Piermarco Cannarsa
email: cannarsa@mat.uniroma2.it

Helene Frankowska
email: helene.frankowska@imj-prg.fr

Infinite Horizon Optimal Control: Transversality Conditions and Sensitivity Relations

We investigate the infinite horizon problem in optimal control with a general-not necessarily discountedrunning cost. The dynamic programming approach associates to it the value function V that may be discontinuous and may take infinite values. In general, the first order necessary optimality conditions have the form of a maximum principle, that may be abnormal and does not include any transversality condition. We show that a normal maximum principle and a sensitivity relation involving the Fréchet subdifferential are valid for a dense subset of initial conditions. In particular, a transversality condition holds true on this subset. Furthermore, if V is continuous, then for every initial state a, possibly abnormal, maximum principle can be written with a transversality condition at the initial state involving a limiting (in the normal case) or horizontal (in the abnormal case) superdifferentials of V . Finally, when V is locally Lipschitz with respect to the state variable, we prove a normal maximum principle together with sensitivity relations and a transversality condition involving generalized gradients of V . These relations simplify the investigation of the limiting behaviour of the costate.

I. INTRODUCTION

Infinite horizon optimization attracted attention of researchers in diverse fields including electrical engineering, economics/finance and operations research/management science. In economics, the first appearance of model of maximizing levels of consumption over successive generations is due Ramsey [START_REF] Ramsey | A mathematical theory of saving[END_REF]. The classical infinite horizon problem is as follows : for some λ > 0

W (x 0 ) = inf ∞ 0 e -λt (x(t), u(t)) dt, (1)
where e -λt is a discount factor, (x, u) are trajectory-control pairs of the following control system

x (t) = f (x(t), u(t)), u(t) ∈ U for a.e. t ≥ 0 x(0) = x 0 (2)
and controls u(•) are Lebesgue measurable. Despite a very good understanding of necessary optimality conditions for finite horizon problems, this topic is still challenging in the infinite horizon context, due to the fact that on one hand the "natural" transversality conditions do not hold in many situations, on the other hand, because, even in the absence of constraints, the maximum principle may be abnormal. We refer to [START_REF] Aseev | The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons[END_REF] for an extended overview of the literature devoted to transversality conditions and normality of the maximum principle for the infinite horizon problem, examples and counterexamples, and for important bibliographical comments and also to [START_REF] Aseev | Maximum principle for infinite-horizon optimal control problems with dominating discount[END_REF], [START_REF] Aseev | Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions[END_REF], [START_REF] Pickenhain | On adequate transversality conditions for infinite horizon optimal control problems -a famous example of Halkin[END_REF] for a further discussion.

Given an optimal trajectory-control pair (x, ū) and T > t 0 , consider the finite horizon Bolza problem minimize T 0 e -λt (x(t), u(t)) dt [START_REF] Aseev | Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions[END_REF] over all trajectory-control pairs (x, u) of the system x (t) = f (x(t), u(t)), u(t) ∈ U for a.e. t ∈ [0, T ] x(0) = x 0 .

Then the necessary optimality condition takes the form of the maximum principle: if (x, ū) restricted to [0, T ] is an optimal trajectory-control pair for the above Bolza problem, then the solution p T := p of the adjoint system -p (t) = D x f (x(t), ū(t)) * p(t) -e -λt x (x(t), ū(t)) [START_REF] Aubin | Shadow prices and duality for a class of optimal control problems[END_REF] with p(T ) = 0, satisfies on [0, T ] the maximality condition p(t), f (x(t), ū(t)) -e -λt (x(t), ū(t)) = max u∈U ( p(t), f (x(t), u) -e -λt (x(t), u)) a.e.

(

) 5
The transversality condition p(T ) = 0 is due the fact that there is no cost term depending on x(T ) in [START_REF] Aseev | Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions[END_REF].

If for any i ≥ 1, the restriction of (x, ū) to [0, i], is optimal for the Bolza problem with T = i, then taking a converging subsequence of {p i } one gets a solution of (4) satisfying the maximality condition (5) a.e. in [0, ∞[. Though the transversality condition does disappear in this approach, some additional assumption on f, , λ alow to conclude that lim t→+∞ p(t) = 0. In a way, zero may appear to be a natural candidate for the transversality condition at infinity. However, in general, the restriction of (x, ū) to the time interval [0, T ] is not optimal. A remedy to this difficulty could be including the constraint x(T ) = x(T ). This results, however, in possibly abnormal maximum principles for finite horizon problems, and, in fine, leads to necessary optimality conditions without the cost function in (4), [START_REF] Aubin | SET-VALUED ANALYSIS[END_REF]. Also in this approach the transversality condition at time T does disappear, becoming -p(T

) ∈ N {x(T )} (x(T )) = R n .
Many authors contributed to proofs of normal maximum principles for the infinite horizon problem involving some transversality conditions, see [START_REF] Halkin | Necessary conditions for optimal control problems with infinite horizons[END_REF], [START_REF] Michel | On the transversality condition in infinite horizon optimal problems[END_REF], [START_REF] Carlson | THEORY AND APPLICATIONS[END_REF], [START_REF] Ye | Nonsmooth maximum principle for infinite-horizon problems[END_REF], [START_REF] Sagara | Value functions and transversality conditions for infinitehorizon optimal control problems[END_REF], [START_REF] Aseev | The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons[END_REF]- [START_REF] Aseev | Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions[END_REF] and the bibliographies contained therein for variational approaches and [START_REF] Aubin | Shadow prices and duality for a class of optimal control problems[END_REF], [START_REF] Pickenhain | On adequate transversality conditions for infinite horizon optimal control problems -a famous example of Halkin[END_REF], [START_REF] Tauchnitz | The Pontryagin maximum principle for nonlinear optimal control problems with infinite horizon[END_REF] which use duality theory for weighted Sobolev spaces L p (0, +∞; R n ) with respect to the measure e -λt dt (or more general measures).

Recently, because of some economic applications, also the nonautonomous infinite horizon problem without the discount factor attracted attention of several researchers. In such case, W may become discontinuous and may take infinite values. When variational methods are used, some authors modify the notion of optimal solution or, alternatively, introduce new variables and cost to obtain relevant necessary optimality conditions for the finite horizon problems and then, by passing to a limit, to get a maximum principle for the infinite horizon problem.

Let us underline that in the finite horizon setting, the transversality condition at final time is one of ingredients of the maximum principle. For the infinite horizon problem the transversality condition at infinity follows from assumptions on data and the adjoint equation: one first gets a solution of the adjoint equation satisfying the maximality condition and then investigates how this solution behaves at infinity. For this reason, it seems being more relevant to impose transversality conditions at the initial state, as it was done in [START_REF] Aubin | Shadow prices and duality for a class of optimal control problems[END_REF], where a discounted problem with linear dynamics was studied using the abstract Green formula.

We consider the nonautonomous optimal control problem

V (t 0 , x 0 ) = inf ∞ t0 L(t, x(t), u(t)) dt (6)
over all trajectory-control pairs (x, u), subject to the system

x (t) = f (t, x(t), u(t)), u(t) ∈ U (t) a.e. x(t 0 ) = x 0 , (7)
where

R + = [0, +∞[, t 0 ∈ R + , x 0 ∈ R n , L : R + × R n × R m → R + , f : R + × R n × R m → R n are given mappings, U (•)
is a set-valued map with nonempty images in R m . Selections u(t) ∈ U (t) are supposed to be measurable. The above setting subsumes the classical infinite horizon optimal control problem when f and U are time independent, L(t, x, u) = e -λt (x, u) for some mapping : R n × R m → R + and t 0 = 0. Furthermore, in this case, for all t ≥ 0, V (t, x 0 ) = e -λt W (x 0 ).

To derive necessary optimality conditions, we rely solely on the dynamic programming principle and do not modify the notion of solution. Unlike the approach in [START_REF] Michel | On the transversality condition in infinite horizon optimal problems[END_REF], [START_REF] Sagara | Value functions and transversality conditions for infinitehorizon optimal control problems[END_REF], [START_REF] Ye | Nonsmooth maximum principle for infinite-horizon problems[END_REF], where dynamic programming was used in the presence of a discount factor, we neither introduce new variables nor modify the cost function. Instead, for a lower semicontinuous value function and for a dense set of initial states we get a normal maximum principle by exploiting the "forward sensitivity relations" derived in [START_REF] Cannarsa | From pointwise to local regularity for solutions of Hamilton-Jacobi equations[END_REF], [START_REF] Cannarsa | Local regularity of the value function in optimal control[END_REF]. If V (t 0 , •) is continuous, then for every x 0 ∈ R n we obtain a, possibly abnormal, maximum principle with a transversality condition involving the limiting or horizontal limiting superdifferentials of V (t 0 , •) at x 0 . Our results imply, in particular, that if for an initial condition x 0 the set of limiting horizontal supergradients of V (t 0 , •) at x 0 is empty, then the derived maximum principle is normal. If the value function is locally Lipschitz, then the maximum principle is normal and the sensitivity relations can also be deduced from [START_REF] Bettiol | Improved sensitivity relations in state constrained optimal control[END_REF] to investigate the behavior of the costate and the Hamiltonian (along the trajectory-costate pair) at infinity.

We would like to underline that introducing the value function into necessary optimality conditions is an additional property of the costate, usually absent in the formulation of maximum principles. Adding such sensitivity relations may even give necessary and sufficient optimality conditions for the finite horizon problems, cf. [START_REF] Cannarsa | Some characterizations of optimal trajectories in control theory[END_REF]. On the other hand, as we show below, they also allow to deduce the behavior of the costate at infinity in a quite straightforward way. For this reason our approach simplifies the earlier investigations of the infinite horizon maximum principles. At the same time it does not apply to local minimizers. The previous works, [START_REF] Aubin | Shadow prices and duality for a class of optimal control problems[END_REF], [START_REF] Sagara | Value functions and transversality conditions for infinitehorizon optimal control problems[END_REF], [START_REF] Ye | Nonsmooth maximum principle for infinite-horizon problems[END_REF], involving the value function in transversality conditions have addressed problems with a discount factor and locally Lipschitz value functions, while we are able to state some results also when W is merely lower semicontinuous or continuous and there is no discount factor.

The outline of the paper is as follows. Section II concerns some preliminaries and notations. In Section III we provide sufficient conditions for V to be lower semicontinuous, continuous or locally Lipschitz and give a relaxation theorem. Section IV discusses maximum principles and sensitivity relations. In subsection A we state a normal maximum principle and a sensitivity relation involving the Fréchet subdifferential of the lower semicontinuous value function. Subsection B deals with continuous (around x 0 ) mapping V (t 0 , •), where we obtain a, possibly abnormal, maximum principle with a transversality condition at x 0 . Finally Subsection C is devoted to maximum principles, sensitivity relations and transversality conditions at infinity for locally Lipschitz value function. Because of the lack of space, we do not include detailed proofs of our results, but only indicate by what arguments they are obtained. Complete proofs will be published elsewhere.

II. PRELIMINARIES

Denote by B(0, R) the closed ball in a finite dimensional space centered at zero with radius R > 0. Let K ⊂ R n and x ∈ K. The contingent cone to K at x consists of all vectors v ∈ R n such that there exist sequences h i → 0+ and vectors

v i → v satisfying x + h i v i ∈ K.
The limiting normal cone to K at x ∈ K is defined using the Peano-Kuratowski upper limit, see for instance [START_REF] Aubin | SET-VALUED ANALYSIS[END_REF],

N L K (x) = Limsup y→ K x T K (y) -
, where → K denotes the convergence in K and T K (y) -is the negative polar of T K (y). It is well known that if x lies on the boundary of K, then N L K (x) is not reduced to zero. For ϕ : R n → R ∪ {±∞} the domain of ϕ is the set dom(ϕ) of all x such that ϕ(x) = ±∞. Denote by epi(ϕ) and hyp(ϕ) resp. the epigraph and the hypograph of ϕ.

The Fréchet subdifferential of ϕ at x ∈ dom(ϕ), denoted by ∂ -ϕ(x), consists of all vectors p satisfying

(p, -1) ∈ T epi(ϕ) (x, ϕ(x)) -. If p ∈ R n is so that (p, -1) ∈ N L epi(ϕ) (x, ϕ(x)), then it is called a limiting subgradient of ϕ at x, while if (p, 0) ∈ N L epi(ϕ) (x, ϕ(x)), then p is called
a limiting horizontal subgradient of ϕ at x. The sets of all limiting and limiting horizontal subgradients of ϕ at x are denoted by ∂ L,-ϕ(x) and ∂ ∞,-ϕ(x) respectively. The limiting supergradients are defined in a similar way:

every p ∈ R n satisfying (-p, +1) ∈ N L hyp(ϕ) (x, ϕ(x)) is called a limiting supergradient of ϕ at x. If (-p, 0) ∈ N L hyp(ϕ) (x, ϕ(x)
), then p is called a limiting horizontal supergradient of ϕ at x. The sets of all limiting and limiting horizontal supergradients of ϕ at x are denoted by ∂ L,+ ϕ(x) and ∂ ∞,+ ϕ(x) respectively.

If ϕ is locally Lipschitz at x, then the sets of limiting horizontal subgradients and supergradients at x are empty and the set co ∂ L,-ϕ(x) is the generalized gradient of ϕ at x, denoted by ∂ϕ(x), where co stands for the convex hull.

For a, b ∈ R we shall use the following notations a ∧ b = min{a, b}, a ∨ b = max{a, b}. For a matrix A, denote by A * its transpose.

III. VALUE FUNCTION

Consider the nonautonomous infinite horizon optimal control problem ( 6)-( 7) with data as described in the introduction. A Lebesgue measurable selection u(t) ∈ U (t) a.e. in R + is called a control and the set of all controls is denoted by U. Note that to state ( 6)- [START_REF] Cannarsa | Some characterizations of optimal trajectories in control theory[END_REF] we need only controls defined on [t 0 , +∞[. However, without any loss of generality, we may assume that controls are defined on [0, +∞[.

Assumptions (H1):

i) There exist locally integrable functions c, θ :

R + → R + such that for a.e. t ∈ R + , |f (t, x, u)| ≤ c(t)|x| + θ(t), ∀ x ∈ R n , u ∈ U (t);
ii) For every R > 0, there exists a locally integrable function c R : R + → R + and a modulus of continuity ω R : R + ×R + → R + such that for a.e.

t ∈ R + , ω R (t, •) is nondecreasing, lim r→0+ ω R (t, r) = 0 and for all x, y ∈ B(0, R), u ∈ U (t), |f (t, x, u) -f (t, y, u)| ≤ c R (t)|x -y|, |L(t, x, u) -L(t, y, u)| ≤ ω R (t, |x -y|); iii) For all x ∈ R n , the mappings f (•, x, •), L(•, x, •) are
Lebesgue-Borel measurable ; iv) There exists a locally integrable function β : R + → R + and a locally bounded nondecreasing function φ :

R + → R + such that for a.e. t ∈ R + , L(t, x, u) ≤ β(t)φ(|x|), ∀ x ∈ R n , u ∈ U (t); v) U (•)
is Lebesgue measurable and has closed nonempty images; vi) For a.e. t ∈ R + , and for all x ∈ R n the set f (t, x, u), L(t, x, u) + r : u ∈ U (t) and r ≥ 0 is closed and convex. Under the above assumptions to every control u(•) and (t 0 , x 0 ) ∈ R + × R n corresponds a locally absolutely continuous solution x(•) of the system in [START_REF] Cannarsa | Some characterizations of optimal trajectories in control theory[END_REF] defined on R + . The couple (x, u) is called a trajectory-control pair. If the integral in [START_REF] Bettiol | Improved sensitivity relations in state constrained optimal control[END_REF] does not converge for any u ∈ U, then set V (t 0 , x 0 ) = +∞.

The extended function V : [0, +∞[×R n → R + ∪ {+∞} defined by ( 6)-( 7) is called the value function of the infinite horizon problem. For any t 0 ∈ R + , x 0 ∈ R n such that V (t 0 , x 0 ) < +∞, a trajectory-control pair (x, ū) is optimal for the infinite horizon problem at (t 0 , x 0 ) if for every trajectory-control pair (x, u) satisfying x(t 0 ) = x 0 we have

∞ t0 L(t, x(t), ū(t)) dt ≤ ∞ t0 L(t, x(t), u(t)) dt.
The proof of the following Proposition is standard. Proposition 1: Assume (H1). Then V is lower semicontinuous and for every (t 0 , x 0 ) ∈ dom(V ), there exists a trajectory-control pair (x, ū) satisfying

V (t 0 , x 0 ) = ∞ t0 L(t, x(t), ū(t)) dt.
Consider the relaxed infinite horizon problem

V rel (t 0 , x 0 ) = inf ∞ t0 n i=0 λ i (t)L(t, x(t), u i (t)) dt (8) over all trajectory-control pairs of    x (t) = n i=0 λ i (t)f (t, x(t), u i (t)) u i (t) ∈ U (t), λ i (t) ≥ 0, n i=0 λ i (t) = 1 x(t 0 ) = x 0 , (9)
where u i (•), λ i (•) are Lebesgue measurable on R + for i = 0, ..., n.

Clearly V rel ≤ V. The following result can be proved by using finite horizon relaxation theorems and careful choice of relaxed controls. The detailed proof will appear elsewhere.

Theorem 1: Assume (H1) i)-v) with ω R (t, r) = cR (t)r, for a locally integrable cR : R + → R + , and that, for a.e. t ∈ R + and all x ∈ R n , the set

f (t, x, u), L(t, x, u) | u ∈ U (t) is compact. If for every t ≥ 0, V rel (t, •) : R n → R is continuous, then V rel = V on R + × R n .
In particular, if a trajectory-control pair (x, ū) is optimal for (6)- [START_REF] Cannarsa | Some characterizations of optimal trajectories in control theory[END_REF], then it is also optimal for the relaxed problem (8)- [START_REF] Cannarsa | Local regularity of the value function in optimal control[END_REF].

Remark 1: a) Notice that if U (t) is compact for a.e. t ≥ 0 and f, L are continuous with respect to u, then the above compactness assumption holds true. b) Sufficient conditions for continuity of V rel (t, •) can be deduced from Theorem 2 below. c) Theorem 1 allows to avoid convexity requirement in assumption (H1) vi).

We provide next three results concerning continuity and local Lipschitz continuity of the value function. Their proofs are quite technically involved and will be published elsewhere. They are based on the dynamic programming, the Gronwall lemma and carefully chosen assumptions.

Theorem 2: Let (H1) hold with time independent c(t) ≡ c ≥ 0 and θ(t) ≡ θ ≥ 0. Assume also that c R (t) ≡ δ ≥ 0 for every R > 0 and that, for all x, y ∈ R n , a.e. t ≥ 0 and all u ∈ U (t),

|L(t, y, u) -L(t, x, u)| ≤ ω(t, |x -y|)× L(t, x, u) ∧ L(t, y, u) + h(t, |x| ∨ |y|) , (10)
where ω : R + × R + → R + , h : R + × R + → R + enjoy the following properties: ω, h are Lebesgue-Borel measurable, nondecreasing with respect to the second variable, ω is bounded, lim r→0+ ω(t, r) = 0 for a.e. t > 0 and

∞ 0 h(t, (R + θt)e ct )dt < +∞ ∀ R ≥ 0. (11)
If dom(V) = ∅, then dom(V) = R + × R n and V (t, •) is continuous for any t ≥ 0. To prove the above theorem it is enough to show that ( 10), [START_REF] Clarke | The relationship between the maximum principle and dynamic programming[END_REF] yield upper semicontinuity of V (t, •). Under further restrictions, V turns out to be locally Lipschitz.

Lemma 1: Assume (H1) with time independent c(t) ≡ c ≥ 0, θ(t) ≡ θ ≥ 0, c R (t) ≡ δ ≥ 0 for every R > 0 and suppose that, for all x, y ∈ R n , a.e. t ≥ 0 and all u ∈ U (t)

|L(t, y, u) -L(t, x, u)| ≤ k(t, |x| ∨ |y|)|x -y|, (12)
where k :

R + × R + → R + is Lebesgue-Borel measurable, k(t, •) is nondecreasing, and ∞ 0 e δt k(t, (R + θt)e ct )dt < +∞ ∀ R ≥ 0. (13)
If dom(V) = ∅, then V (t, •) is locally Lipschitz continuous on R n for all t ≥ 0 and for all x 1 , x 2 ∈ B(0, R)

|V (t, x 2 ) -V (t, x 1 )| ≤ e -δt K t (R) |x 2 -x 1 | (14)
where for all t ≥ 0

K t (R) := ∞ t e δτ k τ, M t (τ, R) dτ < +∞ (15)
and M t (τ, R) := [R + θ(τ -t)]e c(τ -t) . Corollary 1: Under the assumptions of Lemma 1, fix any (t 0 , x 0 ) ∈ R + × R n and let (x, ū) be any trajectory-control pair satisfying x(t 0 ) = x 0 . Then for all t ≥ t 0 and x 1 , x 2 ∈ B(x(t), 1) we have

|V (t, x 2 ) -V (t, x 1 )| ≤ e -δt K t0 (1 + |x 0 |) |x 2 -x 1 |. (16)
Theorem 3: Under all the assumptions of Lemma 1, suppose that β(•) is nonincreasing. If dom(V) = ∅, then dom(V) = R + × R n , V is locally Lipschitz continuous, and for every T ≥ t ≥ 0, R > 0 and for all t 1 , t 2 ∈ [t, T ],

x 1 , x 2 ∈ B(0, R), |V (t 2 , x 2 ) -V (t 1 , x 1 )| ≤ e -δt K t (R) |x 2 -x 1 | + N t (T, R)|t 2 -t 1 |, where N t (T, R) = e -δt K t (R)[ θ+c M t (T, R)+β(t)φ M t (T, R) ].
Remark 2: As in Corollary 1, even the Lipschitz constant of V with respect to time and space can be estimated along any trajectory-control pair. More precisely, for any given trajectory-control pair (x, ū) with x(t 0 ) = x 0 and all

(t i , x i ) ∈ R + × R n (i = 1, 2) satisfying |x i -x(t)| ≤ 1 and t ≤ t i ≤ t + 1 for some t ≥ t 0 , we have that |V (t 2 , x 2 ) -V (t 1 , x 1 )| ≤ e -δt K t0 (1 + |x 0 |){|x 2 -x 1 | + [θ +c M t (t + 1, R t ) + β(t)φ(M t (t + 1, R t ))]|t 2 -t 1 |}, where R t = 1 + |x(t)| ≤ 1 + M t0 (t, |x 0 |). Since M t (t + 1, R t ) ≤ 1 + θ + M t0 (t, |x 0 |) e c ≤ 1 + θ e c + |x 0 | + θ(t -t 0 ) e c(t-t0+1) ,
it is easy to realise that if δ > c, then the Lipschitz constant of V at (t, x(t)) when t → ∞ can be estimated from the above upon the behaviour at infinity of the function t → e -δt β(t) φ t(1 + θ) e ct .

Example 1: Given λ > 0, a closed nonempty set U ⊂ R m , f : R n × R m → R n , : R n × R m → R + , consider the classical infinite horizon problem: (1)-( 2) Assume that f satisfies assumptions (H1) with time independent c(t) ≡ c, θ(t) = θ and c R (t) ≡ δ for all R > 0. Suppose in addition that for all u ∈ U , (0, u) ≤ M for some M ≥ 0 and

| (x, u) -(y, u)| ≤ C[1 + (|x| ∨ |y|) r ]|x -y|
for some constants C, r ≥ 0. Then, taking

β(t) = e -λt and φ(s) = M + C 1 + s r ∀ x, y ∈ R n ,
it is easy to check that Theorem 3 can be applied provided that λ > δ + c r.

Indeed, ( 12) holds true with k(t, s) = Ce -λt (1 + s r ), which in turn satisfies [START_REF] Michel | On the transversality condition in infinite horizon optimal problems[END_REF] owing to [START_REF] Tauchnitz | The Pontryagin maximum principle for nonlinear optimal control problems with infinite horizon[END_REF]. The above assumptions were used in [START_REF] Aubin | Shadow prices and duality for a class of optimal control problems[END_REF] to study analogous problems for linear f and convex compact U . In particular, if

θ := sup u∈U (0, u) + sup x∈R n , u∈U |f (x, u)| < ∞
and there exist 0 ≤ δ < λ, δ 1 ≥ 0 such that f (•, u) and (•, u) are resp. δ-Lipschitz and δ 1 -Lipschitz for every u ∈ U , then, taking k(t, s) = e -λt δ 1 s, c = 0, φ(s) = θ + δ 1 s, we deduce from Theorem 3 that the value function V satisfies

|V (t 2 , x 2 ) -V (t 1 , x 1 )| ≤ e -δt K t (R) |x 2 -x 1 |+ e -δt K t (R)[θ + e -λt (θ + δ 1 (R + θ(T -t)))]|t 2 -t 1 | for all x 1 , x 2 ∈ B(0, R) and t 1 , t 2 ∈ [t, T ],
where

K t (R) = δ 1 ∞ t e (δ-λ)τ (R + θ(τ -t))dτ ≤ K 0 (R).
Notice that we can always suppose that δ > 0. Remark 2 implies then that for any trajectory x of the control system, Lipschitz constant of V at (t, x(t)) decreases exponentially to zero when t → ∞.

IV. MAXIMUM PRINCIPLES

We shall need the following assumption : (H2) For every R > 0, there exists a locally integrable function α R : R + → R + such that for a.e. t ∈ R + and all x, y ∈ B(0, R),

|L(t, x, u) -L(t, y, u)| ≤ α R (t)|x -y|, ∀ u ∈ U (t).

The Hamiltonian H

: R + × R n × R n → R is defined by H(t, x, p) := sup u∈U (t) ( p, f (t, x, u) -L(t, x, u)).
Then H(t, x, •) is convex and under assumptions (H1), (H2) for a.e. t ≥ 0, the supremum is attained for all

(x, p) ∈ R n × R n . Moreover H(t, •, p) is Lipschitz on B(0, R) with Lipschitz constant c R (t)|p| + α R (t).

A. Sensitivity Relation for LSC Value Function

If f (t, •, u) and L(t, •, u), are differentiable, denote D x f and L x their (partial) Jacobian and gradient with respect to x. Recall that if V (t, •) is lower semicontinuous, then ∂ -

x V (t, x 0 ) = ∅ on a dense subset of points x 0 of dom(V (t, •)), where ∂ - x V (t, x 0 ) denotes the partial Fréchet subdifferential of V (t, •) at x 0 .

Theorem 4: Assume (H1), (H2) and let (x, ū) be optimal at

(t 0 , x 0 ) ∈ dom(V ) with ∂ - x V (t 0 , x 0 ) = ∅. If f (t, •, u) and L(t, •, u) are differentiable for a.e. t ∈ R + and all u ∈ U (t), then for every p 0 ∈ ∂ - x V (t 0 , x 0 ) the solution p(•) of the adjoint system -p (t) = D x f (t, x(t), ū(t)) * p(t) -L x (t, x(t), ū(t)) p(t 0 ) = -p 0 satisfies the maximality condition p(t), f (t, x(t), ū(t)) -L(t, x(t), ū(t)) = H(t, x(t), p(t)) (18)
a.e. in [t 0 , +∞[ and the sensitivity relation

-p(t) ∈ ∂ - x V (t, x(t)) ∀ t ≥ t 0 .
Furthermore, if assumptions of Lemma 1 hold true with δ > 0, then p(t) converges exponentially to zero when t → ∞. Proof: We use the approach developed in [START_REF] Cannarsa | Local regularity of the value function in optimal control[END_REF] for the finite horizon Bolza problem and the uniqueness of solution of the adjoint system. Details will appear elsewhere.

The above necessary optimality condition holds true only when the subdifferential ∂ - x V (t 0 , x 0 ) is nonempty and involves elements of this subdifferential. An alternative way is discussed in the next section, but before, for the sake of completeness, we state one more result that follows directly from nonsmooth maximum principles for the finite horizon Bolza problem, but does not include any transversality condition and may be abnormal.

Below, let ∂ x f denote the partial generalized Jacobian of f with respect to x and ∂ x L the partial generalized gradient of L with respect to x.

Theorem 5: Assume (H1), (H2) and let (x, ū) be optimal at some (t 0 , x 0 ) ∈ R + × R n . Then i) either there exists a solution p( Proof: The proof is a bit lengthly so we only indicate the main idea. The complete proof will appear elsewhere. For every integer i ≥ t 0 we introduce a new finite horizon Bolza problem minimize {V (i, x(i))

+ i t0 L(t, x(t), u(t)) dt | u ∈ U}
over all trajectory-control pairs (x, u) of [START_REF] Cannarsa | Some characterizations of optimal trajectories in control theory[END_REF]. Observe that the restriction of (x, ū) to the time interval [t 0 , i] is optimal for this new Bolza problem and V (i, •) is lower semicontinuous. So we can write a, possibly abnormal, nonsmooth maximum principle with a costate p i satisfying on [t 0 , i] the conclusions of theorem. For this aim we adapt [19,Theorem 6.2.1] to our framework. If {p i (t 0 )} i contains a bounded subsequence, then it is enough to extract a subsequence from {p i } i converging almost uniformly to some p satisfying i) or ii). Otherwise, replacing p i by p i /|p i (t 0 )|, we take a subsequence converging almost uniformly to some p as in ii). The last statement results from the Filippov theorem, [START_REF] Aubin | SET-VALUED ANALYSIS[END_REF].

The above theorem lacks the transversality condition and also it does not exclude the abnormality of the maximum principle. For this reason its conclusion is less informative than the one of Theorem 4.

B. Transversality Condition for Continuous Value Function

Actually, if V (t 0 , •) is continuous, then a maximum principle holds true at all points x 0 ∈ R n .

Theorem 6: Let (H1), (H2) hold and (t 0 , x 0 ) ∈ dom(V ). Assume that V (t 0 , •) is continuous on a neighborhood of x 0 and let (x, ū) be optimal at (t 0 , x 0 ). Then i) either there exists a solution p(•) of ( 19) satisfying the transversality condition

-p(t 0 ) ∈ ∂ L,+ x V (t 0 , x 0 )
and the maximality condition (18) a.e. in [t 0 , +∞[ ; ii) or there exists a nonvanishing solution p(•) of ( 20) satisfying the abnormal maximum principle (21) a.e. in [t 0 , +∞[ and the transversality condition

-p(t 0 ) ∈ ∂ ∞,+ x V (t 0 , x 0 ). Proof: For any integer i ≥ t 0 , consider the problem minimize{ i t0 L(t, x(t), u(t)) dt -V (t 0 , x(t 0 ))} over all trajectory-control pairs (x, u) of x (t) = f (t, x(t), u(t)) for a.e. t ∈ [t 0 , i] x(i) = x(i).
Since -V (t 0 , •) is continuous near x 0 we can write a, possibly abnormal, nonsmooth maximum principle on [t 0 , i] to get a costate p i . The proof ends by applying the same arguments as those of the proof of Theorem 5.

C. Sensitivity Relations for Locally Lipschitz Value Function

We show here that if the value function is locally Lipschitz with respect to the second variable, then every optimal trajectory-control pair of the infinite horizon problem satisfies the maximum principle and a sensitivity relation involving the partial generalized gradient of the value function.

Theorem 7: Assume (H1) i) -v), (H2) and that for all large T > 0, the mapping V (T, •) is locally Lipschitz. Then for every t ≥ 0, V (t, •) is locally Lipschitz with the local Lipschitz constants depending only on the magnitude of t.

Moreover, if (x, ū) is optimal at some (t 0 , x 0 ) ∈ R + ×R n , then there exists a solution p(•) of the adjoint inclusion (19), satisfying the maximality condition (18) a.e. in [t 0 , +∞[ and the sensitivity relations

-p(t 0 ) ∈ ∂ x V (t 0 , x 0 ), -p(t) ∈ ∂ x V (t, x(t)) for a.e. t > t 0 .
Furthermore, if assumptions of Lemma 1 hold true with δ > 0, then p(t) converges exponentially to zero when t → ∞.

Proof: Consider the finite horizon problems as in the proof of Theorem 5. It is enough to apply [START_REF] Clarke | The relationship between the maximum principle and dynamic programming[END_REF] to get costates p i satisfying the maximum principle and sensitivity relations as above on [t 0 , i]. Taking the limits for an appropriately chosen subsequence of {p i } i ends the proof.

The next result, whose proof is similar to the above, but uses [START_REF] Bettiol | Improved sensitivity relations in state constrained optimal control[END_REF] instead of [START_REF] Clarke | The relationship between the maximum principle and dynamic programming[END_REF], involves a different adjoint inclusion and provides one more sensitivity relation. It helps to study the limit at infinity of H(t, x(t), p(t)), whenever x is an optimal trajectory and p is a corresponding costate.

Theorem 8: Assume (H1) i)-v), (H2) with bounded c(•), θ(•), β(•) and that for every (t, x) ∈ R + × R n the set {(f (t, x, u), L(t, x, u)) : u ∈ U (t)} is closed. If for all large T > 0, V (T, •) is locally Lipschitz, then V is locally Lipschitz on [0, ∞[×R n .

Let (x, ū) be optimal at some (t 0 , x 0 ) ∈ R + × R n . Then there exists a locally absolutely continuous function p : [t 0 , +∞[→ R n satisfying the Hamiltonian inclusion (-p, x) (t) ∈ ∂ (x,p) H(t, x(t), p(t)), for a.e. t ≥ t 0 , (22) the transversality condition Furthermore, if assumptions of Lemma 1 hold true with δ > 0, then p(t) converges exponentially to zero when t → ∞.

-p(t 0 ) ∈ ∂ L,+ x V (t 0 , x 0 ), (23)
Moreover if assumptions of Theorem 3 are satisfied, δ > c and ess -lim t→∞ e -δt β(t)φ(t(1 + θ)e ct ) = 0, ∀ R > 0, then also ess-lim t→∞ H(t, x(t), p(t)) = 0, where ess-lim denotes the essential limit.

V. CONCLUSIONS AND FUTURE WORK

We have studied the value function of the infinite horizon problem without a discount factor and provided sufficient conditions for its lower semicontinuity, continuity and local Lipschitz continuity. In this very general framework we derived maximum principle and sensitivity relations that greatly simplify the investigation of normality of the maximum principle and of the behaviour at infinity of the costate. The direct approach we have developed allows to generalize several earlier results and links maximum principle to dynamic programming in a very general framework. Our future work will concern the infinite horizon optimal control problem under state constraints, useful in many applied models.

  Moreover, if for all large t > t 0 , V (t, •) is locally Lipschitz at x(t) with a Lipschitz constant independent from t, then the normal maximum principle i) holds true.Furthermore, if there exist integrable mappings C f : R + → R + and C L : R + → R + such that for a.e. t ≥ 0 and all u ∈ U (t), f (t, •, u) is C f (t)-Lipschitz and L(t, •, u) is C L (t)-Lipschitz, then lim t→∞ p(t) does exist.

		•) of the adjoint inclusion
	-p (t) ∈ ∂ x f (t, x(t), ū(t)) * p(t) -∂ x L(t, x(t), ū(t)) (19)
	satisfying the maximality condition (18) a.e. in [t 0 , +∞[ ;
	ii) or there exists a nonvanishing solution p(•) of the
	adjoint inclusion
	-p (t) ∈ ∂ x f (t, x(t), ū(t)) * p(t)	(20)
	satisfying a.e. in [t 0 , +∞[ the abnormal maximum principle
	p(t), f (t, x(t), ū(t)) = max u∈U (t)	p(t), f (t, x(t), u) . (21)


Stuart J Patterson
email: spatterson@rvc.ac.uk

Charlene ; ; Clarke
email: clarke.charlene0303@gmail.com

Tim H ; Clutton-Brock

Michele A Mill

Sven D C Parsons
email: svenparsons@hotmail.com

Dirk U Pfeiffer
email: dirk.pfeiffer@cityu.edu.hk

Timothée Vergne
email: timothee.vergne@envt.fr

Julian A Drewe
email: jdrewe@rvc.ac.uk

M A ; Miller
email: miller@sun.ac.za

Cristina Esmeralda

Di Francesco

Camilla Smoglica

Simone Angelucci

Combining Analytical Approaches and Multiple Sources of Information to Improve Interpretation of Diagnostic Test Results for Tuberculosis in Wild Meerkats

Keywords: diagnostics, interpretation, wildlife, tuberculosis, meerkats

Simple Summary: Diagnostic tests used in animals are ideally extensively trialled to ensure that practitioners have confidence in the results; the ideal target should be 100% of infected animals testing positive (sensitivity), and 100% of uninfected animals testing negative (specificity). In these trials, a gold standard is necessary, against which the diagnostic tests may be compared. Commonly, tests of wild animals are not extensively trialled due to shortages of samples and the rarity of gold standard tests. This study uses samples collected for the purpose of diagnosing Mycobacterium suricattae infection in meerkats and estimates the sensitivity and specificity of available tests, both individually, and in combination. A small number of post-mortems (gold standard) were carried out, providing definitive evidence of infected animals against which to compare the tests. In addition, test results were unconventionally compared to survival times and clinical characteristics, aiming to quantify the prognostic capabilities of the tests. The study therefore not only estimates the required parameters against which to assess these tests, but also provides a model for future assessment of diagnostic tests in imperfect field scenarios. Wildlife diseases are increasingly recognised as important to society, and so methods to better quantify these are increasingly important.

Abstract: Diagnostic tests are used to classify individual animals' infection statuses. However, validating test performance in wild animals without gold standard tests is extremely challenging, and the issue is further complicated in chronic conditions where measured immune parameters vary over time. Here, we demonstrate the value of combining evidence from different diagnostic approaches to aid interpretation in the absence of gold standards, large sample sizes, and controlled environments. Over a two-year period, we sampled 268 free-living meerkats (Suricata suricatta) longitudinally for Mycobacterium suricattae (a causative agent of tuberculosis), using three ante-mortem diagnostic tests based on mycobacterial culture, and antigen-specific humoral and cell-mediated immune responses, interpreting results both independently and in combination. Post-mortem cultures confirmed M. suricattae infection in 22 animals, which had prior ante-mortem information, 59% (13/22) of which were test-positive on a parallel test interpretation (PTI) of the three ante-mortem diagnostic assays (95% confidence interval: 37-79%). A similar ability to detect infection, 65.7% (95% credible interval: 42.7-84.7%), was estimated using a Bayesian approach to examine PTI. Strong evidence was found for a near doubling of the hazard of death (Hazard Ratio 1.75, CI: 1.14-2.67, p = 0.01), associated with

Introduction

In order to have confidence in the results of a diagnostic test, the user must have knowledge of the test's performance characteristics. Diagnostic tests are evaluated for their ability to correctly classify animals as infected (sensitivity), or as uninfected (specificity) [START_REF] Bruel | The evaluation of diagnostic tests: Evidence on technical and diagnostic accuracy, impact on patient outcome and cost-effectiveness is needed[END_REF], characteristics normally defined by comparing test results with a reference test (sometimes called a gold standard [START_REF] Pfeiffer | Interpretation of diagnostic tests[END_REF][START_REF] Dohoo | Veterinary Epidemiological Research[END_REF]). Whilst these parameters may be defined by large-scale studies in domestic species, sample sizes in wildlife are often limited due to difficulties accessing or capturing animals and, in some cases, by ethical considerations [START_REF] Stallknecht | Impediments to wildlife disease surveillance, research, and diagnostics[END_REF]. Here, we evaluate the performance of the available diagnostic tests for tuberculosis (TB) in wild meerkats (Suricata suricatta) and use combinations of tests and life history information to increase our understanding of the results.

A diagnostic test works by determining the presence or quantity of a relevant biological marker, such as antibodies, in order to assign a positive or negative result. In a chronic disease such as TB, the abundance of these markers may vary over time. Therefore, as TB progresses from initial infection to clinical disease, a particular diagnostic test's performance may change with the abundance of the targeted biological entity [START_REF] Pollock | Mycobacterium bovis infection and tuberculosis in cattle[END_REF][START_REF] Vordermeier | The interferon-gamma field trial: Background, principles and progress[END_REF][START_REF] Welsh | Influence of pathological progression on the balance between cellular and humoral immune responses in bovine tuberculosis[END_REF][START_REF] Ritacco | Reciprocal cellular and humoral immune responses in bovine tuberculosis[END_REF]. A test based on measuring presence of bacilli, for example, might work best at a later disease stage when bacterial replication has taken place, rather than at the point of initial infection. This means that a test may perform well at only certain stages of infection, and if relied upon blindly, may be misinterpreted. A combination of tests targeting different biological parameters may better reflect true infection status compared to a single test, especially when applied across a population with individuals at different stages of a chronic disease.

In meerkats, TB is caused by infection with Mycobacterium suricattae, a member of the Mycobacterium tuberculosis complex [START_REF] Parsons | Novel cause of tuberculosis in Meerkats, South Africa[END_REF]. Persistently swollen lymph nodes (particularly the submandibular) are typical signs of advanced disease in this species, leading to a presumptive diagnosis of TB [START_REF] Drewe | Pathology of Mycobacterium bovis infection in Wild Meerkats (Suricata suricatta)[END_REF]. Transmission has been linked with suppurating skin wounds and oral excretion [START_REF] Drewe | Who infects whom? Social networks and tuberculosis transmission in wild meerkats[END_REF]. Given the highly social behaviour of these animals, these routes mean that infection may easily be transmitted throughout a social group [START_REF] Patterson | Social and environmental factors affect tuberculosis related mortality in wild meerkats[END_REF]. Diagnostic tests may therefore be justifiably targeted at either the group-or the individualanimal level. Where individual test sensitivity is low, testing multiple individuals gives a more reliable estimation as to whether there is truly infection within a group.

The Kalahari meerkat project (KMP) is a long-term ecological study in the Northern Cape of South Africa [START_REF] Clutton-Brock | Costs of cooperative behaviour in suricates (Suricata suricatta)[END_REF]. Tuberculosis was first identified at the study site in 1999; since then, clinical signs have been observed in meerkats in the majority of years, affecting 6% of individuals born in the study population [START_REF] Patterson | Social and environmental factors affect tuberculosis related mortality in wild meerkats[END_REF]. The KMP therefore offers a unique opportunity to study TB and its transmission in a natural setting. Each meerkat is individually identifiable [START_REF] Clutton-Brock | Cooperation, control, and concession in Meerkat groups[END_REF], and all animals are habituated to close observation which enables repeated sampling over time. It is the Project's policy to euthanase meerkats with advanced stages of TB at the point at which lymph node swellings burst, in an effort to control spread of disease. Previous records indicate that animals that have reached this stage and are not euthanased, seldom live for more than a month, and never recover. Given the potential applications of TB research that could be carried out at the site, it is important to have a thorough understanding of how to interpret available diagnostic tests. Previous TB diagnostic assays used in meerkats have focused on antibody responses and mycobacterial culture [START_REF] Drewe | Accuracy of Three diagnostic tests for determining Mycobacterium bovis infection status in live-sampled wild Meerkats (Suricata Suricatta)[END_REF]. Cultures of tracheal washes, multiantigen print immunoassay (MAPIA), and a commercially available lateral flow immunoassay rapid test (RT) were shown to be of diagnostic value when used in combination, but had limitations in terms of either sensitivity (culture and RT) or specificity (MAPIA) when interpreted independently [START_REF] Drewe | Accuracy of Three diagnostic tests for determining Mycobacterium bovis infection status in live-sampled wild Meerkats (Suricata Suricatta)[END_REF]. More recently, a test of cell-mediated immunity (CMI), based upon interferon-gamma (IFNγ) inducible-protein 10 (IP-10), has been developed for use in meerkats [START_REF] Clarke | Development and evaluation of a diagnostic cytokine-release assay for Mycobacterium suricattae infection in meerkats (Suricata suricatta)[END_REF], and a new version of the lateral flow device (the dual-path platform, DPP; Chembio Diagnostics Systems, Inc., Medford, New York, NY, USA) has been developed [START_REF] Greenwald | Highly accurate antibody assays for early and rapid detection of tuberculosis in african and asian elephants[END_REF]. The sensitivity and specificity characteristics of the IP-10 release assays (IPRA) and the DPP serologic assay are unknown. Due to the known pattern of disease response in other species (e.g., cattle) [START_REF] De La Rua-Domenech | Ante mortem diagnosis of tuberculosis in cattle: A review of the tuberculin tests, γ-interferon assay and other ancillary diagnostic techniques[END_REF], tests of CMI are typically able to detect infection at an earlier stage than culture or antibody tests.

This study investigated the performance of the IPRA, the DPP, and mycobacterial culture to determine M. suricattae infection status both with and without a reference test. We asked how these test results relate to true infection status by investigating associations between tests, individual animal survival, and the likelihood of clinical disease being present. Finally, we determined whether an individual's age influences its diagnostic test result, and whether this has implications for test reliability. These approaches made use of the same test dataset, but in different ways, giving multiple insights into how the tests may be used and the limitations of the available methods.

Materials and Methods

Data Collection

Nine meerkat social groups at the KMP were selected at random for longitudinal sampling (Groups B, F, J, N, P, Q, R, U, and Z). From September 2014 to September 2016, 268 meerkats were sampled in five sampling periods (Blocks 1-5), each block lasting for three months. Within each block, all animals within the sample groups were scheduled to be sampled once. In block 1, samples from a further 44 individuals were available for testing from a parallel study [START_REF] Clarke | Development and evaluation of a diagnostic cytokine-release assay for Mycobacterium suricattae infection in meerkats (Suricata suricatta)[END_REF] involving an additional 6 groups, and samples previously collected (n = 10) from a captive population described by Clarke et al. [START_REF] Clarke | Development and evaluation of a diagnostic cytokine-release assay for Mycobacterium suricattae infection in meerkats (Suricata suricatta)[END_REF], were utilised. Ethical approval for the study was granted by the University of Pretoria's ethics committee in May 2014 (project reference ECO20-14).

Throughout this two-year study, each group was visited a minimum of three occasions per week. At each visit, all individuals within a group were visually examined for signs of illness or injury. Individuals were identified by both a subcutaneous microchip, and a unique combination of hair dye markings regularly applied to the fur. Presence of each animal was recorded, and an individual was recorded as dead if it had not been seen for a period of three months. Animals were also visually checked for swollen lymph nodes as an indicator of likely disease. Observations contributing to the study therefore continued until 31 December 2016 to allow for this lag. In addition to these life history data, field sampling was carried out in order to collect blood and tracheal wash samples. Captures, anaesthesia, and sampling were performed as described by Drewe [START_REF] Drewe | Accuracy of Three diagnostic tests for determining Mycobacterium bovis infection status in live-sampled wild Meerkats (Suricata Suricatta)[END_REF]. Age was calculated at the point of sampling, based on recorded birth dates for each individual.

Euthanasia of individuals with draining lymph nodes was performed under anaesthesia. Blood samples and tracheal wash samples were collected, after which 2 mL of sodium pentobarbital (Euthanase, Kyron, Johannesburg, South Africa) was injected into the jugular vein. Post-mortem examinations were performed on all euthanased animals in the field. Samples of liver, lung, spleen, and sub-mandibular lymph nodes were frozen at -80 • C prior to being transported to the laboratory. Further samples were similarly submitted from additional organs and lymph nodes with gross abnormalities on post-mortem examination. All samples were transported on ice for full laboratory processing.

Laboratory Processing

Both tracheal washes and post-mortem tissue samples were processed as previously described [START_REF] Drewe | Accuracy of Three diagnostic tests for determining Mycobacterium bovis infection status in live-sampled wild Meerkats (Suricata Suricatta)[END_REF] and cultured in Mycobacterial Growth Indicator Tubes (MGIT, Becton Dickinson, Franklin Lakes, NJ, USA) for 12 weeks in a biosafety level 3 laboratory (Stellenbosch University). Ziehl-Neelsen stain positive cultures were genotyped by PCR targeting selected regions of difference [START_REF] Parsons | Novel cause of tuberculosis in Meerkats, South Africa[END_REF][START_REF] Warren | Differentiation of Mycobacterium tuberculosis complex by PCR amplification of genomic regions of difference[END_REF].

Bovigam ® PC-HP peptide (Prionics, Schlieren-Zurich, Switzerland) was used as a source of M. bovis antigens in the cytokine release assay. IP-10 assays were performed as described by Clarke et al. [START_REF] Clarke | Development and evaluation of a diagnostic cytokine-release assay for Mycobacterium suricattae infection in meerkats (Suricata suricatta)[END_REF]. An adjusted test result was obtained by subtracting the Optical Density (OD) reading for the saline stimulated sample, OD nil , from that for the PC-HP stimulated sample, OD PCHP . A test cut-off value for OD PCHP-nil of 0.038 was used based on a prior study, and OD PWM was used as a positive control [START_REF] Clarke | Development and evaluation of a diagnostic cytokine-release assay for Mycobacterium suricattae infection in meerkats (Suricata suricatta)[END_REF]. Serological analysis was performed using a rapid lateral flow test, DPP, using 5 µL of stored plasma, according to manufacturer's instructions. To quantitate test results as relative light units (RLU), an optical reader (Opticon DPP test reader, Chembio Diagnostic Systems, Inc., Medford, New York, NY, USA) was used to measure the reflectance of a control line and two test lines containing the antigens MPB83 (line 1) and ESAT-6/CFP-10 recombinant protein (line 2). A diagnostic cut-off value of 5.0 RLU was used. Reactivity to either test line in the DPP was considered as a positive antibody result.

Data Analyses

Each ante-mortem test result was expressed as either positive or negative as described above, and a parallel interpretation was added, which was defined as positive if one or more of the three individual tests (serology, IPRA, or culture) returned a positive result. An "ever positive" category was also created in which an animal was deemed negative (uninfected) until its first positive parallel test result, after which it remained categorised as positive at all subsequent time points. The motivation for each analysis, and the samples included in each, are outlined in Table 1.

Analysis for association with risk

Only the first sample from individuals with a known date of birth were included. The time-period was limited to minimise the impact of changes over time.

September 2014-March 2015 inclusive [START_REF] Patterson | Social and environmental factors affect tuberculosis related mortality in wild meerkats[END_REF] Variables found to have a significance of p < 0.2 in the univariable analysis were incorporated into a multivariable analysis in a forward-stepwise process, with each addition to the model compared to the simpler version using analysis of deviance for a Cox model [START_REF] Therneau | Modelling Survival Data: Extending the Cox Model[END_REF]. A gamma-frailty term was included for the social group in which the individual was resident [START_REF] Hougaard | Frailty models for survival data[END_REF].

Sensitivity Estimates in the Presence, and Absence, of a Reference Test

Sensitivity of each test (ability to correctly identify infection) was estimated both in the presence and absence of a reference test (Table 1, Aims 1 and 2) and a kappa statistic was calculated to estimate test agreement (Table 1, Aim 3). For latent class analysis, three populations with different TB prevalences were defined: (1) 0% prevalence-a negative control population (n = 10) consisting of the previously sampled captive animals [START_REF] Clarke | Development and evaluation of a diagnostic cytokine-release assay for Mycobacterium suricattae infection in meerkats (Suricata suricatta)[END_REF] with no TB disease history; (2) unknown prevalence-samples from the wild population, with unknown prevalence (n = 171); and (3) 100% prevalence-a positive control population consisting of wild animals, confirmed to be diseased following post-mortem cultures (n = 11). In these populations, the cross-detection of meerkats with the three different diagnostic tests (serology, IPRA, and culture) was modelled following a latent class modelling approach. The analysis was implemented in a Bayesian framework using the WinBUGS software (MRC, Cambridge, UK, [START_REF] Lunn | WinBUGS-A Bayesian modelling framework: Concepts, structure, and extensibility[END_REF]) embedded within the R package (R Core Team 2014) using the R2WinBugs library [START_REF] Sturtz | R2WinBUGS: A package for Running WinBUGS from R[END_REF]. Uniform distributions (0,1) were assumed for all sensitivity and specificity priors. A prior for the wild population prevalence was given by the beta (1.5, 2.3) distribution, to fit with a prior estimation of approximately 30% prevalence. Diagnostic tests were assumed to be independent due to their differing biological mechanisms of action. Two simulation chains of 1,000,000 iterations were run, with the first 100,000 iterations discarded to allow for burn-in of the chain. The chains were then thinned, taking every hundredth sample to reduce autocorrelation among the samples. Convergence was assessed by checking the trace plots for all monitored parameters [START_REF] Gelman | Bayesian Data Analysis[END_REF].

Survival Analyses

Time-dependent Cox regression [START_REF] Enderlein | Analysis of Survival Data[END_REF][START_REF] Van Dijk | The analysis of survival data in nephrology: Basic concepts and methods of Cox regression[END_REF] models were constructed in R using the Survival package [START_REF] Therneau | Modelling Survival Data: Extending the Cox Model[END_REF] to evaluate time to death, and time to the appearance of clinical signs. Individuals entered analysis either at the start of the study period, 1 September 2014, or at their date of birth if this was later. Each individual's time in the study was divided into time periods from the time of one sampling event until the next. The event "death" was created to describe either the point of death where this was observed, or the date that the animal was last seen if it had not been recorded for a minimum of three months. Within each time period, the animal's age (categorised as under months, 6-12 months, or >12 months), sex, dominance status, social group, and test statuses were recorded. A univariable analysis was carried out to examine the effects of age, dominance status, sex, and test status on the time to two events: appearance of a persistently enlarged lymph node, and loss of the animal from the study population. For time to loss, analysis was carried out by both including and excluding animals that were euthanased as this intervention had an impact upon the end time point, and the time of the final sampling point. When analysing the time until the appearance of a swollen lymph node, animals that had been observed with a swelling prior to their first sampling episode were excluded.

Relationships with Time, Age, and Predicted Risk

All individuals were assigned to one of four risk categories based upon age and group history of disease, using the findings outlined by Patterson et al. [START_REF] Patterson | Social and environmental factors affect tuberculosis related mortality in wild meerkats[END_REF]. Descriptive statistics were calculated to describe how test results related to risk category, how they changed for each individual that was sampled repeatedly over time, and the relationship of test status with age.

Results

Sensitivity Estimates in the Presence, and Absence, of a Reference Test

Over the course of the study, M. suricattae was cultured from post-mortem samples of 23 meerkats, in six different social groups (Table 2). Diagnostic test results were available for all three tests for 22 of these animals, and of these, parallel test interpretation was the most sensitive measure, correctly identifying 59% (95% confidence interval [95CI]: 37-79%) of culture-positive animals. Sensitivity estimates were greater when tests were carried out closer to the time of death. At all time points, previous test history increased sensitivity, with 82% (95CI: 59-95%) of culture-positive individuals having tested positive at least once on a minimum of one diagnostic test. Limited sample size led to wide confidence intervals when evaluating individuals further away from the euthanasia time point (Table 2).

Table 2. Sensitivity estimates for three diagnostic tests, when used alone and in combination, compared to a reference test. Ante-mortem test results are presented for 23 individuals that were later confirmed as definitively infected by post-mortem culture. Animals were tested using tracheal wash culture, DPP assay, and IFNγ inducible-protein 10 release assays (IPRA). All 23 animals were tested at the point of euthanasia, and results from previous sampling points are provided at time points relative to euthanasia. An animal was considered positive on parallel interpretation if it had a positive result on at least one of the three individual tests. A positive status to the "ever positive" interpretation was given if any single individual test was positive either at that time point or ever in the animal's test history.

Individual Test Interpretations

Multiple Test Interpretations On 174 occasions, contemporaneous results were obtained for all three of the diagnostic tests. There were no sampling events where all three tests agreed on a positive diagnosis, whilst all three were negative on 103 occasions (59.2%). As only a single tracheal wash cultured positive for M. suricattae (an IP-10 positive animal with a negative DPP result), culture was excluded from the kappa statistic calculations. The kappa statistic for agreement between the serology and the cell-mediated immunity test results was 0.009 (95CI: 0-0.19).

When analysed in the absence of a reference test (Table 1, Aim 2), the IPRA had the highest sensitivity of the three tests (58.6%; 95% Credible Interval [95Cred]: 35.1-80.6%), and the culture had the greatest specificity (99.2%, 95Cred: 95.7-99.9%) (Table 3). Using a parallel interpretation of three tests, sensitivity increased to 65.7% (95Cred: 42.7-84.6%), at the expense of a fall in specificity to 81.9% (95Cred: 65.2-96.7%) (Table 3). Table 3. Bayesian sensitivity and specificity estimates for three diagnostic tests, when used alone and in combination, in the absence of a reference test. The model for these estimates was based upon tests taken from three populations; a captive population with assumed zero prevalence (n = 10), the wild population of unknown prevalence (n = 171), and a group of post-mortem confirmed cases with 100% prevalence (n = 11). The dual-path platform VetTB (tuberculosis) assay was used for serology, an IP-10 (inducible-protein 10) release assay for the test of cell-mediated immunity, and a tracheal wash sample was provided for culture. The parallel interpretation was considered positive if one or more of the individual tests was positive. The model estimated a prevalence of 32.2% (95% confidence interval: 4.8-69.2%) in the wild population. Convergence of the Bayesian model was considered good.

Serology (DPP)

Cell

Survival Analyses

A total of 126 animals were included in the survival analysis (Table 1, Aims 4 and 5). Fourteen animals from this cohort were euthanased, a further 104 were lost to follow up and assumed dead, and the remaining animals lived beyond the study period. Clinical signs of TB were observed in 22 (17.5%) of these individuals. There was no evidence of an effect of age, sex, nor dominance status on the survival time (Table 4). Strong evidence was found for the effects of serology and culture test statuses (p = 0.04 and p = 0.01, respectively), and for the result of parallel test interpretation (p = 0.01), both when including and excluding euthanased animals. Only very weak evidence was found for an effect of the CMI-test status (p = 0.11). However, the multivariable model showed evidence of confounding, as the significance of the diagnostic test parameters disappeared with the addition of sampling groups, suggesting that the two were associated (Table 4). The Cox proportional hazards assumption was met for all eight models. The analysis of time until an observed persistently enlarged lymph node revealed a hazard ratio of 3.91 (95%CI: 0.89-17.23, p = 0.07) for individuals that were positive on the serological test (Table 1). Table 4. Survival analysis of time to death. Results of univariable and multivariable analysis using time-dependent Cox regression. Hazards were calculated for the likelihood of a meerkat being lost from the study population. Model (a) includes all individuals (of whom 118 experienced the event, death), and on the following page, model (b), excludes euthanased individuals.

Model a-All

Relationships with Time, Age, and Predicted Risk

For 44 individuals, IPRAs were performed on more than one occasion (range 2-5, median, three tests per individual). Of 29 individuals that initially tested negative, 15 (52%, 95CI: 33-70%) became positive at a subsequent test. Ten animals (66.7%, 95CI: 39-87%) initially testing positive, tested negative at a later test. In six of these events (60%), the test result was influenced by a high OD nil result in the upper quartile of all OD nil values.

Serology was performed on multiple occasions (range 2-6, median 3) for 52 individuals. Four individuals (8%, 95CI: 3-19%) converted from negative to positive over a pair of sampling points. Only a single positive animal was presented for retesting at a later time point-this individual subsequently tested negative.

For 44 individuals, there were multiple sampling points (range 2-5, median 3), at which all three diagnostic tests were performed (Figure 1). Using a parallel test interpretation, nine out of 15 positive individuals (69%, 95CI: 33-83%) converted to a negative result (on all three tests) between consecutive tests (Table 1, Aim 6). Post-mortem examinations of those 15 individuals were not possible.

of sampling points. Only a single positive animal was presented for retesting at a later time point-this individual subsequently tested negative.

For 44 individuals, there were multiple sampling points (range 2-5, median 3), at which all three diagnostic tests were performed (Figure 1). Using a parallel test interpretation, nine out of 15 positive individuals (69%, 95CI: 33-83%) converted to a negative result (on all three tests) between consecutive tests (Table 1, Aim 6). Post-mortem examinations of those 15 individuals were not possible. No association between age and test result (Table 1, Aim 7) was found in the data from 704 serological tests (Supplementary Figure S1). Confidence intervals for prevalence in the 0-3 month age category did not overlap with the 9-12, the 12-15, nor the 18-21 month categories, but no other such differences were observed. An increase in the mean OD PCHP-nil was observed between the 2014 and 2015 seasons and 2016 for one group (group N in Supplementary Figure S2), but at all other times, there were no differences between results over time.

The predicted-risk analysis (Table 1, Aim 8) incorporated 159 sampling events, at which 153 serology results were obtained, 112 IPRA results, and 159 culture samples. No differences were observed in the proportions testing positive for either culture or serology between any of the four risk categories (Figure 2). Parallel interpretation and IPRA both differentiated between the categories with 100% (95CI: 62.9-100%) prevalence in the highest risk category (both risk factors) and 17.0% (95CI: 8.5-30.3%) and 15.1% (95CI: 7.2-28.1%), respectively in the lowest risk category (neither risk factor).

results over time.

The predicted-risk analysis (Table 1, Aim 8) incorporated 159 sampling events, at which 153 serology results were obtained, 112 IPRA results, and 159 culture samples. No differences were observed in the proportions testing positive for either culture or serology between any of the four risk categories (Figure 2). Parallel interpretation and IPRA both differentiated between the categories with 100% (95CI: 62.9-100%) prevalence in the highest risk category (both risk factors) and 17.0% (95CI: 8.5-30.3%) and 15.1% (95CI: 7.2-28.1%), respectively in the lowest risk category (neither risk factor).

Discussion

This work used a range of techniques to investigate the performance of three diagnostic tests (IPRA, DPP, and mycobacterial culture) in order to better understand how best to utilise them in studies of tuberculosis in meerkat populations. While addressing this, we highlight characteristics of diagnostic tests that should remind practitioners of the care needed when interpreting such data. The results demonstrate that through the use of multiple methods in combination, confidence can be obtained in diagnostic tests, despite an absence of controlled environments.

Variation in Results

There were no occasions at which all three diagnostics tests were positive, and overall test agreement was low (Kappa statistic for agreement between serology and IPRA was 0.009, 95CI: 0-0.19), likely due to the tests detecting different stages of disease. The tests identify discrete stages of the disease process [START_REF] De La Rua-Domenech | Ante mortem diagnosis of tuberculosis in cattle: A review of the tuberculin tests, γ-interferon assay and other ancillary diagnostic techniques[END_REF], thus a positive cell-mediated response does not make a contemporaneous serological or culture result any more likely. In addition to discrepancy between contemporaneous results, there was a trend toward increased sensitivity the closer an animal was to the end-stage of disease (Table 2). It is important that practitioners are aware of these differing test performances and consider these when interpreting results, not just tuberculosis in meerkats, but chronic diseases across a range of species. Failure to do so will lead to underestimations of disease prevalence.

When interpreting results, consideration should also be given to whether the research question is targeted at diagnosis of an individual or a group. Comparison of IPRA results for four social groups (Supplementary Figure S2) demonstrated a shift in test results in one group (N) shortly before disease was observed in the group in late 2016. Wide variation in results in two social groups is also of interest. The pattern in group Z in both 2014 and 2015 is similar to that seen in group F shortly before the latter group died out due to clinical TB. No disease was observed in group Z, but high numbers were lost coinciding with a drought in late 2015, and these results may possibly reflect a role of sub-clinical infection interacting with harsh environmental conditions. A substantial proportion of the individuals in one group (group Z) were older animals, and one speculated reason for the higher IPRA results in that group in 2014-15 may have been an association between age and antigen-specific cell-mediated immunity. However, across all samples, no association was found between an individual's age, and their IPRA result. Combining the IPRA results of individuals appears to hold promise for interpreting group-level disease status.

Test Performance

Of the individual diagnostic tests used, the IPRA was the most sensitive (45%, 95CI: 25-67%, compared with the reference test and 58.6%, 95Cred: 35.1-80.6% by Bayesian estimation) and least specific (Bayesian estimation: 85.0%, 95Cred: 68.4-99.4%) of the three diagnostic tests used. Inducible-protein 10 studies in cattle and human have shown very high specificities of 100% and 97% respectively, with sensitivities of 96% and 81%, [START_REF] Parsons | Antigen-specific IP-10 release is a sensitive Biomarker of Mycobacterium bovis infection in cattle[END_REF][START_REF] Ruhwald | A multicentre evaluation of the accuracy and performance of IP-10 for the diagnosis of infection with M. tuberculosis[END_REF] much higher than seen in this study. Sensitivity of the IPRA in meerkats was reduced by spontaneous release of IP-10 in the unstimulated samples, a phenomenon previously reported in cattle [START_REF] Parsons | Antigen-specific IP-10 release is a sensitive Biomarker of Mycobacterium bovis infection in cattle[END_REF]. Again, considered alone, serological sensitivity estimates in this study were low. A range of sensitivity estimates for TB serology in wildlife are present in the literature [START_REF] Cousins | A review of tests available for use in the diagnosis of tuberculosis in non-bovine species[END_REF][START_REF] Chambers | Review of the diagnosis of Tuberculosis in non-bovid wildlife species using immunological methods-An update of published work since 2009[END_REF], and the estimates here (12%, Table 3) fall well below published data. Using the same serological test, sensitivities and specificities have been found to be 29.5% and 88% in European badgers (Meles meles), 65.1% and 97.8% in white-tailed deer (Odocoileus virginianus), and 57.5% and 96.7% in alpaca (Vicugna pacos), respectively [START_REF] Greenwald | Improved serodetection of Mycobacterium bovis infection in badgers (Meles meles) using multiantigen test formats[END_REF][START_REF] Lyashchenko | Rapid detection of serum antibody by Dual-Path platform VetTB assay in white-tailed deer infected with Mycobacterium bovis[END_REF][START_REF] Rhodes | Evaluation of gamma interferon and antibody Tuberculosis tests in Alpacas[END_REF].

We asked how well the three available tests were able to correctly identify infected and uninfected meerkats (sensitivity and specificity). Two methods were used to estimate sensitivity of the parallel test; testing against a reference test (59%; 95CI: 37-79%), and a Bayesian approach in the absence of knowledge about true infection status (65.7%; 95Cred: 42.7-84.6%). Only a small number of confirmed TB cases were available for the reference test, thus there was benefit in having a second, similar, estimate of sensitivity from a different method. It was estimated that the specificity of this interpretation was 81.9% (95Cred: 65.2-96.7%) by using a Bayesian approach. Specificity could potentially have been reduced by cross-reactivity with other Mycobacterium spp., which may be present in this environment. Parallel interpretation improved sensitivity; serial interpretation (requiring all results to be positive in order to classify the animal as infected) was not examined, having previously been shown to be extremely limited in TB diagnostics [START_REF] Buzdugan | Diagnosis of tuberculosis in groups of badgers: An exploration of the impact of trapping efficiency, infection prevalence and the use of multiple tests[END_REF]. While interpreting results in a series will always bring about some level of reduction in sensitivity, it is particularly likely to do so when the tests are detecting different stages of disease and have poor test agreement.

Prognostic Indicators

Gold standard validation of a diagnostic test compares results to an accepted standard. In the medical literature, the importance of measuring tests' performances against clinical characteristics has recently been highlighted [START_REF] Kerkhoff | Validating novel diagnostic assays for tuberculosis in the context of existing tools[END_REF], and this study used prognostic indicators as a proxy for confirmed disease presence. Two methods were used to assess how results obtained related to outcomes.

Firstly, the results, from those animals considered most likely to be infected, were checked to see if more positive test results were obtained than for animals at greater risk. Patterson et al. [START_REF] Patterson | Social and environmental factors affect tuberculosis related mortality in wild meerkats[END_REF] showed that increasing age, and a previous history of disease within a social group, increased the risk of visual signs of clinical TB. It was therefore expected that those at risk animals would have the greatest levels of infection. The proportion of animals testing positive on IPRA was greatest when two risk factors were present (increased age and group history), and least when no factors were present (Figure 2). The parallel interpretation was heavily influenced by the IPRA results, whereas serology and culture failed to differentiate between the risk categories. This suggests that the IPRA is the more useful of the individual tests for diagnosis at a group level.

Secondly, in terms of providing meaningful results, some evidence has been provided for an association between survival and both serology and culture results (p = 0.038 and p = 0.013, respectively, Table 4). This relationship disappears in multivariable analysis, almost certainly as a result of the confounding effect of social group; it is likely that infection statuses are shared between members of these socially intimate animals. Relative to the IPRA, stronger evidence was found for serology and culture results being associated with survival, with hazard ratios of 2.4 and 14.8, respectively. This suggests that positive status for these tests might be associated with end-stage disease, as positive results occur shortly before the end of an animal's life. Serology was the only variable where any relationship with time to the appearance of persistently swollen lymph nodes was observed, and the evidence was weak (p = 0.072, Table 1). Evidence is likely to be weak due to the small number of animals testing positive on this test. Observer bias is unlikely to have influenced the recording of these swollen lymph nodes, as the observers were not aware of test results.

Conclusions

Given the characteristics of the three tests, there are multiple ways in which their results could be interpreted, depending upon specific research or management questions. Culture and serology were estimated to have the greatest specificity, be the best predictors of all-cause mortality, and there may also be a weak association with the appearance of clinical signs. These findings were similar to studies in cattle showing that antibodies and bacterial load tend to increase only late in TB pathogenesis [START_REF] Vordermeier | The interferon-gamma field trial: Background, principles and progress[END_REF][START_REF] Ritacco | Reciprocal cellular and humoral immune responses in bovine tuberculosis[END_REF]. Rua-Domenech et al. [START_REF] De La Rua-Domenech | Ante mortem diagnosis of tuberculosis in cattle: A review of the tuberculin tests, γ-interferon assay and other ancillary diagnostic techniques[END_REF] stated that serological tests were not an alternative to CMI-based tests in cattle, but rather could be used as a confirmatory test in parallel with the skin test. Similarly, in the current study, serology and culture appear to be specific, confirmatory tests of disease. However, the sensitivity of these tests is very poor and visual reporting of disease signs identify many more cases than these tests combined. As a surveillance tool, diagnostic assays which are less sensitive than visual signs provide little advantage. In contrast, the IPRA appears to have greater sensitivity and would be a better surveillance tool for infection. Even in cattle, where far greater resources have been put into diagnostic test development and ante-mortem surveillance, post-mortem testing at slaughterhouses can still make a significant contribution to the identification of cases missed ante-mortem [START_REF] Abbate | Prevalence of Bovine Tuberculosis in slaughtered Cattle in Sicily, Southern Italy[END_REF][START_REF] Pascual-Linaza | Efficiency of slaughterhouse surveillance for the detection of bovine tuberculosis in cattle in Northern Ireland[END_REF]. There may be limitations using the IPRA in late-stage disease; therefore, it should be interpreted in parallel with the other tests to optimise overall sensitivity, an interpretation shown to be effective in studies of other species [START_REF] Himsworth | Comparison of test performance and evaluation of novel immunoassays for Tuberculosis in a captive herd of wood bison naturally infected with Mycobacterium bovis[END_REF]. Parallel interpretation involves a loss of specificity but given the high specificities of the serology and culture, this reduction appears to be small. While greater sensitivities and specificities have been found for well-studied species such as badgers, humans, cattle, bison and deer, the estimates for the combined tests of 65.7% and 81.9% are comparable to other published test performance statistics [START_REF] Cousins | A review of tests available for use in the diagnosis of tuberculosis in non-bovine species[END_REF]. For further studies of tuberculosis in meerkats focussing on individuals, a parallel interpretation of these three diagnostic methods is recommended.

Specific recommendations have been made within this study for how best to use combinations of currently available diagnostic tests when studying TB in meerkats. Weaknesses in individual test performances, and the complexities of this chronic disease contribute to potential pitfalls in interpretation, which have been highlighted. An important message from this study is that despite imperfect conditions, valuable information can be obtained about test performance when utilising multiple different methods of evaluation and incorporating outcomes into those analyses. Such approaches are recommended when

  95% confidence interval. DPP: dual-path platform; IPRA: IP-10 release assays.

Figure 1 .Figure 1 .

 11 Figure 1. Parallel testing (using culture, serology, and a test of cell-mediated immunity) of 44 individuals tested on multiple occasions. Each row represents an individual animal, and each circle, a separate sampling point. Positive tests are shown in red, and negative in blue. The sampling points are shown in the order in which they were collected, with the earliest on the left, and the most recent sample shown on the right. Although the time between sampling points varied, the minimum interval between two samples was three months. Nine individuals changed from positive to negative

Figure 2 .

 2 Figure 2. Association between predicted risk of infection and diagnostic test results. The percentage of animals testing positive to each test is shown in each of four risk categories, with 95% confidence intervals. Risk categories are defined by age (animals aged over 2 years were categorised as Old, and those below 2 years as Young) and social group history of disease (groups having had clinical cases being "Pos", and those without being "Neg"). Culture results are based upon a tracheal wash culture, cell-mediated immunity (CMI) results are those from the IPRA (IP-10 release assays), and serology results are based upon the DPP (dual-path platform) assay. A test was considered positive on parallel test interpretation if it tested positive to one or more of the individual tests. The results of tests of cell-mediated immunity and the parallel test interpretation reflect an individual's risk of being infected, whereas no difference is seen between risk groups when testing with serology or culture. As indicated by the *, for both the CMI and the parallel tests, there is a significant difference (p < 0.05) between the highest risk group (Old/Pos) and both the Young/Pos and the Young/Neg groups.

Figure 2 .

 2 Figure 2. Association between predicted risk of infection and diagnostic test results. The percentage of animals testing positive to each test is shown in each of four risk categories, with 95% confidence intervals. Risk categories are defined by age (animals aged over 2 years were categorised as Old, and those below 2 years as Young) and social group history of disease (groups having had clinical cases being "Pos", and those without being "Neg"). Culture results are based upon a tracheal wash culture, cell-mediated immunity (CMI) results are those from the IPRA (IP-10 release assays), and serology results are based upon the DPP (dual-path platform) assay. A test was considered positive on parallel test interpretation if it tested positive to one or more of the individual tests. The results of tests of cell-mediated immunity and the parallel test interpretation reflect an individual's risk of being infected, whereas no difference is seen between risk groups when testing with serology or culture. As indicated by the *, for both the CMI and the parallel tests, there is a significant difference (p < 0.05) between the highest risk group (Old/Pos) and both the Young/Pos and the Young/Neg groups.

Table 1 .

 1 Inclusion criteria and justification for each analysis conducted as part of this study.

	Aim of the Analysis	Approach Used	Inclusion Criteria	Time Period	References
			All animals from which M. suricattae
	1. To determine the ability of each testing regime to correctly identify infected animals (test sensitivity)	Comparison with a reference test	was confirmed on culture of collected prior to death were post-mortem material, at any point during the study. All samples	September 2014-September 2016 inclusive (Full)	[3]
			included.
			The first sample collected for each
	2. To determine each testing regime's ability to correctly identify animals as infected (sensitivity) or uninfected (specificity)	Latent Class Analysis	individual was used. The time-period was limited to minimise the impact of changes over time on	September 2014-March 2015 inclusive	[20-23]
			prevalence.
				Full	[25]
	5. To investigate whether test results were predictive of clinical signs of disease	Survival analysis of time to showing clinical signs	All samples collected over the study used. period from wild individuals were	Full	[25]
	6. To investigate whether individuals' test results fluctuate over time	Analysis of consistency over time	All individuals sampled on more than one occasion were included.	Full	-
	7. To investigate whether test result is influenced by age	Analysis for association with age	All samples from individuals with a known date of birth were included.	Full	-
	8. To Investigate whether test results were
	detecting infection more often in the most
	susceptible animals

3. To quantify the agreement between the results from different tests

Calculation of the kappa statistic All sampling events where there was more than one test were used. Full

[START_REF] Mchugh | Interrater reliability: The kappa statistic[END_REF]

4. To investigate whether test results were predictive of mortality Survival analysis of time to death All samples collected over the study period from wild individuals were used.

Animals (n = 126)

	Univariable	Hazard Ratio	95% Confidence Interval	p-Value
	Dominance	No		0.966
		Yes	0.99	0.64-1.53
	Sex	Female		0.813
		Male	1.05	0.711.54
	Age	<6 months		0.678
		6-12 months	1.03	0.55-1.92
		>12 months	0.85	0.49-1.45
	Serology	Neg a		0.039
		Pos b	2.44	1.04-5.70
	Cell-Mediated immunity	Neg a		0.107

Table 4 .

 4 Cont.

			Model a-All Animals (n = 126)
	Univariable	Hazard Ratio	95% Confidence Interval	p-Value
		Pos b	1.45	0.92-2.28
	Culture	Neg a			0.013
		Pos b	14.84	1.76-125.5
	Parallel	Neg a			0.010
		Pos b	1.75	1.14-2.67
	Multivariable		Hazard Ratio	95% Confidence Interval	p-value	Likelihood ratio for model, p value
	Serology + Frailty Sampling group	Neg a				<0.0001
		Pos b	1.29	0.51-3.31	0.59
	Variance of frailty term	1.126
	Cell-Mediated
	immunity + Frailty	Neg a
	Social group					<0.0001
		Pos b	1.10	0.65-1.87	0.72
	Variance of frailty term	1.939
	Culture + Frailty Social group	Neg a				<0.0001
		Pos b	1.92	0.22-16.60	0.55
	Variance of frailty term	1.075
	Parallel + Frailty Social group	Neg a				<0.0001
		Pos b	1.30	0.80-2.11	0.29
	Variance of frailty term	1.028
			Model b-Excluding euthanised animals (n = 112)
	Univariable		Hazard Ratio	95% Confidence Interval	p-value
	Dominance	No			0.780
		Yes	1.06	0.69-1.65
	Sex	F			0.820
		M	1.05	0.70-1.56
	Age	<6 months			0.897
		6-12 months	0.93	0.48-1.79
		>12 months	0.88	0.51-1.53
	Serology	Neg a			0.116
		Pos b	2.09	0.83-5.27
	CMI	Neg a			0.09
		Pos b	1.48	0.93-2.34
	Culture	Neg a			0.008
		Pos b	18.66	2.13-163.60

Table 4 .

 4 Cont.

			Model a-All Animals (n = 126)
	Univariable	Hazard Ratio	95% Confidence Interval	p-Value
	Parallel	Neg a			0.011
		Pos b	1.75	1.14-2.68
	Multivariable		Hazard Ratio	95% Confidence Interval	p-value	Likelihood ratio for model, p value
	Serology + Frailty Sampling group	Neg a				<0.0001
		Pos b	1.04	0.38-2.86	0.94
	Variance of frailty term	1.126
	CMI + Frailty Social group	Neg a				<0.0001
		Pos b	1.17	0.68-1.99	0.57
	Variance of frailty term	1.809
	Culture + Frailty Social group	Neg a				<0.0001
		Pos b	2.37	0.26-21.39	0.44
	Variance of frailty term	1.013
	Parallel + Frailty Social group	Neg a				<0.0001
		Pos b	1.33	0.81-2.17	0.26
	Variance of frailty term	0.972

a Neg, Negative. b Pos, Positive.

Acknowledgments:

We thank Marta Manser of the University of Zurich, and Dave Gaynor, site manager for the Kalahari Research Trust for providing access to habituated animals and field facilities, and Tim Vink for assistance with the database. We are grateful to the Northern Cape Department of Environment and Nature Conservation for permission to carry out research and to the owners of farms neighbouring the Kuruman River Reserve for allowing us to work on their land. Thanks to the field managers at the time of the study: Sky Bischoff-Mattson, Lyndsey Marris, Chris Duncan, and Laura Meldrum, as well as to Helen Spence-Jones, research assistant, and all the volunteers at the KMP who contributed towards collection of observational data.

Data Availability Statement: Data presented in this study is publicly available at https://figshare. com/s/45bd5369a97b4c2489bf (accessed on 25 October 2021).

working with wildlife datasets, which, due to real-world conditions, are often collected with imperfect sampling methods, using unvalidated tests, and yet are uniquely valuable.

Supplementary Materials:

The following are available online at https://www.mdpi.com/article/10 .3390/ani11123453/s1. Supplementary Table S1: Survival analysis of time until the appearance of visual signs of tuberculosis, Supplementary Figure S1. Relationship between serological test result and meerkat age, Supplementary Figure S2. IFNγ Inducible-Protein 10 Release Assay results for individuals in 4 social groups (F, N, Q, and Z).

Author Contributions: Conceptualization, S.J.P. and J.A.D.; methodology, S.J.P.; formal analysis, S.J.P. and T.V.; investigation, S.

Conflicts of Interest:

The authors declare no conflict of interest.