESS with respect to the resident host's phenotype. The fitness of a mutant host genotype is still given by expression (S7) or (S8) above (respectively for the SIAS and SIARA models), but parameters δi, δa, αi and αa are now functions of the resident host's genotype and other parameters as determined in Section 3. Under this scenario, any ESS for the host will automatically be associated with a corresponding ESS for the pathogen, resulting in a Co-Evolutionary Stable Strategy (CoESS).
Here I will give a few examples of PIPs for the host, as I did in section 4, but also considering the two models of pathogen evolution: plastic virulence or fixed virulence.