^{*}

Conceived and designed the experiments: FD TL SG. Performed the experiments: FD. Analyzed the data: FD. Wrote the paper: FD TL SG.

The authors have declared that no competing interests exist.

The spread of drug-resistant parasites erodes the efficacy of therapeutic treatments against many infectious diseases and is a major threat of the 21st century. The evolution of drug-resistance depends, among other things, on how the treatments are administered at the population level. “Resistance management” consists of finding optimal treatment strategies that both reduce the consequence of an infection at the individual host level, and limit the spread of drug-resistance in the pathogen population. Several studies have focused on the effect of mixing different treatments, or of alternating them in time. Here, we analyze another strategy, where the use of the drug varies spatially: there are places where no one receives any treatment. We find that such a spatial heterogeneity can totally prevent the rise of drug-resistance, provided that the size of treated patches is below a critical threshold. The range of parasite dispersal, the relative costs and benefits of being drug-resistant compared to being drug-sensitive, and the duration of an infection with drug-resistant parasites are the main factors determining the value of this threshold. Our analysis thus provides some general guidance regarding the optimal spatial use of drugs to prevent or limit the evolution of drug-resistance.

The widespread use of antimicrobial drugs during the 20^{th} century greatly
contributed to the increase in human life expectancy

The development and the analysis of mathematical models have played a major role in
our understanding of the evolutionary dynamics of drug resistance. These
epidemiological models allowed to compare various treatment strategies such as
different treatment doses

The issue of resistance management is however not restricted to human infectious
diseases. For example, drug-resistance decreases treatments efficiency in livestock

The underlying concept comes from population genetics studies

We first study a parasite life-cycle with direct parasite transmission between
hosts. At time

The disease is transmitted locally by direct contact between an infected and a
susceptible individual, with a transmission parameter

The treated area is represented with a dark gray filling. Subfigure (a)
shows the effects of treatment on the basic reproductive ratios

The environment is linear, and divided into treated and untreated areas, of width

As illustrated on

Both infected and uninfected hosts migrate. The distribution of the distances of
migration (i.e. the kernel of migration) of both hosts is assumed to be
symmetric (i.e. with mean 0) and with variance

This formulation clarifies the feed-back of demography on evolution (i.e. the
selection

Following on earlier studies

First, when the migration range is restricted to the nearest neighbors (i.e.

Equation (8) shows that the critical size

In contrast, when there is very long-range migration (high

Instead of treating every infected individual in a restricted area corresponding
to a proportion

This outcome depends on the total size of the environment scaled by the
migration range parameter

In the above section we focused on a scenario with parasite transmission by
direct contact among hosts. In the following we consider a more complex parasite
life cycle involving two different host species. In particular, we focus on
vector-borne transmission such as in malaria, leshmaniosis, trypanosomiasis and
many other human infections (the model holds for any disease involving the
sequential infection of two different hosts, and can be readily extended to
other two-stage life-cycles, with air-borne or water-borne transmission for
instance). Hereafter, we call the first host “human”, and
the second host “vector”. Both humans and vectors can
migrate, though at potentially different ranges (with parameters

We find two critical sizes

This figure represents the frequency of drug-resistance at equilibrium,
as a function of the proportion of treated individuals (

As in the single-host life cycle, the fate of drug-resistant parasites depends on
the intensities of selection in treated (

The drug-sensitive invasion condition,

In this study, we analyze the interplay between epidemiological and evolutionary
dynamics of a drug-resistant parasite strain in a one-dimensional environment.
Following on and extending earlier population genetics studies on clines

In our direct transmission model, as pointed out earlier by Nagylaki

Second, as emphasized by Nagylaki

A third factor determining of the critical size is the intensity of selection for the
invading strain, in each environment (

The basic reproductive ratios are classically used in epidemiological models to
evaluate the costs of drug-resistance

Our model can be used to explore new strategies of resistance management. Yet,
various criteria can be used to define an optimal strategy

Suppose that only a limited stock of treatment is available: only a part of the
population can be treated. Two (extreme) strategies are considered: treating
everyone in a limited area of width

For our set of epidemiological parameters, when the migration range is large
(

When migration is more local (

There is thus a critical migration range, above which the heterogeneous strategy
may better limit the spread of drug-resistance. This critical migration range
can be visualized in

To illustrate further this point, let us take the example of two vector-borne
diseases, malaria and trypanosomiasis. Even though we give here the example of
two human diseases, recall that our models are general enough to be applicable
to a wide range of parasites and hosts, including other animals and plants,
provided the use of adequate parameters. Anopheles mosquitoes, malaria vectors,
are known to migrate at longer ranges

Of course, more quantitative recommendations for minimizing parasite prevalence
and the evolution of drug-resistance would require a fully parameterized model
of these two systems, as well as relaxing several simplifying assumptions. In
particular, it would be worth extending our model to dispersal in two spatial
dimensions, and taking diploidy and the effects of dominance into account. Both
extensions have already been studied in a population genetics context by
Nagylaki

In this paper we bridge the gap between the epidemiology and the population
genetics of drug-resistance

We study two models, corresponding to two types of parasite transmission. The parasites are modeled as asexual and haploid. In all models, the individuals are infected by one strain only at a time, and this strain cannot be displaced by the other (there is no coinfection or superinfection). No mutation is explicitly modeled: we assume that the drug-resistant strain pre-exists the treatment, and we study the outcome of competition with the resident strain. We assume that the hosts total density is constant in space and time. However, the total density of parasites varies.

We focus on simple spatial patterns of treatment: a pocket of treatment in an
infinite untreated region (

The migration is modeled using the diffusion approximation. We assume that there
is no directional preference (the mean of the dispersal kernel is zero), and
that the standard deviation of the migration kernel is

In our model with direct transmission of the parasites, only the hosts diffuse,
independent of their infectious status; the parasites move with infected hosts.
The densities of each parasite strain depend on time (

The boundary conditions are periodic and reflecting:

Let

Finding the critical size of the treated (resp. untreated) area comes to studying
the stability of the drug-resistant free (resp. drug-sensitive free)
equilibrium. The method for the stability analysis with the direct transmission
model, under the low migration approximation, is similar to the one already
described in

The sets of Partial Differential Equations (PDEs) can be numerically solved using the Method of Lines implemented in Mathematica's NDSolve function. For each set of parameters, two simulations are run, with different initial conditions, corresponding to the invasion of the drug-sensitive strain in an environment dominated by the drug-resistant strain, and reciprocally. If there are bistabilities, the ultimate outcomes of the two simulations are different.

Reciprocal Invasion Plot. It is the same plot as in ^{U}_{WT} = 0.06,
β_{R} = 0.055,
β^{T}_{WT} = 0.05,
γ^{U}_{WT} = 1,
γ_{R} = 1.25,
γ^{T}_{WT} = 1.5

(0.20 MB TIF)

Reciprocal Invasion Plot. The parameters are different than in ^{U}_{WT} = 0.05,
β_{R} = 0.08,
β^{T}_{WT} = 0.09,
γ^{U}_{WT} = 1,
γ_{R} = 2,
γ^{T}_{WT} = 2.5

(0.20 MB TIF)

(0.25 MB PDF)

We thank Minus van Baalen, Maciek Boni, Mark Kirkpatrick, Ophélie Ronce and François Rousset for insightful discussions. We also thank two anonymous referees for comments which helped clarify the paper.