Askan Can
email: can-askan.mavi@univ-amu.fr.

Mavi

Can harmful events be another source of environmental traps?

Keywords: Environmental damage, Harmful event, Occurrence hazard, Tipping points, Multiple equilibria, Environmental traps, Adaptation, Mitigation. JEL Classification : O13, D81, Q2, Q54

This paper aims to present a new explanation for environmental traps through the presence of endogenous hazard rate. We show that adaptation and mitigation policies affect the occurrence of environmental traps differently. The former could cause environmental traps, whereas the latter could help society avoid such traps by decreasing the probability of a harmful event occurring. As a result, we present a new trade-off between adaptation and mitigation policies different than the usual dynamic trade-off that is highlighted in many studies and is crucial to developing countries. Contrary to the literature, when an economy is in a trap, an economy with a high environmental quality equilibrium tends to be more conservative in terms of resource exploitation than an economy with a low environmental quality equilibrium, which implies a heterogeneous reaction against the endogenous hazard rate.

Introduction

A social planner should consider methods for avoiding damage as a result of hazardous events. A direct response requires action to reduce the probability of a harmful event taking place. In many cases, mitigation activities reduce the risk of a harmful event by improving environmental quality. 1 Still, mitigation activities cannot eliminate risk completely. When risk is unavoidable, a possible action could be to alleviate the negative consequences of damage caused by a harmful event. The measures taken to reduce losses due to the harmful event can be categorized as adaptation activities. The management of adaptation and mitigation activities raises an interesting dynamic trade-off that can be described as an "adaptation and mitigation dilemma" in the field of environmental economics literature [START_REF] Zemel | Adaptation, mitigation and risk: An analytic approach[END_REF], [START_REF] Tsur | Policy tradeoffs under risk of abrupt climate change[END_REF], [START_REF] Crépin | Regime shifts and management[END_REF]).

To elaborate on these concepts, we consider concrete examples. Improvements in energy efficiency through activities such as carbon capture and storage and reforestation address the root causes by decreasing greenhouse gas emissions and reducing the risk of a harmful event occurring. Therefore, such activities can be referred to as mitigation activities 2 . Differently, installing flood defenses and developing irrigation systems aim to reduce the damage inflicted by a harmful event rather than stop the event from occurring. Hence, such activities can be classified as adaptation activities. In this context, adaptation plays a proactive role, which means that it has no noticeable effect prior to the harmful event's occurrence [START_REF] Smit | An anatomy of adaptation to climate change and variability[END_REF] ; [START_REF] Shalizi | To mitigate or to adapt: Is that the question? Observations on an appropriate response to the climate change challenge to development strategies[END_REF]. In this example, the problem is to decide on an optimal combination of risk-reducing and damage-reducing measures within a given budget.

In this paper, we study the optimal management of natural resources and its link with adaptation and mitigation policies in a simple growth model under endogenous hazard rate depending on the environmental quality level 3 . For example, the rates of harmful events such as droughts, crop failures, and floods 4 are linked to the exploitation of natural capital.

The contribution of this paper is threefold. The first contribution of the paper is to analyze the implications of the endogenous hazard rate on the occurrence of the multiplicity of equilibria (i.e., the environmental trap). We show that one of the reasons for environmental traps is the endogenous hazard rate.

When an economy faces an endogenous hazard rate, a second trade-off arises between consumption and the endogenous hazard rate other than the usual intertemporal trade-off between present and future consumption. An economy with serious environmental quality problems is expected to be impatient due to the high endogenous hazard rate. Therefore, agents tend to increase their consumption at earlier dates, which again stresses the environmental quality over time. This trade-off between consumption and harmful events results in a vicious cycle of a "low level of environmental quality and consumption" in the long run and leads to an environmental trap. Since the multiplicity of equilibria can trap an economy to a lower welfare level, it is legitimate to try to avoid it. In that sense, differently from the existing literature, our paper aims to present qualitative implications of the endogenous hazard regarding the long term dynamics of an economy.

The possibility of multiple stationary equilibria in growth models with endogenous hazard 5 has been mentioned by [START_REF] Tsur | The Management of Fragile Resources: A Long Term Perspective[END_REF] but not examined in greater depth. In this paper, not only do we offer an economic explanation of environmental traps, but we also prove the existence of the multiplicity of equilibria by giving explicit mathematical conditions (see Appendix A*).

Our study relates also to the substantial literature on resource exploitation under uncertainty. The consideration of uncertain events for optimal management began with the work of [START_REF] Cropper | Regulating activities with catastrophic environmental effects[END_REF], who finds that the depletion of resources is either faster or slower than expected if the available resource stock is uncertain. [START_REF] Clarke | Consumption/pollution tradeoffs in an environment vulnerable to pollutionrelated catastrophic collapse[END_REF] find that the endogenous hazard rate either increases or decreases the pollution stock when there is a single occurrence event that indefinitely reduces utility to a constant level.

Our paper differs substantially from these studies, since we treat an economy facing recurrent harmful events. Having recurrent events in the model allows us to offer a more realistic setup.

2

Mitigation activities can also be seen as tools used to avert a harmful event [START_REF] Martin | Averting Catastrophies: The Strange Economics of Syclla and Charybdis[END_REF]).

3

We use the terms environmental quality and natural resource stock interchangeably.

4

The depletion of forests in a region increases the probability of floods because the soil loses its ability to absorb rainfall.

5

There is also a large amount of literature on growth models with endogenous discounting which points out the possibility of multiple equilibria [START_REF] Das | Optimal growth with decreasing marginal impatience[END_REF], [START_REF] Drugeon | Impatience and long-run growth[END_REF], [START_REF] Schumacher | Endogenous discounting via wealth, twin-peaks and the role of technology[END_REF]). However, these papers include neither uncertainty components nor environmental aspects.

The existing literature attempts to determine the reaction of an economy to uncertain events. A common argument is that uncertainty pushes the economy to become more precautious by conserving natural resources (Polasky et al. (2011), de Zeeuw and[START_REF] De Zeeuw | Regime shifts and uncertainty in pollution control[END_REF], [START_REF] Ren | The optimal management of renewable resources under the risk of potential regime shift[END_REF] ).

The second contribution of this paper is to show that there can be a heterogeneous reaction to uncertain events. We show that an economy with a high environmental quality level is more precautious than one with a low environmental quality level. In other words, an economy with a low environmental quality equilibrium adopts an "aggressive" exploitation policy relative to an economy with a high environmental quality equilibrium.

Apart from the literature on resource exploitation under uncertainty, recent contributions have been made regarding the implications of harmful events on the long-term behavior of an economy. A paper by van der Ploeg (2014) focuses on how a first-best optimal carbon tax should be adjusted over time when an economy faces harmful event probability. van der Ploeg and de Zeeuw (2017) extend the framework offered by van der Ploeg (2014) and show that a positive saving response can help the economy to dampen the discrete change in consumption at the time of the harmful event. A recent study by [START_REF] Akao | A theory of disasters and long-run growth[END_REF] offers a general framework that can be used to justify the empirical evidence that shows the positive correlation between disasters and long-term economic performance.

Our study differs also from this branch of the literature, since these studies focus on the social cost of carbon stock but say nothing about the necessary policy response in terms of adaptation and mitigation policies in order to deal with harmful events.

The third (and most significant) contribution of this paper is that it presents a new trade-off between adaptation and mitigation different from the dynamic trade-off highlighted by numerous studies [START_REF] Bréchet | Adaptation and mitigation in long-term climate policy[END_REF], Le [START_REF] Kama | Supplementing Domestic Mitigation and Adaptation with Emissions Reduction Abroad to Face Climate Change[END_REF], [START_REF] Millner | Adaptation to Climate Change and Economic Growth in Developing Countries[END_REF]). [START_REF] Zemel | Adaptation, mitigation and risk: An analytic approach[END_REF] and [START_REF] Tsur | Policy tradeoffs under risk of abrupt climate change[END_REF] investigated the time profile of the optimal mix of adaptation and mitigation in a simple growth model with uncertainty. However, these studies did not consider the multiplicity of equilibria and its implications regarding adaptation and mitigation policies. Our main result is to show that adaptation increases and mitigation decreases the possibility of environmental traps.

What is the mechanism behind the new trade-off between adaptation and mitigation? Adaptation capital is shown to decrease the optimal steady-state level of environmental quality, as agents worry less about the consequences of a harmful event. Then, because the endogenous hazard rate increases, the trade-off between present consumption and the endogenous hazard rate becomes more important, which is likely to raise multiple equilibria. Contrary to this mechanism, mitigation activity improves environmental quality, and the trade-off between present consumption and harmful event probability becomes weaker.

Another contribution is our analysis of how the dynamic trade-off between adaptation and mitigation affects the occurrence of environmental traps. Recent papers by [START_REF] Tsur | Policy tradeoffs under risk of abrupt climate change[END_REF] and [START_REF] Zemel | Adaptation, mitigation and risk: An analytic approach[END_REF] focus on the dynamic trade-off between adaptation and mitigation but overlook its qualitative implications regarding the multiplicity of equilibria. We show that the unit cost of adaptation changes the optimal mix of adaptation and mitigation policies over time but also affects the occurrence of the multiplicity of equilibria.

The remainder of this paper is organized as follows: Section 2 presents the benchmark model. Section 3 describes the model with adaptation and mitigation policies and explains in depth the effects of adaptation and mitigation on the occurrence of environmental traps. Section 4 provides numerical illustrations. Section 5 concludes the paper.

Model

Let S (t) represent the environmental stock available or environmental quality at time t, e.g, the stock of clean water, soil quality, air quality, forests, biomass. We refer to a broad definition of environmental quality which encompasses all environmental amenities and existing natural capital that have an economic value 6 .

Obviously, disamenities such as waste and pollution stemming from consumption c (t) decrease environmental quality stock. The stock S (t) evolves in time according to

Ṡ (t) = R (S (t)) -c (t) (1)
With a given initial state S (0), an exploitation policy of environmental stock c (t) generates the state process S (t) according to equation ( 1) and provides the utility u (c (t)). Similar to Le [START_REF] Kama | A Note on the Consequences of an Endogeneous Discounting Depending on the Environmental Quality[END_REF], we use a framework where consumption comes directly from environmental services and causes environmental damage.

We make use of the following assumptions.

A.1 The regeneration of environmental quality is characterized by R (.) : R + → R + , R (S) > 0, R SS (S) < 0, R S (S) ≶ 0 and R S = 0 where S is the carrying capacity of the natural resource stock.

A.2

The utility function u (.) : R + → R + is twice continuously differentiable with the following properties ; u c (c) > 0, u cc (c) < 0 , ∀c and lim c→0 , u c (c) = ∞. [START_REF] Schumacher | Endogenous discounting and the domain of the felicity function[END_REF] shows that models with endogenous discounting should use a utility function with a positive domain because the utility function appears in the optimal path of consumption. Indeed, since our model focuses on recurrent events such as droughts and floods, the utility function does not appear in the Keynes-Ramsey rule. Therefore, changing the utility function from having a negative domain to a positive domain does not qualitatively change the solutions.

Not only does the environmental stock S (t) represent a source of consumption for the economy, but it also affects the endogenous hazard rate. Forests, for example, considerably influence the environmental conditions in any given area and help to decrease the endogenous hazard rate (see [START_REF] Dasgupta | The Environment and Emerging Development Issues[END_REF] Let T be the event occurrence time of an harmful event and let F (t) = P r {T ≤ t} and f (t) = F (t) denote the corresponding probability distribution and density functions, respectively. The endogenous hazard rate h (S) is related to F (t) and f (t) with respect to

h (S (t)) ∆ = P r {T ∈ (t, t + ∆ | T > t)} = f (t) ∆ 1 -F (t) ( 2
)
where ∆ is an infinitesimal time interval. We have h (S (t)) = -dln(1-F (t))

dt

. The term h (S (t)) ∆ specifies the conditional probability that a harmful event will occur between [t, t + ∆], given that the event has not occurred by time t. A formal specification for probability distribution and density functions gives

F (t) = 1 -exp - t 0 h (S (τ )) dτ and f (t) = h (S (t)) [1 -F (t)] (3)
6

We exclude mining and oil industries from our definition of natural capital.

7

An interesting real-world example is the reforestation project in Samboja Lestari conducted by the Borneo Orangutan Survival Foundation. The project helped to increase rainfall by 25% and to avoid droughts by lowering the air temperature by 3 degrees Celcius to 5 degrees Celcius . [START_REF] Boer | Planting sugar palms borassus flabellifer as a means to offset the CO2 load of apenheul primate park, apeldoorn, the netherlands[END_REF], [START_REF] Normile | Restoring a biological desert on borneo[END_REF] We assume that the hazard function converges to a constant h when the natural resource stock tends to infinity.

A.3

The endogenous hazard rate h (S) : R + → R + is twice continuously differentiable with the following properties ; h (S) > 0, h S (S) < 0.

Since S (t) is a beneficial state, the endogenous hazard rate h is a non-increasing function (a higher environmental quality stock entails a lower occurrence probability). We do not make any assumptions on for the second derivative of the endogenous hazard rate.

Recurrent events are repeated events such as droughts and floods that occur frequently in our current society and entail an immediate damage ψ. In addition, the economy is always under the risk of more events occurring later, with value function V (S). The post-value function describing the economy with recurrent is defined as

ϕ (S) = V (S) -ψ (4)
Given the uncertain arrival time T , the exploitation policy c (t) yields the following payoff

T 0 u (c (t)) e -ρt dt + ϕ (S (T )) e -ρT ( 5
)
where ρ is the social discount rate. Taking expectations of the expression (5) according to the distribution of T and considering (3) gives the expected payoff

V (S) = max c(t) ∞ 0 u (c (t)) + h (S (t)) V (S (t)) -ψ exp - t 0 [ρ + h (S (τ ))] dτ dt (6)
The maximization problem (6) at hand is deterministic, yielding a value that corresponds to the maximum expected value of the uncertainty problem (see Appendix A.1). The uncertain time T of the harmful event arrival and assuming a constant value of harmful damage ψ introduce a modelling asymmetry. However, we can interpret the term ψh (S) as the expected harmful damage.

The solution of maximizing (6) with respect to the evolution of environmental stock (1) leads to the Keynes-Ramsey rule (see Appendix (A) for details),

ċ = - u c (c) u cc (c) R S (S) -ρ - ψh S (S) u c (c) (7)
When harmful events are recurrent, the economy adopts precautionary behavior due to the presence of the precaution term ψh S (S) uc(c) in equation ( 7). 8 By using equations ( 1) and ( 7), the steady state of the economy can be described by the function G (S).

G (S) = R S (S) -ρ - ψh S (S) u c (R (S)) (8)
Note that the roots of the function G (S) are steady-states of the economy.

Proposition 1. (i) The necessary condition for multiple equilibria in an economy exposed to an endogenous hazard rate is the following:

If A = {S : G (S) = 0}, then card (A) > 1.
If the hazard rate is exogenous, the multiplicity of equilibria is not a possible outcome.

8

This result is similar to that presented by de Zeeuw and [START_REF] De Zeeuw | Regime shifts and uncertainty in pollution control[END_REF], by which the authors show that recurrent events induce a precautionary behavior.

(ii) The sufficient conditions to have an environmental trap (i.e, multiple equilibria) for recurrent events are G S > 0 and ∃S < S, G (S) < 0 where S is a value between S mid and S high and also

G S (S) = R SS (S) - ψh SS (S) u c (R (S)) + ψh S (S) u cc (R (S)) R S (S) (u c (R (S))) 2 > 0 (9) Proof. See Appendix (B)
The cardinality (with notation card (A)) of a set is a measure for the number of elements of the set A 9 .

The first part of the sufficient conditions holds for the existence of low steady state. The second part ensures the existence of a high steady state (see Appendix (B) for the existence of three different steady states).

Without the precaution term ψh S (S) uc(c) , the function G S (S) becomes a negative term due to A.1 and the economy always admits a unique equilibrium. The explanation for the mathematical expressions in ( 9) are as follows: The first term relates to the convexity of regeneration function, which is a negative term. The second term represents the convexity of the endogenous hazard rate 10 . The more sensitive the endogenous hazard rate is to marginal changes of environmental quality, the greater the chance that G S (S) > 0. The third term relates to the concavity of the utility function. A more concave utility function (i.e., higher risk

aversion) results in decreases in G S (S) if R S (S) > 0.
Overall, the multiple equilibria condition depends on the close relationship between the endogenous hazard rate and consumption. The precaution term ψh S (S) uc(c) in equation ( 7) is the expression representing the second trade-off between present consumption and harmful event risk which is the source of environmental traps. If the constant penalty rate is equal to zero, the trade-off disappears11 . Therefore, the economy admits a unique equilibrium.

Lemma 1. The low and high steady-states S low and S high are saddle path stable. However, S mid could have complex eigenvalues12 .

Proof. See Appendix (D)

This means that the economy could converge to either a high or low environmental quality equilibrium.

Once the economy reaches either a high or low equilibrium, it remains in this state permanently. When an economy reaches a low-quality equilibrium, it is said to be "trapped" in a state where the consumption and environmental quality are low relative to an economy with a high-quality equilibrium.

Phase Diagram Analysis

Considering equation ( 7), we can observe that the steady state curve ċ = 0 is non-linear on a phase plane (S, c) due to the endogenous hazard rate . Therefore, multiplicity of equilibria is a possible outcome13 . We present two different phase diagram configurations14 on a plane (S, λ) where λ is the shadow price for environmental quality. The first case is for multiple equilibria with a unique path. This means that, given 9 For example, when we note card (A) = 3, it means that the set contains three elements. 10 Assume that the endogenous hazard rate has a convex form.

an initial value of S, there is only one reachable steady state and it is optimal to pursue this steady-state.

The second case is where, given an initial value of S, multiple growth paths exist. We conduct the phase diagram analysis on a plane (S, λ) and not (S, c) because differentiating Hamiltonian with respect to a costate variable (here λ) directly gives the isocline Ṡ = 0 (See Appendix (A) for details). Then, it is easier to find the value of different optimal trajectories and compare them to find the global optimum15 .

Of course, our analysis in following sections concentrates on the case of multiple equilibria with a unique growth path, since all steady-states are optimal and relevant. However, it is worthwhile to present the phase diagram with multiple growth paths.

First, we present the dynamics of the co-state and state variables (see Appendix (A))

λ = -λ R S (S) -ρ - ψh S (S) λ (10) Ṡ = R (S) -c (λ) (11)

Multiple equilibria: Unique growth path

Figure 1 shows that S mid acts as a threshold. An economy starting with an initial stock of natural resources S 0 < S mid always converges to S low , while an economy which starts with S 0 > S mid converges to S high . There are two convergence groups and the choice between them is made depending on history (initial condition).

The phase diagram with a unique path is as follows For the sake of clarity, we present trajectories for both equilibria separately and show that different optimal converging paths exist to both low and high steady states. Figure 2 17 shows that depending on the initial conditions, either the economy converges to a steady state or diverges from a steady state. The phase diagram analysis clearly shows that the existence of different convergence groups rules out the possibility of global stability in the model.

When there is a multiplicity of equilibria, an important issue concerns the stability of the equilibrium and the corresponding trajectories converging to different equilibria. [START_REF] Wagener | Skiba points and heteroclinic bifurcations, with applications to the shallow lake system[END_REF] analyses the shallow lake dynamics and shows that there could be heteroclinic connections between different saddle points. In the existence of heteroclinic bifurcations, only one of the saddle points may be relevant (see Figure 3 in [START_REF] Wagener | Skiba points and heteroclinic bifurcations, with applications to the shallow lake system[END_REF]). In this study, Lemma 1 rules out the possibility of heteroclinic bifurcations.

Multiple equilibria: Multiple growth paths

When the Hamiltonian is not concave in S, the uniqueness of Pontryagin paths is no longer ensured.

Therefore, starting with an initial value of natural resource stock, several paths may exist, and these may lead to different saddle points (i.e., there may be multiple reachable equilibria) (see [START_REF] Palivos | Endogenous fertility, multiple growth paths, and economic convergence[END_REF]). In this case, the values of different trajectories, starting from a given initial condition of S, should be compared to determine the global optimum [START_REF] Skiba | Optimal Growth with a Convex-Concave Production Function[END_REF], [START_REF] Davidson | Non-Convexities in Continuous-Time Investment Theory[END_REF]).

As illustrated in Figure 3 there are three steady states B, D, H. 17 The parameters used for the simulation is as follows; σ = 2.1, ρ = 0.0095, g = 0.01, S = 1, η = 1.75 h = 0.03, ψ = 140.

V (S 0 ) = H (S 0 , λ 0 ) ρ ( 12
)
where H is the Hamiltonian associated with the infinite horizon program presented in (6) (See Appendix (A) for details). From ( 12), the choice between AH and V B is immediate. Given that H λ = Ṡ = R (S)c (λ) > 0 for all λ above Ṡ = 0 locus and that λ A 0 < λ V 0 (i.e, c A 0 > c V 0 ), the optimal solution is to take path AH. Lemma 1 implies that it is impossible to be optimal for one country to start at S 0 < S low and to move toward H while another country starts at S 0 > S high and moves toward B.

For the case with multiple reachable steady states, one would like to know whether there is a threshold at which the economy is indifferent toward going to either of these steady states. This leads us to perform an analysis to find a Skiba point. [START_REF] Wagener | Skiba points and heteroclinic bifurcations, with applications to the shallow lake system[END_REF] in which it is shown that a Skiba point may not be associated with an unstable equilibrium. Departing from the initial condition S 0 = 0.69464, the value of the optimal path on the left and on the right of Figure 4 (see thick dashed lines) is the same20 . The existence of an indifference point is understandable; Either the economy prefers having higher consumption with a lower natural resource stock or vice versa.

Proposition 2. S * high -S * N R > S * low -S * N R
where S * N R , S * high and S * low are the optimal steady-state value of natural resources in an economy without risk and the optimal steady-state for high and low environmental quality in an economy with risk, respectively. This means that a high environmental quality equilibrium economy adopts a more conservative exploitation policy than low environmental quality equilibrium economy.

Proof. See Appendix (E)

According to the precaution term in equation ( 7), when there are recurrent events, the endogenous hazard rate always makes an economy more precautious in terms of resource exploitation (see de [START_REF] De Zeeuw | Regime shifts and uncertainty in pollution control[END_REF], [START_REF] Tsur | The Management of Fragile Resources: A Long Term Perspective[END_REF], [START_REF] Polasky | Optimal management with potential regime shifts[END_REF], [START_REF] Clarke | Consumption/pollution tradeoffs in an environment vulnerable to pollutionrelated catastrophic collapse[END_REF]). As seen in Figure 5, the novelty of this proposition with respect to the literature is that the endogenous hazard rate leads to different levels of precaution (different steady states) against harmful events when there is a multiplicity of equilibria.

High level of precaution

Can environmental policy cause/prevent an environmental trap?

The focus of our study is to determine the effects of adaptation and mitigation policies on environmental traps. To facilitate an understanding of their mechanisms, we prefer to analyze separately the effects of adaptation and mitigation policies on environmental traps.

An economy with an adaptation-only policy

A social planner could reduce the damage ψ via adaptation capital K A . This way of modeling adaptation is consistent with [START_REF] Zemel | Adaptation, mitigation and risk: An analytic approach[END_REF] and [START_REF] Tsur | The Management of Fragile Resources: A Long Term Perspective[END_REF]. However, this differs from the work of [START_REF] Bréchet | Adaptation and mitigation in long-term climate policy[END_REF] and Le [START_REF] Kama | Supplementing Domestic Mitigation and Adaptation with Emissions Reduction Abroad to Face Climate Change[END_REF], where the adaptation capital directly affects the damage function for all time t. In our model, adaptation plays a proactive role. Hence, the tangible benefits of adaptation can be gained only if a harmful event occurs. However, this is not to say that investing in the adaptation decision does not make a difference. Its contribution is accounted for by the objective function of the social planner. Investing at rate A contributes to adaptation capital K A which is reduced by constant depreciation δ > 0. The evolution of the adaptation capital is as follows

KA (t) = A (t) -δK A (t) (13)
with K A (0) given. Damage function ψ (K A ) decreases when adaptation capital K A increases. We assume that A.4 The damage function is characterized by ψ (.):

R + → R + , ψ (0) = ψ, ψ (∞) = ψ, ψ (K A ) > 0, ψ K A (K A ) < 0 and ψ K A K A (K A ) > 0
When there is no adaptation capital, the inflicted damage will be a constant term. Moreover, it is realistic to assume that damage reduction has a limit, as we cannot eliminate the negative effects of a harmful event by accumulating adaptation capital. For that reason, we assume that the damage function is constrained within an upper ψ and a lower bound ψ.

Investing in adaptation capital has the cost

Q 1 (A) = φ A A 2
2 with the unit cost of adaptation φ A which has a convex form. The cost function for adaptation enters in the utility function as a social cost. This is in line with [START_REF] Zemel | Adaptation, mitigation and risk: An analytic approach[END_REF] and [START_REF] Tsur | The Management of Fragile Resources: A Long Term Perspective[END_REF]. Consumption c (t) provides utility u (c (t)). The expected payoff is

V (S, K A ) = max {c(t),A(t)} ∞ 0 u (c (t)) -Q 1 (A (t)) +h (S (t)) [V (S (t) , K A (t)) -ψ (K A (t))]] exp - t 0 [ρ + h (S (τ ))] dτ dt (14)
V (S, K A ) stands for the value of the maximization problem. The optimal policy is to maximize (14) subject to (1) and ( 13) (see Appendix F). Optimal dynamics for consumption and adaptation are as follows

ċ = - u c (c) u cc (c) R S (S) -ρ - ψ (K A ) h S (S) u c (c) (15) Ȧ = Q 1 A (A) Q 1 AA (A) ρ + δ + h (S) ψ K A (K A ) Q 1 A (A) (16)
When an economy invests in an adaptation policy, we have an additional equation ( 16), which shows the optimal dynamics of adaptation investment.

In order to assess the effect of adaptation capital on environmental traps, we calculate the necessary conditions for multiple equilibria (see Appendix (F)).

GS (S) = RSS (S) - hSS (S) ψ (KA) uc (R (S)) + hS (S) ψ (KA) ucc (R (S)) RS (S) (uc (R (S))) 2 - ψK A (KA) hS (S) uc (R (S)) dKA dS =Z 1 > 0 (17)
where the term Z 1 stands for the effect of adaptation capital on the necessary condition for the environmental trap.

To assess the effect of this term, we must understand how environmental quality changes the adaptation capital level at steady state. To do this, we write equation ( 16) at the steady-state

Q 1 A (δK A ) (ρ + δ) = -ψ K A (K A ) h (S) (18)
which defines the function K A (S). The right-hand side -ψ K A (K A ) h (S) can be interpreted as the marginal benefit of adaptation capital, and the left-hand side

Q 1 A (δK A ) (ρ + δ)
is the marginal cost of adaptation capital. Optimal steady states are located on this curve. We can find the slope of the steady state curve by taking the total derivative of equation ( 18)

dK A dS = h S (S) ψ K A (K A ) (ρ + δ) Q 1 A (δK A ) + ψ K A K A (K A ) h (S) < 0 ( 19
)
Proposition 3. A higher adaptation capital K A (S) makes the necessary condition G S (S) > 0 easier to hold, meaning that the possibility of multiple equilibria increases with adaptation capital.

Proof. See Appendix (F)

Because we have dK A dS < 0, the term Z 1 is negative; this means that the possibility of multiple equilibria increases with an increase in adaptation capital.

According to equation ( 19), a higher level of environmental quality requires a lower adaptation capital level at a steady state. The numerator in (19) can be considered the variation of the marginal benefit of adaptation with respect to environmental quality, which is a negative term. This means that the marginal benefit of adaptation decreases as the environmental quality level increases. The first and second terms in the denominator represent the marginal cost of adaptation capital and concavity of penalty function with respect to adaptation capital.

Another interpretation of this result can be as follows: Because agents expect to face less damage with an adaptation policy and can more easily bear the negative consequences of a harmful event, they tend to care less about environmental quality.

How can we explain the multiplicity of equilibria when the economy implements an adaptation policy?

When there is investment in adaptation capital, environmental quality decreases as shown in ( 19), and the endogenous hazard rate increases. As a result, due to a higher impatience level, an economy becomes less precautionary in exploiting natural resources. It follows that the endogenous hazard rate amplifies again, and the need for adaptation capital increases again21 . This mechanism, causing multiple equilibria, explains how an adaptation-only policy may trap a country into a low welfare state.

Indeed, the trade-off between present consumption and a harmful event becomes more significant when an adaptation policy is in place because the endogenous hazard rate increases.

An economy with a mitigation-only policy

We now show that an economy implementing only a mitigation policy could escape an environmental trap. Because consumption comes from environmental assets, improving environmental quality (mitigation) increases the consumption level in the long run.

Investing at rate M for mitigation improves environmental quality by Γ (M ). Then, in the presence of mitigation activity, environmental quality evolves according to

Ṡ (t) = R (S (t)) + Γ (M (t)) -c (t) (20)
where Γ (M ) represents the effects of mitigation activities such as reforestation, desalination of water stock, enhancing carbon sinks, etc. Mitigation is defined as a "human intervention to reduce the sources or enhance the sinks of greenhouse gases." (IPCC (2014), p.4) In this sense, reforestation can be considered as a means to enhance carbon sinks because forests allow for carbon sequestration.

The specification of the mitigation variable is similar to that proposed in [START_REF] Chimeli | Total factor productivity and the environmental Kuznets curve[END_REF].

Alternatively, function Γ (M ) can be considered an "environmental protection function". The expenditures for environmental protection may be directed not only toward pollution mitigation but also toward the protection of forests and the recovery of degraded areas. As such, mitigation activity can be seen as a means of improving environmental quality.

To keep the model as simple as possible, we choose to consider mitigation as a flow variable. We use the following assumption

A.4

The mitigation function given by Γ (.) : R + → R + is twice continuously differentiable, with the following properties ; Γ M (M ) > 0, Γ M M (M ) < 0.

The mitigation function is assumed to be an increasing and concave function. Mitigation activity can be considered a complement to the regeneration of the environment. Investing in mitigation activity has the cost Q 2 (M ) = P M M with the unit price for mitigation P M that enters in the utility function as a social cost. In an economy with mitigation activity, the expected payoff is

V (S) = max {c(t),M (t)} ∞ 0 u (c (t)) -Q 2 (M ) +h (S (t)) V (S (t)) -ψ exp - t 0 [ρ + h (S (τ ))] dτ dt (21)
Assuming a concave function for mitigation activity Γ (M ) gives optimal interior solutions and simplifies calculations.

The optimal policy is to maximize (21) subject to (20) (see details in Appendix (H)). Optimal dynamics for consumption and mitigation are as follows

ċ = - u c (c) u cc (c) R S (S) -ρ - ψh S (S) u c (c) (22) Ṁ = - u cc (c) u c (c) Γ M (M ) Γ M M (M ) R S (S) -ρ - ψh S (S) u c (c) (23)
In fact, mitigation activity renders the trade-off between present consumption and the endogenous hazard rate less binding. It follows that there is a lower risk of an economy being trapped in a multiplicity of steady states.

Due to the concavity of the mitigation function, we notice that the signs of ċ and Ṁ are different. It

follows that the optimal level of mitigation activity comes at the cost of lower consumption. This also explains that the economy causes less harm to the environment when mitigation activity is invested in.

Consequently, the presence of mitigation activity weakens the trade-off between present consumption and the endogenous hazard rate.

We provide the necessary condition for multiple equilibria by using equations ( 22) and ( 23) (Appendix (G) for details)

G S (S) = R SS (S) - ψh SS (S) u c (c) + ψh S (S) u cc R S (S) (u c (c)) 2 + ψh S (S) u cc R S (S) (u c (c)) 2 dM dS =Z 2 > 0 (24)
In the above equation, Z 2 represents the effect of mitigation on the necessary condition for multiple equilibria.

We look at how environmental quality levels change the steady state level of mitigation investment. To do that, we write the first-order condition for mitigation activity (see equations (G.3) and (G.4) in Appendix (G))

u c (R (S) + Γ (M )) Γ M (M ) = Q 2 M (M ) = P M (25)
and take the total derivative of ( 25) to find

dM dS = - u cc (c) R S (S) Γ M (M ) u cc (c) (Γ M (M )) 2 + u c (c) Γ M M (M ) < 0 (26)
The economic reasoning behind equation ( 26) is as follows: When environmental quality is high, the social planner decreases mitigation. In other words, the economy needs less mitigation as environmental quality improves.

Proposition 4. Mitigation activity M decreases the possibility of multiple equilibria, since the necessary condition for multiple equilibria G S (S) > 0 is less likely to hold.

Proof. See Appendix (G)

Because we have dM dS < 0, the term Z 2 is negative and decreases G S (S). This implies that it is more difficult to ensure the multiple-equilibria condition (24).

To better understand how mitigation can prevent an environmental trap, suppose that mitigation activity could reduce the endogenous hazard rate to zero. It follows that the trade-off between present consumption and a harmful event, which causes multiple equilibria, disappears. Based on this, one can understand that mitigation activity weakens the trade-off between present consumption and a harmful event.

Obviously, mitigation activity not only protects the economy against an environmental trap but also increases the steady-state level of environmental quality, as expected. Indeed, since consumption comes from natural resource rents, mitigation policy allows the economy to increase its consumption level. Hence, the impatience level of low-income countries decreases, allowing them to postpone their consumption and to make far-sighted decisions. In summary, a mitigation policy could break the vicious cycle of low consumption and environmental quality that can be triggered by the implementation of an adaptation-only policy. Then, one can conclude that social planners should couple an adaptation policy with a mitigation policy.

Numerical analysis

The section aims to illustrate the theoretical findings obtained in previous sections for the effect of adaptation and mitigation on the occurrence of multiple equilibria in the previous sections. For the numerical part, we calibrate the benchmark economy according to New Zealand's economy, which is a rare country in that it publicly announces the likelihood of harmful events such as floods and droughts.

By doing a calibration exercise, we produce a model that matches the EPI index 22 and the endogenous hazard rate that is computed with real data for New Zealand 23 . We use the following functional specifications.

An economy with mitigation-only policy

We present a multiple-equilibria economy without a policy and an economy with a mitigation policy. The most important message portrayed by Figure 6 is that mitigation activity enables an economy to avoid the occurrence of multiple equilibria as shown by Figure 6b. The reason for this result is that mitigation activity reduces the endogenous hazard rate. It follows that the trade-off between present consumption and the endogenous hazard rate which is the source of the multiplicity of equilibria weakens. Hence, the economy avoids the multiplicity of steady states.

Figure 6b also shows that mitigation activity increases environmental quality. It is easy to see on Figure 6b that the unique steady-state level of environmental quality is greater than that of all three steady-state levels of environmental quality on Figure 6a.

Figure 6a shows that, on the one hand, an economy with an initial stock of environmental quality S 0 < S mid converges to S low . On the other hand, the economy converges to S high if the initial stock of environmental quality is S 0 > S mid . We can conclude that S mid is unstable and can be considered as a threshold25 separating two stable steady state S low and S high .

Figure 6b shows that the economy avoids the multiplicity of equilibria by investing in mitigation. The economy admits a unique equilibrium and regardless of the initial stock of environmental quality, there is convergence to the unique steady-state.

The economy with adaptation-only policy

We present the benchmark model without a policy and an economy with an adaptation policy. For this numerical exercise, we calibrated the benchmark economy according to New Zealand's economy. As in the previous section, we calibrate the damage parameter ψ and the intrinsic growth rate of the resource stock g such that the steady-state level of environmental quality S equals the EPI score of New Zealand in 2018. As in the previous numerical exercise for mitigation, we set the elasticity of substitution σ to 2 and the pure rate of time preference ρ to 0.015 as in [START_REF] Nordhaus | A Question of Balance: Weighting the Options on Global Warming Policies[END_REF]. Unfortunately, we do not have any available data for the adaptation capital. Therefore, we set parameters ω to 0.95 and γ to 0.9. The occurrence of the multiplicity of equilibria relies on the fact that adaptation capital decreases environmental quality (see equation ( 19)). As a result, the trade-off between present consumption and the endogenous hazard rate becomes more binding and the multiplicity of equilibria occurs. It is worth noting that S high is lower than the environmental quality level in an economy without environmental policy, since adaptation capital tends to decrease environmental quality.

Parameter Calibrated value

An important policy-based message of this result is that an adaptation policy should be coupled with a mitigation policy in order to avoid environmental traps.

Adaptation vs mitigation: what about the multiplicity of equilibria?

An important issue regarding adaptation and mitigation policies is the matter of how the trade-off between adaptation and mitigation affects the occurrence of multiple equilibria. The adaptation/mitigation trade-off is intensively discussed in the existing literature. [START_REF] Buob | To mitigate or to adapt: How to confront global climate change[END_REF] argue that a country's stage of development matters and that high-income countries should invest both in adaptation and mitigation while low-income countries should only invest in adaptation. [START_REF] Bréchet | Adaptation and mitigation in long-term climate policy[END_REF] studied the optimal mix of adaptation and mitigation and conclude that the substitutability between adaptation and mitigation depends on the country's stage of development. A recent study by [START_REF] Tsur | Policy tradeoffs under risk of abrupt climate change[END_REF] focuses on the optimal mix of adaptation and mitigation under harmful event uncertainty, but it remains agnostic about the implications of the trade-off between adaptation and mitigation on the multiplicity of equilibria. We seek to analyze this issue when an economy is subject to an endogenous hazard rate (see Appendix H for calculations).

In this subsection, apart from the calibration, the unit cost of adaptation φ is a free parameter. To assess the role of the trade-off between adaptation and mitigation on the occurrence of multiple equilibria, we present two cases with low and high unit cost of adaptation φ. Consequently, in the first case, the optimal level of adaptation investment is higher than in the second case.

Figure 8a shows the existence of the multiplicity of equilibria in an economy with adaptation and mitigation policies. Figures 8b and8c represent the optimal path of adaptation and mitigation corresponding to the economy with a high environmental quality level in the long run. We observe that both adaptation and mitigation increase from the start date and converge to a stable high-environmental-quality equilibrium26 . Figure 9 shows that when the unit cost of adaptation φ is higher, the economy admits a unique equilibrium.

When we look at Figures 9b and9c, both adaptation and mitigation activities increase over time. Compared to the previous case, the economy shifts its resources from adaptation to mitigation due to the higher unit cost of adaptation φ. We see that the multiplicity of equilibria disappears when the mitigation activity is relatively higher than adaptation. Note that both adaptation and mitigation aim to decrease the disutility of the presence of the endogenous hazard rate. The former does this by decreasing the level of vulnerability to inflicted damage, the latter by decreasing the endogenous hazard rate. As such, the choice between adaptation and mitigation depends on the relative costs of these policies. When the cost of adaptation increases (i.e, a higher unit cost φ), it is evident that the economy starts to give more weight to mitigation than to adaptation in order to cope with the endogenous hazard rate.

As shown, a higher adaptation cost not only affects the dynamic trade-off between adaptation and mitigation but also has qualitative implications, such as those related to the occurrence of environmental traps. When there is more weight given to mitigation activity, the multiplicity of equilibria disappears. The reason for this result is that the trade-off between present consumption and the endogenous hazard rate of harmful events becomes less important.

Conclusion

In this paper, we analyzed the effects of adaptation and mitigation policies on environmental traps in an economy subject to an endogenous hazard rate. This study offers a new explanation for environmental traps via endogenous hazard rate and increases the overall understanding of how environmental policy causes or prevents environmental traps. We believe that this new perspective also provides interesting advice to policymakers regarding the opposite effects of adaptation and mitigation policies.

Our main results show that an adaptation policy can lead the economy into an environmental trap, whereas mitigation helps the economy to avoid environmental traps. We show that a new trade-off appears between adaptation and mitigation with respect to their effects on environmental traps. This trade-off is different from the trade-off between adaptation and mitigation over time mentioned in numerous studies (see [START_REF] Zemel | Adaptation, mitigation and risk: An analytic approach[END_REF], [START_REF] Tsur | Policy tradeoffs under risk of abrupt climate change[END_REF], [START_REF] Bréchet | Adaptation and mitigation in long-term climate policy[END_REF]). The fact that adaptation policy could cause an environmental trap does not mean that social planners should not invest in adaptation activities.

On the contrary, since it is impossible to eliminate the endogenous hazard rate, policymakers should invest in adaptation capital but should couple this policy with mitigation activity to avoid the adverse effects of an adaptation-only policy. This is because mitigation activity weakens the trade-off between present consumption and harmful events by improving environmental quality.

Future research could test the model using empirical methods. Currently, this is very challenging, as there are no available data on adaptation investments. However, it is very desirable and is a crucial part of our future research agenda.

economy tends to care little about the environment, and natural resources are exploited as a result, leading to a unique equilibrium. However, when the damage rate lies between a range of values that respects the condition (A*.6), the trade-off between harmful events and present consumption becomes relevant because the economy is neither very precautionary nor careless about the environment (see Wirl (2004) 28 ).

Example 2: An economy with only an adaptation policy

To prove the existence of multiple equilibria in an economy with an adaptation policy, we start with an example of an economy without any environmental policies that admits a unique equilibrium and uses the following usual functional specifications:

R (S) = gS (1 -S) (A*.7) h (S) = 1 -hS (A*.8) u (c) = log (c) (A*.9) ψ (K A ) = ψ (1 -aK A ) (A*.10) Q (A) = φ A 2 2 (A*.11)
For the sake of analytical tractability, differently from the previous example and from the definition given in A.329 , we use a linear hazard function and show that such a function ensures a unique equilibrium for an economy without an environmental policy. Then, we prove that an adaptation policy results in a multiple-equilibria economy.

Under the specification of functional forms (A*.7), (A*.8), (A*.9), an economy without an environmental policy admits a unique steady-state. When the economy invests in adaptation activity, multiplicity of equilibria occurs if the following condition on harmful damage ψ holds

4g 2 + 36g (g -ρ) -27 (g -ρ) 2 - 4g 2 + 36g (g -ρ) -27 (g -ρ) 2 2 -2 9 g 3 (g -ρ) 8 (g -ρ) g h 2 < ψ < 4g 2 + 36g (g -ρ) -27 (g -ρ) 2 + 4g 2 + 36g (g -ρ) -27 (g -ρ) 2 2 -2 9 g 3 (g -ρ)
8 (g -ρ) g h 2 (A*.12)

Proof. See Appendix (J)

This example clearly shows that an economy which is not initially exposed to multiple equilibria faces the possibility of environmental traps under an adaptation policy if condition (A * .12) is ensured.

Indeed, a policy recommendation based on increasing adaptation capital could cause a multiplicity of steady states and, in turn, trap an economy at a low equilibrium. The proof is simple to follow when one looks at the Figure 10. We denote steady states as 0 < S low < S mid < S < S high < S . A sufficient condition for the existence of S low is that ∃S < S which ensures (i) G S (S) > 0 and (ii) G (S) < 0. Unless the condition (i) G S (S) > 0 is satisfied, G (S) starting from an arbitrary positive value may cross the x-axis just one more time and converge to z, which results in a unique steady state equilibrium. Additionally, the condition (ii) G (S) < 0 is also necessary to ensure that G (S)

crosses the x-axis by S low at least once.

The sufficient condition for the existence of S high is that G S > 0. If this condition does not hold, G (S) does not cross the x-axis for a second time and converges to z without changing sign. Then, there exists a unique equilibrium. Once the condition G S > 0 is satisfied, we notice that G (S) crosses the

x-axis unambiguously by S mid and thereafter converges to z when S approaches S. Under these conditions, we prove the existence of three steady states, one of which is unstable and two of which are stable.

When there is no endogenous occurrence probability, the necessary condition (9) reduces to R SS (S) > 0, which makes multiple equilibria an impossible outcome due to the Assumption 1. Therefore, the model reduces to a standard neoclassical growth model. This completes the proof.

C Slope of the steady-state curve

Using equation (B.3) and the implicit function theorem, we can find the slope of the λ = 0 line.

dλ dS λ=0 = R SS (S) -ψh SS (S) λ -h S (S) ψ (λ) 2 (C.1)
We can remark that the denominator is always negative. However, the sign of the numerator depends on the convexity of the regeneration function and on the hazard function. This is why we present different phase diagram configurations.

D Proof of Lemma 1

The differential system describing the economy can be written as follows:

  ċ Ṡ   =   d ċ dc d ċ dS d Ṡ dc d Ṡ dS   ċ=0, Ṡ=0   c -c * S -S *   d ċ dc = ρ -R S (S) (D.1) d ċ dS = - u c (c) u cc (c) R SS (S) - ψh SS (S) u c (c) (D.2) d Ṡ dc = -1 (D.3) d Ṡ dS = R S (S) (D.4)
We know that for a saddle-stable path system, it is necessary to have one positive and one negative eigenvalue, denoted as µ 1,2 . T r (J) = µ 1 + µ 2 and Det (J) = µ 1 µ 2 . It is sufficient to show that T r (J) > 0 and Det (J) < 0. It is easy to see that T r (J) = ρ > 0. By arranging the terms for the determinant, we can see that the determinant reduces to the multiple steady state condition G (S). We conclude that Det (J) is negative if

G S (S) = R SS (S) -ρ - ψh SS (S) u c (R (S)) + ψu cc (R (S)) (u c (R (S))) 2 < 0 (D.5) det (J) = (ρ -R S (S)) R S (S) - u c (c) u cc (c) R SS (S) - ψh SS (S) u c (c) (D.6) Complex eigenvalues arise if (T r (J)) 2 -4Det (J) < 0. ρ 2 < 4 (ρ -R S (S)) R S (S) - u c (c) u cc (c) R SS (S) - ψh SS (S) u c (c) (D.7)
As Det (J) is shown to be negative for low and high steady states, this prevents these two steady states from having complex eigenvalues. However, the middle steady state might have complex eigenvalues if condition (D.7) above holds.

E Proof of Proposition 2

The aim of this proof is to show mathematically that all steady-state values of environmental quality S take higher values than the unique steady state of the model without uncertainty. This means that the economy always becomes precautionary when exposed to risk relative to an economy without any harmful events. Suppose that condition (A*.6) holds for the existence of three positive real roots. The function G (S)

with risk and the function without risk G W R (S) are as follows:

G (S) = R S (S) -ρ - ψh S (S) u c (c) = 0 (E.1) G W R (S) = R S -ρ (E.2)
With functional forms in section (A*), we have

G (S) = -2g ψh S 3 + 2g ψh S 2 -2gS + (g -ρ) = 0 (E.3) G W R (S) = -2gS + (g -ρ) = 0 (E.4)
Because we use a logistic function for natural resource regeneration, we have S ∈ [0; 1]. We know that

G (0) = G W R (0) = (g -ρ)
. This means that two functions start from the same point. It is easy to see that functions G (S) and G W R (S) never intersect, other than at the point where S equals 0, as G (S) = G W R (S).

Given this information, we can easily show that all three roots of function G (S) take higher values than the unique root of function G W R (S). The additional term -2g ψh S 3 + 2g ψh S 2 in G (S) is always a positive term when S ∈ [0; 1]. This means that G (S) > G W R (S) for S ∈ [0; 1]. This completes the proof.

F Proof of Proposition 3: An economy with adaptation-only policy

The Hamilton-Jacobi-Bellman equation of an economy with only an adaptation policy is as follows

ρV B (S, K A ) = max c,A u (c) -Q 1 (A) + V B S (S, K A ) (R (S) -c) +V B K A (S, K A ) (A -δK A ) -h (S) V B (S, K A ) -ϕ (S, K A ) (F.1)
First order conditions for an internal optimal solution give

u c = V B S (F.2) V B K A = Q 1 A V B S (F.3)
Differentiating (F.1) with respect to S and K A gives

ρV B S = V B SS (R (S) -c) + V B S R S + V K A S (A -δK A ) -h S (S) ψ (K A ) (F.4) where V K A S = ∂ 2 V ∂K A ∂S ρV B K A = V B SK A (R (S) -c) + V B S R S + V K A K A (A -δK A ) -h (S) ψ K A (K A ) -δV B K A (F.5)
The optimal dynamics of consumption and adaptation investment are

ċ = - u c (c) u cc (c) R S (S) -ρ - ψ (K A ) h S (S) u c (c) (F.6) Ȧ = Q 1 A (A) Q 1 AA (A) (ρ + δ) + h (S) ψ K A (K A ) Q A (F.7)
Using A = δK A , at the steady state, we have

Q 1 A (ρ + δ) + h (S) ψ K A (K A ) = 0 (F.8) R S (S) -ρ - ψ (K A ) h S (S) u c (c) = 0 (F.9)
Using equations (F.6) and R (S) = c, we have

G (S) = R S (S) -ρ - ψ (K A ) h S (S) u c (R (S)) (F.10)
Differentiating the equation (F.10) with respect to S yields By equation ( 19), we know that dK A dS < 0. Because the penalty function ψ (K A ) decreases in K A , it is evident that w (S) increases in S.

G S (S) = R SS (S) - h SS (S) ψ (K A ) u c (R (S)) + h S (S) ψ (K A ) u cc (S)) R S (S) (u c (R (S))) 2 - ψ K A (K A ) h S (S) u c (R (S)) dK A dS =Z 1 > 0 > 0 Because the
∆ = -b 1 4d 1 b 2 1 -c 2 1 -18c 1 d 1 -27d 2 1 b 1 -4c 3 1 (I.3)
Then, we have a second degree equation to be solved for the value of b. The discriminant of this second degree equation is ∆ 1 = c 2 -18cd -27d 2 2 -64dc 3 . Equation (I.3) is written as

∆ = -b 1 <0   b 1 -   c 2 1 -18c 1 d 1 -27d 2 1 + (c 2 1 -18c 1 d 1 -27d 2 1 ) 2 + 64d 1 c 3 1 8d 1       b 1 -   c 2 1 -18c 1 d 1 -27d 2 1 -(c 2 1 -18c 1 d 1 -27d 2 1 ) 2 + 64d 1 c 3 1 8d 1     (I.4)
Then, by assuming ∆ 1 > 0, the discriminant ∆ has a positive value if

  c 2 1 -18c 1 d 1 -27d 2 1 -(c 2 1 -18c 1 d 1 -27d 2 1 ) 2 + 64d 1 c 3 1 8d 1   L E < b 1 <   c 2 1 -18c 1 d 1 -27d 2 1 + (c 2 1 -18c 1 d 1 -27d 2 1 ) 2 + 64d 1 c 3 1 8d 1   U E (I.5)
where L E and U E are lower and upper extremes of the inequality (I.5). By replacing the terms a 1 , b 1 , c 1 and d 1 with their corresponding values, condition (I.5) becomes 4g 2 + 36g (g -ρ) -27 (g -ρ) 2 -4g 2 + 36g (g -ρ) -27 (g -ρ)

2 2

-2 9 (g -ρ)

8 (g -ρ) g h < ψ < 4g 2 + 36g (g -ρ) -27 (g -ρ) 2 + 4g 2 + 36g (g -ρ) -27 (g -ρ) By assuming g -ρ > 0, we observe that there is no sign change. This proves all roots are positive for G (S). This completes the proof.

J Proof of Example 2 (only-adaptation)

Equation ( 7) combined with (1) at the steady state can be reformulated as G (S) = -ψh gS 2 + ψh g -2g S + (g -ρ) = 0 (J.1)

It is easy to show that this equation has one positive and one negative real root when g -ρ > 0:

S 1,2 = ψh g -2g ± ψh g -2g 2 + 4 ψh g (g -ρ)

2 ψh g (J.2)

We exclude the negative root because it has no economic meaning. Consequently, the economy has a unique equilibrium without any possibility of multiple equilibria. What happens if this economy starts to invest in adaptation capital? For the sake of analytical tractability, we use a linear hazard and adaptation function. In the numerical analysis, we relax the linearity assumption on functional forms and use more general functional forms to show the robustness of our results.

ψ (K A ) = ψf (K A ) = ψ (1 -aK A ) (J.3) Q (A) = φ A 2 2 (J.4)
where a shows the extent to which the adaptation capital is able to decrease the vulnerability against the inflicted damage ψ and φ stands for a scale parameter for the adaptation cost function. Using equations (F.6), (F.7) and functional forms presented in section (A*), the economy with an adaptation policy ends up with the following G (S) (see Appendix F) G (S) = -hgzS 3 + g + hg z -ψh g S 2 + ψh g -2g -zg S + (g -ρ) = 0 (J.5)

where z = h(a ψ) 2 φδ(ρ+δ) . For analytical tractability, we suppose that ψ = z h . Then, equation (J.5) can be reformulated in a simplified form G (S) = -hgzS 3 + hgzS 2 -2gS + (g -ρ) = 0 This equation is similar to (A*.5) which is shown to have three positive real roots. Following the same method as that used in proof of Proposition 3., the condition of having multiple equilibria in the benchmark economy augmented by the adaptation investments is as follows 30 30 The proof is available upon request. ) is similar to G (S) of the benchmark economy without an environmental policy. The only term that is different in the economy with mitigation policy compared to the benchmark economy is the term c 1 of equation (I.1). The term c 1 is higher in the case with a mitigation policy than in the benchmark economy case.

We look at the effect of a higher value c 1 on the upper and lower extremes of the inequality (I.5). By assuming c 1 < 0 and d 1 > 0, the derivative of the upper and lower extremes are as follows

∂L E ∂c 1 = 1 2 c 2 1 -18c 1 d 1 -27d 2 1 >0 (2c 1 -18d 1 ) <0 (c 2 1 -18c 1 d 1 -27d 2 1 ) 2 1 2 - 1 2      c 2 1 -18c 1 d 1 -27d 2 1 >0 (2c 1 -18d 1 ) <0 + 3.64d 1 c 2 1 (c 2 1 -18c 1 d 1 -27d 2 1 ) 2 + 64d 1 c 3 1 1 2      > 0 ∂U E ∂c 1 = 1 2 c 2 1 -18c 1 d 1 -27d 2 1 >0 (2c 1 -18d 1 ) <0 (c 2 1 -18c 1 d 1 -27d 2 1 ) 2 1 2 + 1 2      c 2 1 -18c 1 d 1 -27d 2 1 >0 (2c 1 -18d 1 ) <0 + 3.64d 1 c 2 1 (c 2 1 -18c 1 d 1 -27d 2 1 ) 2 + 64d 1 c 3 1 1 2      < 0
We observe that the occurrence of multiple equilibria with a mitigation policy is relatively unlikely because there is a smaller range of values for damage ψ that cause multiple equilibria.

  , chapter 1) and Jie-Sheng et al. (2014), Bradshaw et al. (2007)) 7 .

Figure 1 :Figure 2 :

 12 Figure 1: Phase diagram on the plane (S, λ): unique growth path

Figure 3 :

 3 Figure 3: Phase diagram on the plane (S, λ): multiple growth paths

Figure 4 :

 4 Figure 4: Skiba point with trajectories converging to saddle-point equilibria 18 Figure 3 does not stem from a numerical exercice. It only aims to show clearly the optimal trajectories and to understanding which trajectory is a global maximum.

Figure 4

 4 Figure419 is line with the work of[START_REF] Wagener | Skiba points and heteroclinic bifurcations, with applications to the shallow lake system[END_REF] in which it is shown that a Skiba point may not be associated with an unstable equilibrium. Departing from the initial condition S 0 = 0.69464, the value of the

Figure 5 :

 5 Figure 5: Different reaction against the endogenous hazard rate

ω2

  Natural Regeneration Function : R (S) = gS 1 -S S Utility function : u (c) = c 1-σ 1-σ Source : Ren and Polasky (2014) Source : Bommier et al. (2015) S Carrying capacity of environment g Intrinsic growth rate of the resource stock σ Degree of relative risk aversion harmful Damage : ψ (KA) = ψ ω + (1 -ω) e -γK A Endogenous Hazard Function : h(S) = Lower bound of damage when ψ (∞) η Endogeneity level of harmful event γ Elasticity of adaptation w.r.t to damage rate S Carrying capacity of environment Mitigation function : Γ (M ) = M α Cost of adaptation investment : Q1 (A) = φA A 2 Source : Le Kama and Pommeret (2016) α Elasticity of mitigation activity φA Parameter for the change of marginal cost of adaptation

Figure 6 :

 6 Figure 6: The benchmark economy vs the economy with mitigation

Figure 7 :

 7 Figure 7: The benchmark economy vs the economy with adaptation

Figure 8 :

 8 Figure 8: Adaptation vs mitigation trade-off and multiple equilibria

Figure 9 :

 9 Figure 9: Adaptation vs mitigation trade-off and unique equilibrium

Figure 10 :

 10 Figure 10: G (S) function with uncertainty

  dynamical system is four dimensional, the original condition on G (S) is not sufficient to prove the multiplicity of equilibria. Therefore, we give additional conditions and write the function G (S) = 0 given in equation (F.10) in a slighlty different form (R S (S) -ρ) u c (R (S)) h S (S)

Figure 11 :

 11 Figure 11: Multiplicity of equilibria with adaptation capital

  positive discriminant ∆ proves only the existence of real roots and does not indicate whether these roots are positive or not. For this purpose, we use the Descartes rule. To see if the equation G (S) = -2g ψh S 3 + 2g ψh S 2 -2gS + (g -ρ) = 0 has three positive real roots, we write G (-S) G (-S) = -2g ψh (-S) 3 + 2g ψh (-S) 2 -2g (-S) + (g -ρ) = 0 (I.7) This yields G (-S) = 2g ψh S 3 + 2g ψh S 2 + 2gS + (g -ρ) = 0 (I.8) 32

  At the steady state, we haveG (S) = R S (S) -ρ -ψh S (S) u c (R (S) + Γ (M ) -Q 2 (M )) = 0Because we use linear specifications for analytical tractability, we have the optimum level of mitigation investment as M * = 1 P M . Then, we can reformulate the function G (S) asG (S) = -2g ψh S 3 + 2g ψh S 2 + ψh P M -2gS + (g -ρ) = 0 (K.8)where a 1 = -2g ψh , b 1 = 2g ψh , c 1 = ψh P M -2gS and d 1 = g -ρ. Equation (K.8

  Tounderstand the implications of mitigation on the economy with multiple equilibria, we first calibrate the benchmark model with multiple equilibria to match the features of New Zealand's economy. We calibrate the damage parameter ψ and the intrinsic growth rate of the resource stock g such that the high steady-state level of environmental quality S matches the environmental performance index (EPI) score of New Zealand in 2018. We calibrate the important parameters of the endogenous hazard rate (the endogeneity of the hazard η, and the upper bound for hazard rate h) according to the estimates of New Zealand's government,

				Parameter Calibrated value
				S		1.01
				g		0.062
				ψ		1315
				h		0.25
				η		7.8
				α		0.5
				P M		250
			The economy without policy
	G S
	0.004
	0.002
		0.5	0.6	0.7	0.8	S
	0.002	S low	S mid	S high
	0.004

We have the same dynamics as in a classical Ramsey-Cass-Koopmans model, where the economy admits a unique equilibrium.

 12 This lemma is related to the analysis of the multiple equilibria with a unique growth path. This concerns the local stability of steady-

states.13 If the steady state curve ċ = 0 is not non linear but vertical, there is always a unique

equilibrium 14 Of course, there can be different kinds of phase diagrams. However, the analyses for other phase diagram configurations are not different than what we present in this section.

This is particularly important for the second phase diagram that we present in this study.

Note that Figure1does not stem from a numerical exercise and stands for representative purposes in order to show the optimal dynamics on a single graphic.

The parameters used for the simulation is as follows; σ = 2.1, ρ = 0.015, g = 0.09, S = 1, η = 12 h = 0.5, ψ = 715. The functional forms for the simulation of the phase diagram are given in section 4.

The Skiba point is the point where both value functions of optimal trajectories leading to low and to high steady-state are equal. Then, the economy is indifferent toward converging to either of these steady states (see[START_REF] Haunschmied | A DNS-curve in a two state capital accumulation Model: a Numerical Analysis[END_REF]). The numerical calculation with the calibration given in footnote 19 yields the Skiba point S 0 = 0.69464.

Because the endogenous hazard rate increases, the marginal value of adaptation capital increases.

which show that the probability of a flood occurring in any 30-year period is 45%. For the elasticity of substitution, we choose the conventional value used in the literature, which is 2 and the pure rate of time preference ρ is set to 0.015 as in[START_REF] Nordhaus | A Question of Balance: Weighting the Options on Global Warming Policies[END_REF]. We assume that the elasticity of mitigation and the unit price for mitigation activity are equal to 0.5 and to

250, respectively.22 The EPI index gives a score for each country between 0 and 100. Because we use a logistic growth function with carrying capacity set to 1 in our calibration, we have a value of S between 0 and 1. Hence, if a country has a score of 60, we take it as 0.60. The EPI score of New Zealand is 0.75 in 2018.23 We refer to EMDAT International Disaster Database (2015). 24 https://www.ecan.govt.nz/your-region/your-environment/natural-hazards/floods/flood-probabilities/

Note that the numerical example is in line with the phase diagram presented in section 2.1.1.

Similarly, the convergence to S low can be shown.

We assume that the quantities under the square root are positive. This is also confirmed numerically with plausible parameters.

[START_REF] Wirl | Sustainable growth, renewable resources and pollution: Thresholds and cycles[END_REF] explains that the economy faces multiple equilibria when it becomes difficult to steer between an economic goal (accumulating physical capital) and environmental goal (protecting amenities).

In the numerical analysis, we use a penalty function which is in line with the definition given in A.4.

This research is supported by the ANR project GREEN-Econ (Grant ANR-16-CE03-0005) and by MENRT/Ecole Doctorale ED465 Panthéon-Sorbonne. environmental quality in the paper, mitigation activities are intended to increase environmental quality. A possible definition of mitigation activity can be reforestation, which enhances carbon sinks. (IPCC (2007))

Appendix 1 A* Examples with functional forms

This section aims to present examples for the existence of the multiplicity of equilibria which is an issue related to functional forms and to parameter values. The primary purpose of examples is to prove the existence of multiple equilibria rather than giving necessary conditions to ensure its existence.

Example 1: An economy without environmental policy

The regeneration of the environment, the endogenous hazard rate, and the utility function are given as follows: R (S) = g (1 -S) S (A*.1)

where g is the intrinsic growth rate of the environmental quality and 0 < h < 1 shows the extent to which the endogenous hazard rate depends on the environmental quality level. At the steady state, equation ( 7) can be written as

u c (c) = 0 (A*.4) By using the functional specifications, we reformulate equation (A*.4) combined with (1) at the steady state -2g ψh S 3 + 2g ψh S 2 -2gS + (g -ρ) = 0 (A*.5)

The sufficient condition to have three positive real roots for the third-degree polynomial (A*.5) is as

Proof. See Appendix (I)

When the damage rate ψ is too high, the economy adopts precautionary behavior because the marginal benefit of protecting the environment is high. On the other hand, when the damage rate is too low, the

Example 3: An economy with only mitigation policy

To prove that mitigation can rid the economy of multiple equilibria, we start with an example of a benchmark economy with multiple equilibria and use the following usual functional specifications. In order to obtain analytical results, we use the following linear functional forms:

Another simplification is used only for this example in order to ease the calculations. This simplification is that the cost function appears in the dynamic equation of environmental quality Ṡ.

In this example, the optimization problem consists of maximizing the following objective function

with respect to (A*.18).

For the sake of analytical tractability, in the example, we use a linear mitigation function and convex cost function for mitigation. We relax these simplifications in the numerical analysis and adopt the specifications in subsection 3.2 to show that our result is robust, with realistic functional forms. With the presence of mitigation activity, the function G (S) is as follows (see Appendix (K))

Once we have equation (A*.20), we analyze the region of parameter space which defines the values for a parameter (here the damage parameter ψ) within which the multiplicity of equilibria occurs. In Appendix K, we show that the region of parameter where the multiplicity of equilibria appears diminishes when the economoy invests in mitigation activity.

Proof. See Appendix (K)

When the economy implements a mitigation policy, there are few parameter combinations, which causes multiple equilibria.

Appendix 2

A Derivation of (7).

To solve the maximization problem, we write the Hamilton-Jacobi-Bellman equation.

where V B (S) is the value of the maximization program before the event. As also stated in the text, the value of the problem after the event is as follows:

The economy is exposed to inflicted damage after the event. The first-order condition is given by

Computing the derivative of (A.1) with respect to the environmental quality stock S yields

Differentiating equation (A.3) and using equations (A.3) and (A.4) gives

Arranging equations (A.5) and (A.6) gives the Keynes-Ramsey rule

The associated Hamiltonian of the problem is

where λ and µ are the shadow price for natural resources and for the endogenous hazard rate respectively.

Because µ = V (S), it is easy to see that

One can easily understand from optimal control and dynamic programming that we have λ = V B S (S) = u c (c). Then, by using equation (A.6), the dynamics of co-state can be derived λ = -λ R S (S) -ρ -ψh S (S) λ (A.9)

A.1 Derivation of the objective function

We can obtain equation ( 6) by using the Hamilton-Jacobi-Bellman equation. One can reformulate (A.1) as follows

Multiplying both sides of (A.10) by e

(ρ+h((S(τ ))))dτ , we have

Taking the integral of both sides of (A.11) between 0 and ∞ yields

(ρ+h((S(τ ))))dτ (A.12)

B Proof of Proposition 1

The proof starts with an analysis of the limits of a function (let this function be G (S)) that describes the steady state of the economy by a single equation in the long run. Then, in the second part of the proof, we focus on the form of G (S) and related necessary conditions for the existence of multiple equilibria.

(a) In a sense, the function G (S) can be considered equation ċ = 0 in terms of S at the steady state equilibrium. Writing equations ċ = 0 and Ṡ = 0 ;

Note that we limit our analysis between S ∈ 0, S . It can be noticed that the function G (S) starts with a positive value and has negative values when S approaches S. In this framework, S is the environmental quality level when the consumption level is equal to zero. With this information, it is easy to verify lim S→0 G (S) = ∞ and lim S→ SG (S) = z < 0.

(b) In this part of the proof, we show the necessary conditions for the existence of multiple equilibria, which also allows us to represent the function G (S) ;

We remark that lim S→0 n (S) = -∞ and lim S→ Sn (S) = +∞. A sufficient condition for a low steady state is ∃S < S such that n S (S) < 0 with n (S) > w (S). Note that S(< S) is between the low and high steady state. A sufficient condition for high steady state is ∃S > S such that n (S) < w (S).

G Proof of Proposition 4: An economy with mitigation-only policy

We write the Hamilton-Jacobi-Bellman equation for the economy with a mitigation activity.

where V B (S) is the value of the maximization program before the event. As also stated in the text, the value of the problem after the event is as follows:

The economy is exposed to inflicted damage after the event. The first-order conditions are given by

Calculating the derivative of (G.1) with respect to the environmental quality stock S yields

Differentiating equation (G.3) and using equations (G.3) and (G.5) gives

Arranging equations (G.6) and (G.7) gives the Keynes-Ramsey rule

At the steady-state, we have

Note that the optimal steady state level of mitigation M depends on S. Differentiating the equation (G.8) with respect to S yields

H The economy with adaptation and mitigation

We solve the model with both adaptation and mitigation policies. The social planner maximizes the following program

The optimal policy is to maximize (H.1) subject to (H.2) and (H.3).

We write the Hamilton-Jacobi-Bellman equation for the economy implementing both adaptation and mitigation policies.

where V B (S) is the value of the maximization program before the event. As also stated in the text, the value of the problem after the event is as follows:

The economy is exposed to inflicted damage after the event. The first-order conditions are given by

Computing the derivative of (H.4) with respect to the environmental quality stock S and adaptation

Differentiating equation (H.6) and using equations (H.6) and (H.9) gives

Optimal dynamics for consumption, adaptation and mitigation are as follows

I Proof of Example 1

The third degree polynomial equation has the following form

With the functional forms given in the text, we remark that terms b 1 = -a 1 with a 1 = -2g ψh , c 1 = -2g

and d 1 = g -ρ . This simplifies the proof of the existence for three positive real roots. The discriminant of the cubic equation is as follows:

• ∆ > 0, equation (I.1) has three distinct real roots.

• ∆ = 0, equation (I.1) has multiple roots and all three roots are real.

• ∆ < 0, equation (I.1) has one real root and two non-real, complex roots.

Because b = -a, we can reformulate the discriminant (I.2) in the following way

K Proof of Example 3 (only-mitigation)

The optimization problem holds only for the Example 3 and not for the subsection 3.2, since we put the cost function of the mitigation activity in the resource constraint in the example for the sake of analytical tractability. We write the Hamilton-Jacobi-Bellman equation


Hanane Yousfi

Stéphane Ranque

Carole Cassagne

Jean-Marc Rolain

Fadi Bittar
email: fadi.bittar@univ-amu.fr

Identification of repositionable drugs with novel antimycotic activity by screening Prestwick Chemical Library against emerging invasive molds

Keywords: Drugs repurposing, Prestwick chemical library, emerging fungi, multidrugresistant molds, alternative therapeutics

The incidence of severe filamentous fungi infections has increased over the past decade. Some of these filamentous fungi are resistant to available antifungals; it is thus urgent to find new compounds that are active against such life-threatening pathogens. Here, 1280 drugs (Prestwick Chemical Library) were tested against six multidrug-resistant filamentous fungi including, Aspergillus, Fusarium, Scedosporium, Rhizopus and Lichtheimia species. We identified several hits that induce fungal growth inhibition ≥ 70%. Clioquinol, alexidine dihydrochloride, hexachlorophene and thonzonium bromide, displayed a broad activity against all strains tested. This study enriches the potential antifungal options that can be used against multidrug-resistant invasive fungal diseases.

Introduction

While more than 400 fungal species have been implicated as opportunistic pathogens in human infection [START_REF] Pauw | What are fungal infections?[END_REF], only 4 classes of systemic antifungal agents, namely polyenes, azoles, echinocandins and flucytosine, are mainly used in clinical practices [START_REF] Scorzoni | Antifungal therapy: New advances in the understanding and treatment of mycosis[END_REF]. Although these antifungal drugs are usually active against the major clinical fungi, there are limitations to their routine use, including off-target toxicity and drug-drug interaction. Moreover, a poor clinical response has been reported, especially against emerging resistant fungal isolates [START_REF] Nicola | Antifungal drugs: New insights in research & development[END_REF].

Antifungal resistance is a major concern that has been described in most clinical fungal genera. The major human fungal pathogens, including Candida spp., Cryptococcus spp. and Aspergillus spp. were commonly sensitive to antifungal therapy. However, nowadays, they display a relatively high-acquired resistance rate and constitute a serious public health concern. Additionally, rare life-threatening and emerging fungal pathogens, including opportunistic multidrug-resistant (MDR) pathogens, are increasingly being reported [START_REF] Enoch | Invasive fungal infections: A review of epidemiology and management options[END_REF]. These latter include Scedosporium/Lomentospora spp., Fusarium spp. and other Mucorales [START_REF] Geddes-Mcalister | New pathogens, new tricks: Emerging, drug-resistant fungal pathogens and future prospects for antifungal therapeutics[END_REF]. Thus, this study aimed at determining available off-label drugs with potential antifungal activity against emerging multidrug-resistant molds. This promising strategy, which is commonly referred to as a "repurposing approach", is particularly useful to elude drugresistance expansion and prevent treatment failure [START_REF] Corsello | HHS Public Access[END_REF].

Testing of Prestwick Chemical Library against MDR-filamentous fungi

We screened the Prestwick Chemical Library (Prestwick, Illkirch graffenstaden, France), which is a molecule-library containing 1,280 compounds, belonging to 291 different therapeutic classes, which mostly have been approved by the Food and Drug Administration (FDA), to identify molecules that inhibit the growth of a panel of multidrug-resistant fungi.

The chemical library was initially provided in DMSO at a concentration of 10 mM for each drug. Here, all molecules were tested, following serial dilutions, at a fixed concentration of 10 µM (the final DMSO concentration was ≤ 0.5%). This low and fixed concentration (i.e. 10 µM) was used according to previous screening studies in order to find the drugs which display antifungal activities at low concentrations and to avoid the toxicity and the adverse effect that these tested drugs may induce at high concentrations. The fungal inoculum was prepared in High AMB MICs (> 32 mg/l) were recorded for all tested clinical strains, especially for Rhizopus oryzae, Lomentospora prolificans and Fusarium solani, while MICs of azoles were variable but remained higher than 32 mg/l for Fusarium, Lomentospora and Rhizopus species [START_REF] Yousfi | In vitro polymyxin activity against clinical multidrug-resistant fungi[END_REF]. Based on the primary drugs-screening, we identified 3 compounds that were active against R. oryzae; 11 that were active against F. oxysporum, 12 that were active against F. solani, 15 that were active against L. prolificans and A. niger ATCC, 13 that were active against A. calidoustus and, finally, 14 that were active against both A. flavus ATCC and L. corymbifera (Table 1). All retained compounds induced ≥ 70% fungal growth inhibition and were classified, in Table 1, according to the annotated database provided along with the Prestwick chemical library. The identified hits belong to six different therapeutic classes, including antifungals, antibacterials, antiseptics, anthelmintics, antineoplastics and other miscellaneous drugs (Table 1). Not surprisingly, most of the hits were among antifungals ( 47%). Nevertheless, there were some discrepancies between the E-test MIC results obtained for AMB and VRC and the fungal growth inhibition mesured for these compounds against Aspergillus and Fusarium spp. strains. It is worth mentioning that the study by [START_REF] Lamoth | Comparing Etest and Broth Microdilution for Antifungal Susceptibility Testing of the Most-Relevant Pathogenic Molds[END_REF] showed that VRC MICs for Aspergillus spp. tend to be lower with the E-test method as compared to the broth micodilution method [START_REF] Lamoth | Comparing Etest and Broth Microdilution for Antifungal Susceptibility Testing of the Most-Relevant Pathogenic Molds[END_REF]. This may explain the absence of 70% of fungal growth inhibition of Aspergillus spp. even with concentrations higher than VRC E-test MICs. Furthermore, the aforementioned study considered that a variation, within a range of ± 2 dilutions, of AMB and triazole MICs between E-test and broth microdilution methods against Aspergillus and non-Aspergillus strains is acceptable. Among the non-antifungal compounds that were effective against the clinical molds tested, four, namely clioquinol, alexidine dihydrochloride, hexachlorophene, and thonzonium bromide, displayed a broad antifungal spectrum against at least five of the six tested strains (Figure 1). Although these compounds are mainly used as antiseptics, which may limit their systemic uses, our results indicate how difficult it is to identify a drug with a broad spectrum of activity against fungal pathogens.

Clioquinol (5-chloro-7-iodo-8-quinolinol) was initially used as a topical antiseptic and for the treatment of intestinal amoebiasis [START_REF] Bareggi | Clioquinol: Review of its mechanisms of action and clinical uses in neurodegenerative Disorders[END_REF]. In the early 1970s, the systemic administration route of clioquinol was restricted or discontinued because it had been associated with subacute myelo-optic neuropathy in Japanese patients [START_REF] Bareggi | Clioquinol: Review of its mechanisms of action and clinical uses in neurodegenerative Disorders[END_REF]. Currently, topical clioquinol administration is used to treat skin infections [START_REF] Bareggi | Clioquinol: Review of its mechanisms of action and clinical uses in neurodegenerative Disorders[END_REF]. Recently, You et al reported a significant activity of 3% clioquinol cream against a large number of fungal species [START_REF] You | Clioquinol, an alternative antimicrobial agent against common pathogenic microbe[END_REF]. Using modified agar diffusion assay, the authors showed that the inhibition zone was the largest for Candida tropicalis, Candida guilliermondii, Aspergillus terreus and F. solani [START_REF] You | Clioquinol, an alternative antimicrobial agent against common pathogenic microbe[END_REF]. In our study, clioquinol activity testing was performed using the microdilution broth method and showed significant growth inhibition of all tested mold strains except R. oryzae.

Alexidine dihydrochloride (AXD), a bis-guanide molecule, initially identified for its antibacterial properties, also has anti-inflammatory and anti-cancer effects [START_REF] Zhu | Alexidine Dihydrochloride Attenuates Osteoclast Formation and Bone Resorption and Protects Against LPS-Induced Osteolysis[END_REF]. In this study, AXD exerted a broad antifungal activity against A. calidoutus (81%), F. solani (82%), F. oxysporum (81%), L. prolificans (77%) and L. corymbifera (77%). Recently, in accordance with our results, Mamouel et al, reported an antifungal activity of AXD against a wide range of MDR-fungi, including the yeast C. albicans, Candida auris, Cryptococcus neoformans and the filamentous fungi Scedosporium apiospermum, Aspergillus fumigatus, L. corymbifera and also R. oryzae. Moreover, they reported a synergistic interaction between AXD and fluconazole against fluconazole-resistant C. albicans [START_REF] Mamouei | Alexidine Dihydrochloride Has Broad-Spectrum Activities against Diverse Fungal Pathogens[END_REF]. Intriguingly, we found no effect of 10 µM (i.e. 5.8 mg/l) AXD against R. oryzae (Table 1), whereas the aforementioned study found AXD MIC80 = 3 mg/l against R. oryzae strain. This discrepancy may be due to the multi-drug resistance profile (including posaconazole MIC >32 mg/l) of the R. oryzae strain used in our study, which contrasts with the one used in Mamouel et al's study that displayed a posaconazole MIC = 0.25 mg/l. To our knowledge, the activity of AXD against Fusarium species and L. prolificans has not been previously reported. Hexachlorophene (HCP), a chlorinated bisphenol, has significant bacteriostatic properties against several Gram-positive bacteria (including Staphylococcus) and is used as antiseptic in dermatological topical preparations [START_REF] Leśniak | Effects of Hexachlorophene, a Chemical Accumulating in Adipose Tissue, on Mouse and Human Mesenchymal Stem Cells[END_REF]. HCP showed significant fungal growth inhibition against all the tested strains, except A. calidoustus and A. niger (Table 1). The antifungal activity of HCP against difficult-to-treat fungal strains is reported here for the first time to our knowledge. Importantly, the clinical use of HCP has been restricted because studies on mice and human stem cells have demonstrated its chronic deleterious effect and oral or vaginal administration of high HCP doses has been shown to be embryotoxic and teratogenic in rats [START_REF] Leśniak | Effects of Hexachlorophene, a Chemical Accumulating in Adipose Tissue, on Mouse and Human Mesenchymal Stem Cells[END_REF].

Thonzonium bromide (TB), a monocationic detergent, helps the penetration of active ingredients (antibacterial and anti-inflammatory drugs) through topical corticosporin-TC ear drops. Its antifungal activity against planktonic C. albicans strain has been highlighted in previous Prestwick chemical library screening [START_REF] Mamouei | Alexidine Dihydrochloride Has Broad-Spectrum Activities against Diverse Fungal Pathogens[END_REF]. It has been identified by Chan et al as a specific C. albicans ATPase inhibitor [START_REF] Chan | Inhibitors of V-ATPase proton transport reveal uncoupling functions of tether linking cytosolic and membrane domains of V 0 subunit a (Vph1p)[END_REF]. The significant activity of TB against all difficultto-treat mold species tested in our study has not been described; it prompts for further study of its antifungal activity.

Conclusion

Despite the current advances in the diagnosis and prevention of invasive fungal infections, the incidence of mycosis, treatment failure and mortality remain excessively high in immunocompromised patients. This study results point out several off-label drugs that may be considered in the treatment of difficult-to-treat fungal infections. Further investigations of the antifungal properties of the compounds identified in this study, as monotherapy or in combination with other antifungal drugs, will enhance the management of MDR mold infections.

  RPMI-1640 medium (Sigma Aldrich, St Louis, France) according to the Clinical and Laboratory Standards Institute (CLSI) protocol (M38-A, Vol. 22 No. 16). The fungal growth inhibition rate was calculated derived from Optical Density (OD) values measured with the plate reader spectrophotometer (Multiskan spectrum, Thermo Scientific, France) at a wavelength = 405 nm compared to the same untreated strain. The tested strains' incubation time with drugs varied from 48 to 72 hours according to the optimal fungal growth time observed in positive control wells. The six fungal strains that were isolated from various clinical samples were tested twice; these include; Aspergillus calidoustus (bronchial aspiration), Fusarium oxysporum (nails), Fusarium solani (eye), Rhizopus oryzae (sinus biopsy), Lomentospora prolificans (blood) and Lichtheimia corymbifera (eye). All strains were recovered from La Timone University Hospital in Marseille and were in vitro resistant to several available systemic antifungal agents. Aspergillus flavus ATCC 204304 and Aspergillus niger ATCC 200930 were used as internal quality controls. The minimum inhibitory concentration (MIC) of posaconazole (POS), itraconazole (ITR), voriconazole (VRC), isavuconazole (ISA), and amphotericin B (AMB) were determined using the E-test method for each strain.

fungi tested. (%): the proportion of fungal growth inhibition. (-): no or < 70% activity.

Acknowledgments

We thank CookieTrad for English reviewing.

Ethical Approval

Not required

Funding information

This work was supported by the French Government under the «Investissements d'avenir » (Investments for the Future) program managed by the Agence Nationale de la Recherche (ANR, fr: National Agency for Research), (reference: Méditerranée Infection 10-IAHU-03) This work was supported by Région Provence-Alpes-Côte d'Azur and European funding FEDER (fonds européen de développement régional) PRIMMI ((Plateformes de Recherche et d'Innovation Mutualisées Méditerranée Infection).

Transparency declarations

These is no conflict of interest or financial disclosure to declare for all authors.

Compound name

Aspergillus flavus

Aspergillus niger

Aspergillus calidoustus

Fusarium solani

Fusarium oxysporum

Lomentospora prolificans

Rhizopus oryzae

Lichtheimia corymbifera

Antifungals