PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 4, Winter 211 PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS Dedicated to Herb Freedman on the occasion of his seventieth birthday K. P. HADELER ABSTRACT. We consider the problem of identifying the time-dependent transmission rate from incidence data and from prevalence data in epidemic SIR, SIRS, and SEIRS models. We show closed representation formulas avoiding the computation of higher derivatives of the data or solving differential equations. We exhibit the connections between the formulas given in several recent papers. In particular we explain the difficulties to estimate the initial number of susceptible or, equivalently, the initial transmission rate. 1 Introduction Consider the standard epidemic SIR(S) model, usually named after Kermack and McKendrick, (1.1) S = β(t)si +γ(t)r, I = β(t)si ν(t)i, R = ν(t)i γ(t)r, for susceptible S, infected I, and recovered R, with constant population size S +I +R 1, and time-dependent coefficients β >, ν >, and γ. Assume that the model describes the time course of an epidemic or endemic disease on an observation interval [, L]. The transmission rate β, the recovery rate ν, and the rate of loss of immunity γ depend on time. Suppose that the time course of the disease is known, can the parameters be determined from the data? We have three functions S, I and R and three parameter functions β, ν and γ. Since R 1 S I, we have essentially only two functions S and I, the system can be written Copyright c Applied Mathematics Institute, University of Alberta. 343

2 344 K. P. HADELER as (1.2) S = β(t)si +γ(t)(1 S I), I = β(t)si ν(t)i. If either S(t) or I(t) is known, then using (1.2), the other function, I(t) or S(t), can be obtained by solving a linear differential equation. Hence, the other function is determined up to a constant. Thus, the data provide only one function. We expect, that only one of the three coefficient functions β, ν and γ can be determined from the data. In a realistic deterministic model, the rate of loss of immunity γ is a constant (in a stochastic model it would be a random variable independent of time); this parameter can hardly be influenced by medical or social policies. The rate of recovery ν can be affected by medical treatment. If we assume that always the best possible treatment is applied, then also ν should be assumed as a constant. On the other hand, the transmission rate β can rapidly change over time due to seasonal effects, to adaptive behavior of the susceptible population or the application of public health policies. Thus, we have the problem to determine the transmission rate β(t) from the data S(t), I(t), R(t). This problem has been posed and answered to some extent in a seminal paper [4] of 29 which has been published recently [5]. Simpler solutions to the problem and to a number of similar problems have been found in [2], and the connection between the two approaches has been discussed in a recent paper [3]. The goal of the present note is to review and compare the various results, present even simpler proofs, and hopefully provide some additional insight. The paper is organized as follows. In Section 2, we assume that we have complete information about the time course of the susceptible for < t <. In Section 3, we consider the realistic problem that data are given on a bounded time interval. We derive representations of β(t) based on incidence or prevalence data. In Section 4, we try to estimate some constants that appear in the formulas. In Section 5, the approach is extended to a multi-group model. In Section 6, the endemic case is covered (in particular the periodic case), and in Section 7, an SEIR model is treated as a further example. Finally, in Section 8 we explain the approach of [4] which is based on solving a Bernoulli equation and requires higher order derivatives of the data. 2 Recovering the transmission rate on the time axis We assume γ and ν > constant. Then (1.2) becomes the SIR model

3 PARAMETER ESTIMATION IN EPIDEMIC MODELS 345 for an epidemic with variable transmission rate (2.1) S = β(t)si, I = β(t)si νi, where ν is known and β(t) should be determined. In the model system (2.1), the prevalence I can be replaced by the incidence (2.2) w = β(t)si as a dependent variable. Then the equation (2.3) w = S holds, and the system (2.1) assumes the form (2.4) S = w, w = w [ β (t) β(t) +β(t)s ν w S ]. The S, I-system and the S, w-system are equivalent descriptions of the time course of the epidemic as long as β(t) is positive and differentiable. We underline that data may be given as prevalence data (representing I(t)) or incidence data (representing w(t)) and that it is absolutely crucial to take the difference between these two scenarios into account. In the model (2.1), the variables are the susceptible S(t) and the prevalence I(t). Assume that the time course of the susceptible part of the population S(t) is known for all t and that there are some infected, i.e., I(t). We have a decreasing continuously differentiable function S(t) with S( ) = 1. The incidence w(t) is given by (2.2), and the prevalence satisfies the equation (2.5) I +νi = w. For any t, we have ] (2.6) I(t) = e [I(t νt )e νt e ντ S (τ)dτ, t

4 346 K. P. HADELER and hence for t, (2.7) I(t) = e ν(t τ) S (τ)dτ. In the first equation of (2.1) we solve for β, (2.8) β(t) = S (t) S(t)I(t), and get finally a formula for β(t), S (t) (2.9) β(t) = S(t) e ν(t τ) S (τ)dτ which depends only on S and its derivative. The function β(t) is positive for all finite t. From the formula (2.3) we get (2.1) S(t) = 1 w(τ)dτ, and then, using (2.3), a formula that represents the transmission rate β(t) in terms of the incidence w(t), w(t) (2.11) β(t) = [1 w(τ)dτ] e ν(t τ) w(τ)dτ. We have two formulas for β(t), formula (2.9) in terms of the susceptible and formula (2.11) in terms of the incidence. If we replace w in (2.5), then we get β(t) in terms of the prevalence I(t). Hence, the problem to recover the transmission rate from data appears to be solved. Later, we shall see that additional problems arise if data are given only on a bounded time interval. Notice that the basic reproduction number R does not play a role in this context. The approach works if there is an outbreak (the prevalence increases to a maximum and then decreases to zero) and also if there is no outbreak (the prevalence decreases immediately to zero). Now we make the hypothesis that the function S(t) assumes its limits (2.12) lim S(t) = 1, lim S(t) = S (,1) t t

5 PARAMETER ESTIMATION IN EPIDEMIC MODELS 347 exponentially fast. We determine the behavior of the function β(t) as given by (2.9) at the limits t ±. For t, the function S(t) goes to 1. Hence, it remains to discuss the quotient (2.13) e νt S (t) eντ S (τ)dτ. Assume S (t) e λt for t, with λ >. Then β(t) ν + λ for t. Assume that S(t) S +κe µt for t, with µ,κ >. If µ < ν then β(t) (ν µ)/s for t, otherwise, β(t). For comparison, consider the case of constant β with β > ν (basic reproduction number β/ν > 1). Then (from the eigenvalues of the Jacobian matrix) for the exact solution we know λ = β ν and µ = ν βs. As an explicit example, we choose, with ν = 1 and S = κ (,1), 1 1+κet (2.14) S(t) = (1 κ) +κ = 1+et 1+e t. Then (2.15) (2.16) e t w(t) = (1 κ) (1+e t ) 2, [ I(t) = (1 κ) e t log(1+e t ) 1 1+e t ], and (2.17) β(t) = e 2t (1+κe t )[(1+e t )log(1+e t ) e t ]. The function β decreases from β( ) = 2 to β(+ ) =. 3 Data on a bounded interval The situation of Section 2 is not realistic. In practice, there are data on infected cases on a bounded time interval. We assume that the observation interval is [, L] with L >. Typically, there are no source data on the remaining susceptible or recovered individuals. We underline (with respect to concrete data sets)

6 348 K. P. HADELER that it is extremely important to know whether the data are prevalence or incidence (notification) data. Prevalence I(t) and incidence w(t) are connected by the equation (3.1) w(t) = I (t)+νi(t). If the incidence w(t) is known, then the prevalence I(t) can be obtained by solving a differential equation (3.1), provided one value of I, say I(), is known. If the prevalence I(t) is known, then the incidence w(t) can be obtained from (3.1) by differentiation. Recall that numerical differentiation is a rather unstable process. The most natural first step to recover β(t) from any data is to use the equation (2.2) and solve for β(t), (3.2) β(t) = w(t) S(t)I(t). Now we have a very simple situation: If the three functions w(t), S(t) and I(t) are available, then we can recover β(t) from the formula (3.2). But we know that we can express the functions S(t) and I(t) in terms of the function w(t). From the equation (2.3) we get (3.3) S(t) = S() and hence, (3.4) β(t) = w(τ)dτ, w(t) I(t)[S() w(τ)dτ]. From the differential equation (3.1), we get (3.5) I(t) = I()e νt + e ν(t τ) w(τ)dτ. We introduce the expressions (3.3) and (3.5) into the formula (3.2) and get what we call the incidence formula. Incidence Formula: w(t) (3.6) β(t) = [S() w(τ)dτ][i()e νt + e ν(t τ) w(τ)dτ].

7 PARAMETER ESTIMATION IN EPIDEMIC MODELS 349 Notice that the formula (2.11) is a limiting case of the formula (3.6) in the situation, where we know that at the beginning of observation (in this case t = ) we have S = 1 and I =. The expression (3.6) appears as the basic formula for all recovery attempts. It shows that the transmission rate β(t) can be recovered from incidence data provided two more numbers are known, the number of susceptible S() and the prevalence I() at the beginning of observation. We underline that these two numbers cannot be obtained from the incidence data w(t). The estimation of these two numbers requires particular attention, see Section 4. Now, we proceed to other representations of the function β(t). Adding the equations in (2.1), we find (3.7) S +I = νi, and by integration ( (3.8) S(t) = S() I(t) I()+ν ) I(τ)dτ. We use this expression for S(t) and the expression (3.1) for w(t) in (3.6) and get what we call the prevalence formula. Prevalence Formula: (3.9) β(t) = I (t)+νi(t) I(t)[S() (I(t) I()+ν I(τ)dτ)]. This formula allows to recover the transmission rate β(t) from prevalence data I(t). Again, the two numbers S() and I() occur. Since we know I(t), wealsoknowi(). Butalsohere, theestimationofthenumbers() causes problems. Finally, we use the formulas (3.2) and (3.1) at t = to express S() in terms of β(), (3.1) S() = w() β()i() = I ()+νi(). β()i() If we use this formula in equation (3.9), then we get a version of the prevalence formula where S() does not occur, however the unknown quantity β().

8 35 K. P. HADELER Prevalence Formula using β(): (3.11) β(t) = I (t)+νi(t) I(t) β()i() I ()+νi() β()i()[i(t) I()+ν I(τ)dτ]. The incidence formula (3.6) and the prevalence formula (3.9) have been shown in [2]. In [3], it has been shown that the formula (3.11) follows from a more complicated formula in [4]; see Section 8. 4 Estimating the constants It is impossible to estimate S() from incidence or prevalence data unless observation starts at the very beginning of the epidemic (mathematically corresponding to observation at t = ). This fact has been underlined in [4] and [5], and these authors have suggested to estimate β() instead which makes sense in their context since they obtain β(t) as the solution of a differential equation. However, we can avoid the differential equation and, moreover, according to (3.1), the quantities S() and β() are connected by a very simple relation. In [4], it has been suggested to find an estimate for β() by a try and error method. But there is no goal function or optimization principle other than checking whether the graph of the reconstructed function β(t) looks plausible. Since finding β() and S() is mathematically equivalent, we suggest, as we did in [2], to estimate S(). There are two options. Either we are sure that we observe the epidemic from the very onset. Then it will not be a gross mistake to put S() = 1 and I() =. Or we know that the epidemic has gone on for t (,) until observation started at t =. In that case, we can hardly get a direct estimate for the number of remaining susceptible at t =. We can expect, by statistical sampling, to get an estimate for R(), i.e., the number of individuals who have recovered from the infection during (, ). Once we have R(), and also I(), we have S() = 1 I() R(). As we have pointed out, there is no way to get I() from incidence data. 5 Multi-group models The standard multi-group SIR model is an extension of the system (2.1) to n groups. It has the form (5.1) S = S BI, I = S BI DI,

9 PARAMETER ESTIMATION IN EPIDEMIC MODELS 351 where S and I are vectors, and denotes the component wise product of vectors, (x y) i = x i y i. The matrix B(t) = (b jk (t)) is the matrix of transmission rates, and D = (ν j δ jk ) is the diagonal matrix of recovery rates. The vector of incidences is (5.2) w = S BI. Hence, we have the equation (5.3) S = w and also (5.4) I +DI = w. Integrate (5.3), (5.5) S(t) = S() Together with (5.2), we get the equation (5.6) ( S() w(τ)dτ. ) w(τ)dτ (B(t)I(t)) = w(t). The equation has been derived in [3] for n = 2 groups (following the approach of [2] and not that of [4]). If there is only one group, then the equation (5.6) becomes (compare the equations (3.2) (3.6)) (5.7) ( S() ) w(τ)dτ β(t)i(t) = w(t) and has a unique solution. The system (5.6) has n equations for n 2 unknowns, in coordinate notation (5.8) b jk (t)i k (t) = k ( S j () w j (τ)dτ) 1 w j (t). In order to get estimates for B, one must make further assumptions. In [3], the number of four parameters is reduced to two by choosing one value for one intra-group rate and another for all the other rates.

10 352 K. P. HADELER A more standard approach is separability, B = β(t)d T, where β(t) is a column vector of susceptibilities and d T is a row vector of infectivities. Here, we assume B(t) = β(t)e T with e T = (1,...,1). Then b jk (t) = β j (t) for all k, and the problem is reduced to finding the parameter functions β j (t) from the equations (5.9) β j (t) k as I k (t) = ( S j () 1 (5.1) β j (t) = k I k(t) w j (τ)dτ) 1 w j (t) w j (t) S j () w j(τ)dτ. In the case of one group, this formula reduces to the prevalence formula (3.4). 6 The endemic case We compare the epidemic situation (2.1) with a simple model for an endemic disease (6.1) S = µ β(t)si µs, I = β(t)si (ν +µ)i, with demographic replacement. The formula (3.6) can be easily generalized (as can be the formulas (3.9) and (3.11)) to cover also this scenario. The function β(t) can be represented as in formula (3.2), where now, (6.2) (6.3) I(t) = I()e (µ+ν)t + S(t) = S()e µt + e (µ+ν)(t τ) w(τ)dτ, e µ(t τ) [µ w(τ)]dτ. Of particular interest is the periodic case. If µ and ν are constant and β(t) is periodic with minimal period ω such that ω (β(t) ν)dt > then there is a periodic infected solution. It is also known that there may be subharmonic bifurcations, i.e., there may be stable solutions with minimal period 2ω. These play a role in understanding the twoyear cycles in measles [1]. On the other hand, in the model (6.1), if we have a nontrivial periodic solution (S(t), I(t)) with minimal period ω, then from (6.4) β(t) = I (t)+(ν +µ)i(t) S(t)I(t)

11 PARAMETER ESTIMATION IN EPIDEMIC MODELS 353 it follows that β(t) has period ω (which does not exclude the possibility that β(t) has the period ω/2). As an example, we consider the situation that (6.5) S(t) = A+ǫcost. The equation S +I = µ µs (µ+ν)i can be seen as a differential equation for the function I. There is exactly one bounded solution which has the form (6.6) I(t) = B +γcost+δsint. We find B = µ(1 A) µ+ν, Then (6.7) β(t) = 1+µν +µ2 γ = (µ+ν) 2 +1 ǫ, δ = ν (µ+ν) 2 +1 ǫ. [(µ+ν) 2 +1][µ(1 A) µǫcost+ǫsint] (A+ǫcost)[ µ(1 A) µ+ν ((µ+ν)2 +1) (µ 2 +µν+1)ǫcost+νǫsint]. The elements of this 2-parameter family with parameters A and ǫ are feasible (in the sense that S(t),I(t),S(t) + I(t) (,1) for all t) if A (,1) (then also A+B < 1) and ǫ > is sufficiently small. The system (6.1) with β(t) = β (1 + κcost) has been investigated in [6]. An explicit solution seems not to be known. Here, we have an example of an endemic model with periodic transmission rate, where β(t), and also S(t) and I(t) are explicitly known. 7 SEIR model with demographic replacement The following SEIR model has been studied in [5] using the Bernoulli approach (which in this case requires third order derivatives of the prevalence data), and in [3], following an extension of the approach in [2], (7.1) S = µ β(t)si µs, E = β(t)si αe µe, I = αe νi µi.

12 354 K. P. HADELER The parameter α is the exit rate from the exposed compartment. Large values of α correspond to a short exposed phase. Suppose µ,α,ν are given. We want to recover the transmission rate β from prevalence data I(t). Define the function w(t) = β(t)si. In this situation, w(t) is the incidence of entering the exposed phase. Although in a practical situation this function cannot be observed, it is still useful as a mathematical tool. Note that the incidence of entering the infectious phase is αe. From the second equation in (7.1) we have (7.2) w = E +(α+µ)e, and from the last equation in (7.1) (7.3) E = 1 α [I +(ν +µ)i]. Hence, w can be expressed in terms of I, I and I as (7.4) w = 1 α [I +(ν +α+2µ)i +(ν +µ)(α+µ)i]. We have (7.5) β(t) = w(t) S(t)I(t) with (7.6) S(t) = S()e µt + Finally, we get (7.7) β(t) = e µ(t τ) [µ w(τ)]dτ. w(t)e µt I(t)[S()+ eµτ (µ w(τ))dτ] with w given by (7.4), which is the formula of [3], Theorem 2, written in such a way that one can easily take the limit to the SIR model. Indeed, if we scale the exposed phase as Ẽ = αe, (7.8) 1 S = µ β(t)si µs, αẽ = β(t)si Ẽ µ αẽ, I = Ẽ νi µi, andlettingα wefindẽ = β(t)si, andhence, (6.1). Andasα the formula (7.4) reduces to (3.1), and (7.7) becomes the formula (3.11).

13 PARAMETER ESTIMATION IN EPIDEMIC MODELS The approach of Pollicott-Weiss-Wang We describe the approach of [4]. From the second equation in (2.1) (8.1) S(t) = I (t)+νi(t). β(t)i(t) Differentiate (8.2) d I +νi dt β(t)i = S = β(t)si = β(t) I +νi β(t)i I. Calculate the derivative and find a Bernoulli equation (8.3) β pβ Iβ 2 = with (8.4) p = I I I 2 I(I +νi). Solve the Bernoulli equation by reduction to a linear equation, x = 1/β, x +px+i =, (8.5) 1 β(t) = x(t) = x()e P(t) e P(t) e P(τ) I(τ)dτ, with (8.6) P(t) = Finally, (8.7) β(t) = p(τ)dτ. β()e P(t) 1 β() ep(τ) I(τ)dτ. This is the algorithm presented in [4] and also in [5]. In [3], it has been observed that the function p can be written as (8.8) p(t) = [ ( I )] (t) log I(t) +ν,

14 356 K. P. HADELER and hence, the formulas of [4] can be greatly simplified. Indeed, (8.9) e P(t) = e p(τ)dτ = I (t) I(t) +ν I () I() +ν, (8.1) e P(τ) I(τ)dτ = I() I ()+νi() (I (τ)+νi(τ))dτ. Finally, we get the formula (3.11) where the second derivatives do not show up. REFERENCES 1. K. Dietz, The incidence of infectious diseases under the influence of seasonal fluctuations, Lecture Notes Biomath., Springer, 11 (1976), K. P. Hadeler, Parameter identification in epidemic models, Math. Biosci. 229 (211), A. Mummert, Studying the recovery algorithm for the time-dependent transmission rate(s) in epidemic models, J. Math. Biol., published online 2 June M. Pollicott, H. Wang and H. Weiss, Recovering the time-dependent transmission rate from infection data, Arxiv preprint arxiv: (29) and updates. 5. M. Pollicott, H. Wang and H. Weiss, Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem, J. Biol. Dynamics 6 (212), H. L. Smith, Subharmonic bifurcation in an SIR epidemic model, J. Math. Biol. 17 (1983), University of Tübingen, Mathematics/Biomathematics, Auf der Morgenstelle 1, 7276 Tübingen, Germany, address: hadeler@uni-tuebingen.de