Optimal sampling strategies for detecting zoonotic disease epidemics.

The early detection of disease epidemics reduces the chance of successful introductions into new locales, minimizes the number of infections, and reduces the financial impact. We develop a framework to determine the optimal sampling strategy for disease detection in zoonotic host-vector epidemiological systems when a disease goes from below detectable levels to an epidemic. We find that if the time of disease introduction is known then the optimal sampling strategy can switch abruptly between sampling only from the vector population to sampling only from the host population. We also construct time-independent optimal sampling strategies when conducting periodic sampling that can involve sampling both the host and the vector populations simultaneously. Both time-dependent and -independent solutions can be useful for sampling design, depending on whether the time of introduction of the disease is known or not. We illustrate the approach with West Nile virus, a globally-spreading zoonotic arbovirus. Though our analytical results are based on a linearization of the dynamical systems, the sampling rules appear robust over a wide range of parameter space when compared to nonlinear simulation models. Our results suggest some simple rules that can be used by practitioners when developing surveillance programs. These rules require knowledge of transition rates between epidemiological compartments, which population was initially infected, and of the cost per sample for serological tests.

where we denote the number of infected vectors by I V (t), the number of infected hosts by I H (t), the 7 total number of vectors by N V , the total number of hosts by N H , the transmission rate from hosts 8 to vectors by β V,H , and the transmission rate from vectors to hosts by β H,V . 9 For ease of notation, define to be the proportion of the infected vector (host) population. Then rewriting the above system, we 11 have that Similarly, Lastly, for ease of notation, redefine 14 β V,H = α and β H,V = γ.

Then our system becomes
The model variables and parameters are summarized in Table 2 in Text S3. S3.1 Basic analysis of (S16) 1 It is easy to see that system (S16) has two steady states, (V, H) = (0, 0) and (1, 1). Examining the 2 vector field of (S16), we see that

11
Then we may reparameterize our system (S16) as a function of V or as a function of H. We will 12 choose to reparameterize our system as a function of H. Dividing (S16a) by (S16b), we have the 13 auxiliary equation H − γ which we can solve by separation of variables: where c is some constant. Then solutions of (S16) lie within the level sets of the function In particular, given an initial condition (V 0 , H 0 ), the solution (V, H) of the initial value problem 2 (S16),

S3.3 Optimal sampling 5
Suppose that C(s V , s H ) is a strictly increasing cost function where s V denotes the number of vectors sampled and s H denotes the number of hosts sampled. Our goal is to find possible optimal sampling schemes s * = (s * V , s * H ) that maximize the probability of detecting a disease in a single sampling trial at a fixed time t, assuming that the vector and host population dynamics are known. With reference to Table 1 in Text S2, we see that there are three possible sampling schemes. First (Case 2 in Table  1 in Text S2), if there exists some s * such that s * V ≥ 0 and s * H ≥ 0 and then we may choose to sample both the vector and the host populations. Second (Case 4 in Table  1 in Text S2), if there exists some s * such that s * V ≥ 0 and s * H = 0 and then we may choose to sample only the vector population. Third (Case 4 in Table 1 in Text S2), if there exists some s * such that s * V = 0 and s * H ≥ 0 and then we may choose to sample only the host population. Since each of these cases depends on the 1 ratio V H , we now characterize this curve. We will first restate two useful equations and make some easy observations. Then, we will give two lemmas that elucidate some properties of the curve V (H) H .

3
First, by (S17), 1. If not, then there exists some 0 < M < 1 such that lim H→1 V (H) = M . Then by the analysis in 6 Section S3.1, it is easy to see that H 0 = 1, a contradiction to the uniqueness of solutions.

7
Suppose that the initial condition (V 0 , H 0 ) is given, suppose that only one of V 0 or H 0 is positive, and let (V, H) be the solution to this initial value problem. Let Note that by (S19). Then, since at most one of V 0 or H 0 is positive, any initial condition (V 0 , H 0 ) must satisfy exactly one of the above conditions. Since the sign of I(V 0 , H 0 ) implied in the above relations is unique for each class of initial condition (V 0 , H 0 ), we have that We now prove two lemmas that are useful in characterizing the curve V (H) H . Proof. Suppose that Then by (S19), Since the function F (x) = −x − ln(1 − x) is strictly increasing for x ∈ (0, 1), the above equation 14 implies that V (H * ) = H * . We now show that there exists some H * ∈ (0, 1) that solves (S20).

16
In addition, F (0) = 0 and lim H→1 F (H) = ∞. Then, since c, α and γ are constants, there exists S10 some unique H * ∈ (0, 1) that solves (S20) if and only if c = 0 and the sign of c is the same as the 1 sign of 1 α − 1 γ . It remains only to be shown that H * ∈ (H 0 , 1).
and supposeH ∈ (H 1 , H 2 ) such that d dH V (H) H H=H = 0. Since the denominator of (S21) is strictly positive, it must be true that by (S18) and d dH We claim that d dH x 4 is an increasing function for x ∈ [0, 1] and since H < V we have that for all H ∈ (H 1 , H 2 ). Then if there exists someH ∈ (H 1 , H 2 ) ⊆ (0, 1) such that (S22) holds, it is 1 unique.

2
Assuming that the disease starts in either the vector population or the host population (not then the claim holds by (S21).  ≥ 0 for all H ∈ (H 0 , 1). 16 We summarize the above cases in Table 3 in Text S3 and illustrate them in Figures for all H ∈ (H 0 , 1). By the analysis in Section S3.3, we choose to sample only the host population. If then there exist some H 1 , H 2 ∈ (H 0 , 1), H 1 < H 2 such that Then at early stages of the epidemic (while H ∈ (H 0 , H 1 )), we should sample only the host popula-1 tion, at intermediate times (while H ∈ (H 1 , H 2 )) we sample only the vector population, and at late 2 times in the epidemic (while H ∈ (H 2 , 1)) we return to sampling only host population. As in Case 3 1, if H = H 1 or H = H 2 , then we should sample both the vector and host populations.

4
As in the main text, we find that there is a critical time at which we should switch our 5 sampling scheme. We can solve for this critical time numerically.