^{1}

^{2}

The authors have declared that no competing interests exist.

Many infectious diseases exhibit seasonal dynamics driven by periodic fluctuations of the environment. Predicting the risk of pathogen emergence at different points in time is key for the development of effective public health strategies. Here we study the impact of seasonality on the probability of emergence of directly transmitted pathogens under different epidemiological scenarios. We show that when the period of the fluctuation is large relative to the duration of the infection, the probability of emergence varies dramatically with the time at which the pathogen is introduced in the host population. In particular, we identify a new effect of seasonality (the _{0}, the basic reproduction ratio of the pathogen. This theoretical framework is extended to study the probability of emergence of vector borne diseases in seasonal environments and we show how it can be used to improve risk maps of Zika virus emergence.

Seasonality drives fluctuations in the probability of pathogen emergence, with dramatic consequences for public health and agriculture. We show that this probability of pathogen emergence can be vanishingly small

The development of effective control strategies against the emergence or re-emergence of pathogens requires a better understanding of the early steps leading to an outbreak [_{0} > 1 and will go extinct otherwise (_{0} > 1. The probability of emergence _{0} < 1 and, when _{0} > 1, it is equal to [

In a direct transmission model pathogen dynamics is driven by the birth rate λ and the death rate ^{H}, ^{H}), exposed and infected mosquito vectors (^{V}, ^{V}). In the absence of seasonality (i.e. no temporal variation in birth and death rates) the basic reproduction ratio _{0} can be expressed as a ratio between birth and death rates. The probability of emergence _{e} after the introduction of a single infected individual can also be expressed as a function of these birth and death rates. With vector borne transmission this probability of emergence depends on which infected host is introduced (Figure E in ^{V}, and where the index

The above results rely on the assumption that birth and death rates of the infection remain constant through time (i.e. time homogeneous branching process). Many pathogens, however, are very sensitive to fluctuations of the environment. For instance, the fluctuations of the temperature and humidity have been shown to have a huge impact on the infectivity of many viral pathogens like influenza [_{e}(_{0}) when one infected individual is introduced (i.e. _{0} is well known (see e.g. [_{T}(_{T}(_{0} < 1 the pathogen will never produce major epidemics and will always be driven to extinction. When _{0} > 1, however, a pathogen introduced at a time _{0} may escape extinction. In this case the probability of emergence can also be expressed as a ratio of average birth and death rates, but with different weights (see section

In the following we show that very good approximations of the probability of pathogen emergence can be derived from this general expression when the period is very large (or very small) compared to the duration of the infection. These approximations give important insights on the effect of the speed and the shape of the temporal fluctuations of the environment on the probability of pathogen emergence. We use this theoretical framework to determine optimal control strategies that minimize the risk of pathogen emergence. We provide clear cut recommendations in a range of epidemiological scenarios. We also show how this theoretical framework can be extended to account for the effect of seasonality in vector borne diseases. More specifically, we use this model to estimate the probability of Zika virus emergence throughout the year at different geographic locations.

For the sake of simplicity we start our analysis with a directly transmitted disease with a constant clearance rate

Both the speed and the amplitude of the fluctuations of λ(

Pathogen _{0}. In the final portion of the year λ(_{0} = 2.5 and 1.5 and, in B λ(_{0} = 2.5 and 0. Pathogen _{0} can be well approximated by _{0} = 2.2 and _{e} ≃ 0.55; in B, _{0} = 1.75 and _{e} ≃ 0.43).

When the fluctuations are slower, however, the probability of pathogen emergence does depend on the timing of the introduction. The probability of emergence drops with the transmission rate (

Our theoretical framework can be used to identify optimal control strategies. The objective is to minimize the average probability of emergence under the assumption that the introduction time is uniformly distributed over the year:
_{ρ}(_{1} and _{2}, the times at which the control starts and ends, respectively, and _{M} the intensity of control during the interval [_{1}, _{2}]. The cost of such a control strategy is thus: _{2} − _{1})_{M}. For a given investment in disease control _{1}, _{2} and _{M} that minimize the average probability of emergence 〈_{e}

We first answer this question when the fluctuation of transmission is a square wave where λ(_{0} (for a fraction 1 − _{0} − _{0} − 1) the basic reproduction (after control) of the pathogen drops below one and the probability of emergence vanishes. _{e}_{0} is too high). We assume that the investment in control is fixed and equal to _{2} − _{1})_{M} = 0.2 and we explore how different values of _{1} and _{M} affect 〈_{e}_{1} = 0, _{2} = 0.7, _{M} = 2/7) yields an average probability of emergence equal to 〈_{e}_{e}_{e}

In A and B we plot The pathogen birth rate before (black line) and after the optimal control (dashed blue line) which minimizes the mean emergence probability < _{e}> (see also _{(0<t<0.7)}. The sinusoidal wave assumes that λ(

We used the same scenarios as in _{M}(_{2} − _{1}) = 0.2). We explore how the intensity of control (_{M}) and the timing of control (between _{1} and _{2}) affect < _{e} >, the mean probability of pathogen emergence (lighter shading refers to higher values of < _{e} >). For the square wave scenario we identify a range of optimal strategies withing the dotted red curve where < _{e} > is minimized. The optimal strategies used in _{e} > are: 0.166 − 0.366 (square wave) and 0.085 − 0.31 (sinusoidal wave). For the square wave (A), _{0} = 1.5 does not depend on the timing and the intensity of the control. For the sinusoidal wave (B), there is a single strategy minimizing _{0}, namely _{0} = 1.28 for _{1} = 0.15 and _{M} = 1.0, marked with a red cross in B. With the sinusoidal wave there is a single control strategy minimizing < _{e} > for _{1} = 0.07 and _{M} = 0.93 (blue cross in B).

Second, we consider a seasonal environment where λ(_{M} = 1) in a time interval centered on the time at which pathogen transmission reaches its peak (red cross in _{e}

Next we want to expand the above analysis to a more complex pathogen life cycle. Indeed, many emerging pathogens are vector borne [^{V} and ^{V}), exposed and infectious humans (^{H} and ^{H}). The stochastic description of this epidemiological model yields a four dimension multi-type birth-death branching process (see section 2 of ^{H}, ^{H}, ^{V}, ^{V}):
_{i,i+1} denotes the _{i} denotes the _{i} ratio). We show in the

Seasonality can drive pathogen transmission through the fluctuations of the available density of the mosquito vector. Following [_{opt}, the optimal temperature for mosquito reproduction (see sup info). The rate λ_{IH, EV} at which mosquitoes are exposed to the parasite is directly proportional to _{V}/_{H}. In such a fluctuating environment the _{0} is the spectral radius of the next generation operator, see [_{0}. Yet, it is tempting to use _{0}, to obtain an approximation _{e}(_{0}) for large periods. The exact probability of emergence can be efficiently computed numerically thanks to the seminal work of [_{e}(_{0}). However, when seasonality induces more pronounced drops in transmission, we recover the

The top figures (A and B) show the seasonal variations in λ_{IV, EH}, the transmission rate from humans to the vectors because of the fluctuations the density of vectors in two habitats (this illustrates the effect of _{0} on Zika emergence. The dotted black line refers to the naive expectation for the probability of pathogen emergence at time _{0} if all the rates were constant and frozen at their _{0} values (see (_{e}(_{0}

The effect of seasonality on the probability of pathogen emergence depends critically on the duration of the infection 1/

Note that our approach neglects the density dependence that typically occurs after some time with major epidemics. Our probability of pathogen emergence thus provides an upper approximation of the probability emergence. Indeed, with density dependence the size of the pathogen population may be too small to survive even very shallow demographic traps. In section 5 of

Understanding this effect allows us to identify the optimal deployment of control strategies minimizing the average probability of pathogen emergence in seasonal environments. We identified optimal control strategies in different epidemiological scenarios under the assumption that the introduction time is homogeneous (Figures 3, 5, and A, D in

This work can be extended to explore optimal timing of other control strategies. For instance [_{0}. Yet, as pointed out above, the strategy minimizing _{0} may not always coincide with the strategy minimizing 〈_{e}_{0})〉 (see _{0} is minimized (3.71 instead of 3 months before the peak transmission).

So far we focused on control strategies that lower pathogen transmission. Our approach can also be used to optimize control measures that do not act on the transmission rate but on the duration of the infection. For instance, what is the optimal timing of a synchronized effort to use antibiotics to minimize bacterial pathogens emergence? We found that the timing of these treatment days have no impact on _{0} but pathogen emergence is minimized when treatment occurs 1.3 months before the peak of the transmission season. This strategy creates deeper traps and results in a stronger

The above examples show that our analysis has very practical implications on the understanding and the control of emerging infectious diseases in seasonal environments. This theoretical framework could be used to produce maps with a very relevant measure of epidemic risk: the probability of pathogen emergence across space and time (_{e}(_{0}) would be unambiguous and more informative. Our model could thus contribute to development of “outbreak science” [

Experimental test of theoretical predictions on pathogen emergence are very scarce because the stochastic nature of the prediction requires massive replicate numbers. Some microbial systems, however, offer many opportunities to study pathogen emergence in controlled and massively replicated laboratory experiments [

The life cycle of a directly transmited pathogen is governed by its birth and death rates (λ and

In a seasonal environment the birth and death rates are assumed to be functions of time, noted λ(_{e}(_{0}), the probability of pathogen emergence when a single infected host is introduced in the host population at time _{0}.

Let us now consider rates with period _{T} and _{T}. Accordingly, we denote _{e}(_{0}, _{e}(_{0}, _{0} ≤ 1.

If _{0} > 1, we can rearrange formula (_{e}(_{0}, _{T}(

Hence,

Under the assumption that _{e}(_{0}, _{0}. In the following we rescale time so that the _{T}, _{T} become 1 periodic functions defined by
_{0} refers to introduction time between 0 and 1 and by a change of variables we obtain
_{e}(_{0}

We see from

In other words when

We observe on various examples that for large _{e}(_{0}

We are going to give a mathematical formulation to this observation. Define
_{0}) and _{0}) to λ and

In other words a time _{0} is in the _{0} for which the expected size of the population _{0}):

We present: (1) a calculation of the probability of emergence of directly transmitted pathogen for different scenarios of seasonality, (2) a generalisation of our results when the pathogen life cycle goes through multiple stages before completing its life cycle, (3) an exploration of the

(PDF)

We thank Mike Boots and Sébastien Lion for comments on an earlier draft. Philippe Carmona thanks the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment.

This paper was transferred from another journal. As a result, its full editorial history (including decision letters, peer reviews and author responses) may not be present.

Dear Dr Gandon,

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The reviewer indicated the importance of the problem, but raised concerns on the presentation of the arguments and results, and the way the reader would have to construct an understanding of the main idea. We agree with this concern and think that the way the argument is presented may have been itself one reason why we have failed to secure referees (rather than a lack of interest in the subject).

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Reviewer's Responses to Questions

Reviewer #1: see attached

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Dear Dr. Gandon,

Thank you very much for submitting your manuscript "Winter is coming: pathogen emergence in seasonal environments" for consideration at PLOS Computational Biology.

As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by two referees. The first referee had commented on the previous version of the manuscript and had requested clarifications and changes to make the work more accessible. She/he is now satisfied with the changes and provides minor suggestions. The second referee agrees that the work is a valuable and interesting contribution. But his/her major concerns are again with the way the work is presented, in particular the lack of sufficient context motivating the work and placing its relevance into a more tangible and historical context. Specific suggestions are made for the consideration of a number of studies that would help doing this. We happen to agree that the manuscript can be written in a way to reach a broader audience and goes from the technical contribution to deeper context and discussion.

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Reviewer's Responses to Questions

Reviewer #1: This is a very carefully worked out and mature work that covers a lot of ground from periodically forced natural systems, to their optimal control. The authors have a detailed knowledge of stochastic systems and branching processes, and this has led to the fruition of a deep analysis from many fronts. I learnt a lot from reading this manuscript.

1) The authors have made significant new contributions to the study of forced seasonal systems and highlighted some new (sometimes subtle) important dynamical behavious.

2) I have checked all the maths in the SI that is relevant to the equations and models discussed in the main text.

Minor comments:

*Authors should mention that Eqns 5&6 are derived in the relevant Appendix.

*Page 5 Eqn.5 Perhaps give the value of R0, and pE in the main text, for Fig.2E and 2F so the reader can see this works.

*Ref missing in Section 2.3 of SI

*Fig.2 The notation p(toT, T) is used in the SI, but I don't think it is given in the main text. (I am referring here to the rescaling of t0)

Reviewer #2: The topic and idea developed in this work is very interesting, and I find it particularly valuable that the authors emphasize the importance of working with probabilities rather than with deterministic quantities such as R0. Although R0 is useful for many purposes, it presents limitations, especially for low-transmission situations, among others.

Although the development of expressions and the study of probabilities for the occurrence of major outbreaks are not new, there is a lack of concrete applications and discussion of different transmission scenarios. This theoretical study investigates how the interaction of seasonality, duration and arrival time of the infection, affects the probability of a major outbreak. In particular, the authors contribute to the formalization of an effect that is somehow intuitive which they call “the winter is coming effect”.

Unfortunately, however, the authors do not better exploit the intuitive and biological aspects of this effect in a way that would be accessible to a broader audience (that would be very interested in this topic). The way the paper is written is quite technical. It would benefit from a deeper connection to particular infectious diseases or actual transmission results, mainly in the introduction and discussion sections. For example, the authors could compare/connect their results with Otero et al. 2010, who studied the probabilities of major dengue outbreaks based on the first infection arrival time to the city of Buenos Aires. Another example that could be discussed in this paper is the study of flu arrival times for different cities in the US and UK by Truscott & Ferguson 2012, as well as the results obtained by Dalziel et al. 2018 also for flu. In addition, I was surprised that the authors did not cite and connect their probabilities’ expressions with those in the papers by Bartlett 1964 and by Lloyd et. al 2007. Both papers discuss the probability of pathogen emergence, in direct and vector transmitted diseases.

In conclusion, I like the idea and I think it is a compelling topic that a broad spectrum of readers of this journal and beyond would be interested in. The writing is too technical and needs deeper motivation and discussion related to empirical and intuitive aspects, including a clearer connection to results in other studies of transmissible diseases.

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Dear Dr. Gandon,

We are pleased to inform you that your manuscript 'Winter is coming: pathogen emergence in seasonal environments' has been provisionally accepted for publication in PLOS Computational Biology.

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Winter is coming: pathogen emergence in seasonal environments

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