Virulence on the Edge: A Source–Sink Perspective by Holt, Robert; Hochberg, Michael, Michael

Summary

Main Points

If infections are localized, spatial separation increases the degrees of freedom of a host-parasite system permitting a rich array of dynamical behaviours to arise even in a spatially homogenous world (e.g., Hassell et al. 1994)

Local populations might have different carrying capacities or involve anisotropic flows.

This leads to the potential for asymmetries among habitats in the degree of the impact of spatial coupling on local ecological and evolutionary dynamics

Sink populations may readily arise at the edges of species’ ranges, or where habitats that differ greatly are juxtaposed. Given that the genetic variation is present, natural selection might be expected to improve the ability of a species to utilize the sink habitat.

However, there appear to be severe constraints on the ability to have adaptive evolution in sinks (i.e., adaptive evolution to the conditions within the sinks).

leading to a king of evolutionary conservatism in spatially heterogeneous environments.

Yet few studies (of gene flow and selection, md) focus explicitly on the potential implications of source-sink dynamics for our understanding of the evolution of virulence and resistance (for an analysis of comparable issues in predator-prey coevolution along a gradient, see Hochberg and Van Baalen 1998).

Motivating examples

1. Generate some source sink scenarios
2. Develop potential exploratory models for these scenarios and discuss the initial stages of adaption for the host-pathogen interaction in a sink
3. Generalize some conclusions given these simple models

Motivating examples

1. Organic farmer growing corn and sees a fungal blight that reduces crop yield. The question is how much seed to hold back (which might have resistance to this blight).
2. Animal husbandry with a benign pathogen in the animals. What concern should there be regarding a more serious illness emerging.
3. Doctor in a nursery ward with the emergence of nonsocomial pathogen

What unifies these three situations is that they all involve spatial dynamics (in the broad sense of mixing together individuals drawn from distinct populations); depending upon the quantitative details, these scenarios could involve a source-sink structure for either the host of the pathogen

Models of source-sink dynamics

Several simplifying assumptions are made for modelling these dynamics

1. Host exhibits continuous generations and any direct density-dependence is dominated by the effects of the pathogen
2. Once a host is infected so that it is infectious, it cannot be super- or multiply- infected and infectious hosts do not recover/give birth and remain infectious until removed from the population
3. These assumptions mean that co-circulation does not occur

With no potential for host recover, host evolution in response to the pathogen is related to the likelihood of successful infection in the first place and to the production of infected host individuals who themselves are infectious. The transmission parameter, $\beta$, at the heart of a standard epidemiological model defines the rate at which susceptible hosts themselves become infective.

Selection on the host tends to reduce $\beta$ (i.e., selection favors reducing transmission of the pathogen from infectious host to susceptible host).

Some additional definitions in this context:

$S$: the density of immigrant (ancestral) host in the sink habitat $S_{mut}$: the density of a novel type of host $I$ : the density of infected hosts Virulence: the average fitness across all hosts carrying the pathogen, including those whose successful defenses are reducing the pathogen titer toward extinction.

Model 1 Host evolution in a sink with generalist pathogen

The source contains the host alone whereas the sink has both the pathogen and the host. The pathogen is a generalist, so its dynamics are decoupled from from the focal host (e.g., no evolution in this sink). Additionally, assume that density of infected hosts is fixed at I. Here we are not concerned with pathogen persistance. We also have some host immigration (from the source) at some value, $H$. Evolution of virulence is governed by evolution in $\beta$ in the case.

We have some intrinsic growth rate $r$ which is government by births, $b$ and deaths $b$.

$$\begin{align} r = b - d \newline \frac{dS}{dt} = rS - \beta S I + H \end{align}$$

Important conditions

1. If $r \lt 0$ then this habitat is an intrinsic sink, i.e., in the long run, the population will go extinct regardless of the presence of the pathogen.
2. If $0 \lt r \lt \beta I$ then the presence of the pathogen can cause the habitat to become a sink (i.e. an induced sink). The potential for adaptive evolution of the resistance by hosts is different in these intrinsic sinks is different here.

Now adding evolution

$$\frac{dS_{mut}}{dt} = r_{mut} S_{mut} - \beta_{mut} S_{mut} I$$

1. In the case where the habitat is an intrinsic sink (r < 0) then we will have Error in rendering LaTeX so even it is a sink without pressure, then intrinsic sinks do not benefit from natural selection selecting for alleles that confer host resistance (even if they are cost-free). Likelihood of local adaptation in intrinsic sinks via locally favoured mutations is reduced in an intrinsic sink
2. However, in an induced sink, a novel allele can increase in abundance if $r \gt \beta_{mut} I$ with a minimum magnitude of the mutational effect required as $\Delta \beta = \beta - r/ I$. If there is a fitness cost then it must be such that $\Delta \beta$ is exceeded even if $r_{mut} \lt r$

Three main conclusions:

1. Local adaptation to of the host to the pathogen is less likely if productivity in the sink is low, or pressure from the pathogen (I) is high.
2. At larger I, or smaller r, evolution occurs only if mutants with a sufficiently large effect arise
3. We do not expect evolution in the hosts in intrinsic hosts at all

Immigration can increase the local population size and make it more likely that mutants will arise. More immigration also means more variation.

Model 2 Host evolution in a sink with specialist pathogen

Host flows from the sink with a specialist pathogen with a dynamical response such that the infection level depends on host dynamics. This takes the form of the classic SI model with host immigration, H, also included and where some pathogenicity from the infection is also included as, $\alpha$, and where $\mu$ represents the addition of the intrinsic death rate and this pathogen induced death rate.

$$\begin{align} \frac{dS}{dt} = r S - \beta S I + H \newline \frac{dI}{dt} = \beta S I - \mu I \newline \mu = d + \alpha \end{align}$$

Then we have the following equilibrium points:

$$\begin{align} S^{\star} = \frac{\mu}{\beta} \newline I^\star = \frac{r}{\beta} + \frac{H}{\mu} \end{align}$$

Thus if r <0 then $H \gt |r| d/ \beta$ for pathogen persistance meaning that a specialist pathogen must be sufficiently transmissible to persist in an environment that is an intrinsic sink for its host. Additionally, there must be enough hosts coming in to sustain the sink.

The abundance of infected individuals rises until the negative growth rate of the host population just matches the rate of input (immigration) from outside.

We can then re-introduce the evolutionary component:

$$\frac{dS_{mut}}{dt} = r_{mut} S_{mut} - \beta_{mut} S_{mut} I$$

Again, if the habitat is an intrinsic sink for the host ($r_{mut} \lt 0$) then evolution does not promote local adaptation by the host (as $S_{mut}$ also tends to 0). However, if the habitat is an induced sink:

$$\begin{align} \frac{dS_{mut}}{dt} \gt 0 \newline iff: \newline \beta_{mut} \lt \beta \big(1 + \frac{H \beta}{r \mu} \big) \end{align}$$

Thus $r$ and $H$ have opposing effects on local adaptation as

• increasing r increases the range of mutational effects on $\beta$
• Increase $H$ makes adaption more difficult ![[Pasted image 20230904135426.png]]

So key conclusion, local adaptation towards lower transmission rates is more likely for:

1. Habitats with high host productivity (and unlikely in intrinsic sinks)
2. Habitats in which the pathogen initially has a low rate of transmission ($\beta$)
3. Habitats in which infected individuals are short-lived (e.g., high $\mu$)
4. Unproductive source habitats (low H)

Host immigration indirectly increases pathogen abundance and thus increases the mutational effect required to allow the conditions for the lower transmission rate population to persist. Including the cost function ($r_{mut} \lt r$) makes local adaption arrise (See Hochberg and Van Baalen 1998)

Model 3 Host evolution in a sink with specialist pathogen

Habitat contains both host and pathogens and there is cross-habitat infection with infected individuals in the source infecting healthy individuals in the sink. This is comparable to the notion of disease reservoirs as described by May and Anderson (1991).

$$\frac{dI}{dt} = \beta_{source} I_{source} S + \beta I S - \mu I$$

Note that $\beta_{source} I_{source} S$ reflects the force of infection on health, susceptible hosts in the sink caused by infected individuals in the source.

If we assume that $I_{source}$ is constant for the sink and that S is the host carrying capacity, K, then we can make some simplifying assumptions. Thus this habitat will be an instrinsic sink if:

$$\begin{align} \beta K \lt \mu \newline \text{OR} \newline \beta \lt \frac{\mu}{K} \end{align}$$

Factors constributing to being an intrinsic sink for the pathogen

• Host is scarce (low K)
• Transmission rates are low (low $\beta$)
• Infected hosts have high death rates either because hosts have instrinsically high deaths rates (high d) or pathogen is highly virulent (high $\alpha$)

However, because of external inputs, a pathogen can persist in an intrinsic sink:

$$\frac{dI_{mut}}{dt} = \beta_{mut} I_{mut} K - \mu_{mut} I_{mut}$$

Thus a novel strain can survive if $\beta_{mut} \gt \mu_{mut} / K$. Note that if the transmissibility increases it will be selected for if the impact on virulence is not too high.

A novel mutation will be successful if $\mu = \mu_{mut}$ then a novel mutation will be successful if $\Delta \beta = \mu / K -\beta$ . Evolution in a host habitat that is a sink for the pathogen thus occurs only if mutants have large effect upon transmission arise.

In the case where there is a trade-off between mortality and transmissibility, the picture is a little more complicated. Higher values of K (more population abundance) will allow for higher degrees of total mortality by expanding the adapability range into the fitness space.

Thus pathogen is sustained by immigration which is a pathogen sink, then local adaption in rates of transmission and virulence are more likely if:

• The host population is abundant
• The immigrant parasite strain is not too badly maladapted to the local host in the first place

A more virulent pathogen strain is most likely to invade a sink habitat when:

• Mutations occur that confer a substantial difference between the immigrant and mutant pathogen strains. Small changes are less likely to become locally adapted to hosts in the sink
• Host density in the sink, K, is high. (with increasing K, the slope of the dashed line decreases increasing the feasible invasion space for more virulent pathogens)
• The pathogen in the immigrant stream is low in virulence to begin with

A simple message of the above models is that the demographic context of the initial stages of contact with novel hosts may be crucial in predicting emergence. IF the demographic context is that the novel hosts are sinks for the pathogens invading from another species, then adaptation by the pathogen to the host requires mutants of large effect; if such genes are rare, the emergence of the disease as a serious problem may be unlikely

Questions

Code/ Supplemental Data

Implications for Infectious Diseases