HETEROGENEOUS MIXING IN EPIDEMIC MODELS

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number 1, Spring 212 HETEROGENEOUS MIXING IN EPIDEMIC MODELS FRED BRAUER ABSTRACT. We extend the relation between the basic reproduction number and the initial exponential growth rate of an epidemic to models with heterogeneous mixing, and show that an epidemic with heterogeneity of mixing may have a quite different epidemic size than an epidemic with homogeneous mixing and the same reproduction number and initial exponential growth rate. Determination of the final size of an epidemic if there is heterogeneous mixing requires additional data from the initial exponential growth stage of the epidemic. 1 Introduction The age of infection epidemic model, originally introduced by Kermack and McKendrick in 1927 [1] is a very general model in the sense that it includes a great variety of compartmental structures, including exposed periods, treatment, quarantine, and isolation. However, it is restricted to homogeneous mixing, and therefore does not include the possibility of subgroups with different activity levels and heterogeneous mixing between subgroups. It is possible to formulate age of infection epidemic models that include heterogeneous mixing [3, 5]. The initial exponential growth rate observed in an epidemic may be used to estimate the basic reproduction number and the final size of the epidemic [6, 9, 13]. The relation between the initial exponential growth rate can be generalized to the age of infection model [4, 7, 15]. In this note, we show that the same relation extends to age of infection models with heterogeneous mixing. However, we point out that heterogeneous mixing in an epidemic model can result in final size predictions that differ substantially from those of the corresponding homogeneous mixing model. The basic reproduction number is the largest eigenvalue of the This paper is in honour of Herb Freedman s 7th birthday. This research was supported in part by MITACS and NSERC. Copyright c Applied Mathematics Institute, University of Alberta. 1

2 2 FRED BRAUER next generation matrix; if the data used to estimate the initial exponential growth rate includes the distribution of disease cases during the exponential growth stage, it is possible to estimate the corresponding eigenvector and this makes it possible to calculate the final size of an epidemic with heterogeneous mixing. 2 The age of infection model A small variation [1, 8] of the original general Kermack-McKendrick age of infection model [1] in a population of constant total population size N is (1) S = a S N ϕ ϕ(t) = ϕ (t) + = ϕ (t) + t t a S(t τ) ϕ(t τ)a(τ) dτ N [ S (t τ)]a(τ) dτ. Here, a is the number of contacts sufficient to transmit infection per individual in unit time, S represents the number of susceptible members of the population, ϕ(t) represents the total infectivity of all infectious individuals at time t, ϕ (t) represents the total infectivity at time t of all individuals who were already infected at time t =, and A(τ) represents the mean infectivity of all individuals who had been infected τ time units previously, including those who are no longer infectious. The function A(τ) is the product of the function representing the fraction of infected members still infected at infection age τ and the relative infectivity at infection age τ. For this model, it is known [2] that the basic reproduction number, representing the number of secondary infections that would be caused by a single infective in a totally susceptible population, is given by (2) R = a A(τ) dτ. We have suggested in [4] that a general definition of an epidemic should be that the equilibrium in which all members of the population are susceptible is unstable. According to [7, 15], the initial exponential growth rate of infectives in the model (1) is the solution r of the equation (3) a e rτ A(τ) dτ = 1.

3 HETEROGENEOUS MIXING IN EPIDEMIC MODELS 3 Since instability of this equilibrium corresponds to a solution that grows exponentially initially, there is an epidemic in the model (1) if and only if the solution r of (3) is positive. If there is an epidemic described by a model (1) and the initial exponential growth rate r is measured, then so that (4) R = 1 a = e rτ A(τ) dτ = R A(τ) dτ, A(τ) dτ e rτ A(τ) dτ. We assume that the initial infection comes from outside the population being modeled, and then the limiting susceptible population size S is given by the final size relation [2] ln N [ = R 1 S ]. S N Thus an estimate of the initial exponential growth rate r leads to an estimate for the basic reproduction number and thence to an estimate for the final size of the epidemic, provided the infectivity function A(τ) is known. 3 A heterogeneous mixing model The basic age of infection model (1) extends the simple SIR epidemic model by allowing an arbitrary number of stages in the model and arbitrary distributions of stay in each stage. However, it does not include the possibility of subgroups with different activity levels and heterogeneous mixing between subgroups. This possibility can be included in a heterogeneous mixing age of infection model as in [3, 5]. Consider two subpopulations of sizes and, respectively, each divided into susceptibles and infected members with subscripts to identify the subpopulation. Suppose that group i members make a i contacts in unit time and that the fraction of contacts made by a member of group i that is with a member of group j is p ij, i, j = 1, 2. For the properties of the mixing matrix; see [12]. Then p 11 + p 12 = p 21 + p 22 = 1.

4 4 FRED BRAUER A two-group model may describe a population with groups differing by activity levels and possibly by vulnerability to infection, so that a 1 a 2 but A 1 (τ) = A 2 (τ). It may also describe a population with one group which has been vaccinated against infection, so that the two groups have the same activity level but different disease model parameters. In this case, a 1 = a 2 but A 1 (τ) A 2 (τ). An age of infection model with two subgroups is, as described in [5], (5) S 1 S 1 = a 1 [p 11 ϕ 1 + p 12 ϕ 2 ] ϕ 1 (t) = ϕ 1(t) + S 2 t [ S 1(t τ)]a 1 (τ)dτ S 2 = a 2 [p 21 ϕ 1 + p 22 ϕ 2 ] t ϕ 2 (t) = ϕ 2 (t) + [ S 2 (t τ)]a 2(τ)dτ. Here, ϕ i (t) is the total infectivity of infected members of group i. The next generation matrix, in the sense of [14], is a 1p 11 A 1 (τ) dτ a 1 p 12 A 2 (τ) dτ. a 2 p 21 A 1 (τ) dτ a 2 p 22 A 2 (τ) dτ Thus R is the largest root of (6) det p 11a 1 A 1 (τ) dτ λ p 12 a 1 A 2 (τ) dτ p 21 a 2 A 1 (τ) dτ p 22 a 2 A 2 (τ) dτ λ =. The analysis of the model (5) is analogous to the analysis carried out for the homogeneous mixing model (1) in [4]. In order to obtain an expression for the initial exponential growth rate if there is an epidemic,

5 HETEROGENEOUS MIXING IN EPIDEMIC MODELS 5 we first replace the model (5) by the limit system (7) S 1 S 1 = a 1 [p 11 ϕ 1 + p 12 ϕ 2 ] ϕ 1 (t) = [ S 1(t τ)]a 1 (τ) dτ S 2 S 2 = a 2 [p 21 ϕ 1 + p 22 ϕ 2 ] ϕ 2 (t) = [ S 2 (t τ)]a 2(τ) dτ. According to the asymptotic theory of [11], the asymptotic behaviour of (5) is the same as that of the limit system (7) for all initial functions ϕ 1 (t) and ϕ 2 (t) that tend to zero as t. In order to avoid the difficulties posed by the fact that there is a twodimensional subspace of equilibria ϕ 1 = ϕ 2 =, we include small birth rates in the equations for S 1 and S 2 and corresponding proportional natural death rates in each compartment, to give the system (8) S 1 = µn S 1 1 µs 1 a 1 [p 11 ϕ 1 + p 12 ϕ 2 ] ϕ 1 (t) = [ S 1 (t τ)]e µτ A 1 (τ) dτ S 2 S 2 = µ µs 2 a 2 [p 21 ϕ 1 + p 22 ϕ 2 ] ϕ 1 (t) = [ S 2(t τ)]e µτ A 2 (τ) dτ. We then linearize about the unique disease-free equilibrium obtaining S 1 =, ϕ 1 =, S 2 =, ϕ 2 =, (9) u 1 = a 1 [p 11 v 1 + p 12 v 2 ] µu 1 v 1 (t) = [p 11 v 1 (t τ) + p 12 v 2 (t τ)]e µτ A 1 (τ) dτ u 2 = a 2 [p 21 v 1 + p 22 v 2 ] µu 2 v 2 (t) = [p 21 v 1 (t τ) + p 22 v 2 (t τ 1 )]e µτ A 2 (τ) dτ,

6 6 FRED BRAUER and form the characteristic equation (the condition on r that (9) have a nonzero solution for u 1 (, v 1 (), u 2 (), v 2 ()), 2 3 r + µ a 11p 1 a 1p 12 R R a 1p 11 e (r+µ)τ a 1p 12 e (r+µ)τ A 1(τ) dτ 1 A 2(τ) dτ det =. a 2p 21 r + µ a 2p 22 R 6 R 4 a 2p 21 e (r+µ)τ p 22a 2 e (r+µ)τ 7 5 A 1(τ) dτ A 2(τ) dτ 1 There is a double root r = µ <, and the remaining roots of the characteristic equation are the roots of 2 R R p 11a 1 e (r+µ)τ A 1(τ) dτ 1 p 12a 1 e (r+µ)τ 3 A 2(τ) dτ det 4 5 =. R R p 21a 2 e (r+µ)τ A 1(τ) dτ a 2p 22 e (r+µ)τ A 2(τ) dτ 1 Since this is valid for every sufficiently small µ >, we may let µ and conclude that if there is an epidemic, corresponding to an unstable equilbrium of the model, there is a positive root of the characteristic equation 2 R R a 1p 11 e rτ A 1(τ) dτ 1 a 1p 12 e rτ 3 A 2(τ) dτ (1) det 4 5 R R =, a 2p 21 e rτ A 1(τ) dτ a 2p 22 e rτ A 2(τ) dτ 1 and the initial exponential growth rate is equal to this root. In the special case of proportionate mixing, in which p 11 = p 21 and p 12 = p 22, so that p 12 p 21 = p 11 p 22, the basic reproduction number is given by R = p 11 a 1 A 1 (τ) dτ + p 22 a 2 A 2 (τ) dτ, and the characteristic equation (1) reduces to (11) p 11 a 1 e rτ A i (τ) dτ + p 22 a 2 e rτ A i (τ) dτ = 1.

7 HETEROGENEOUS MIXING IN EPIDEMIC MODELS 7 There is an epidemic if and only if R > 1. In the special case of exponential infectivity distributions, P i (τ) = e α1τ, the growth rate equation is the quadratic equation p 11 a 1 r + α 1 + p 22a 2 r + α 2 = 1, and the initial exponential growth rate is the larger of the two roots. In fact, the condition that the basic reproduction number must be greater than 1 implies that the quadratic equation has exactly one positive root. While we have confined the description of the heterogeneous mixing situation to a two-group model, the extension to an arbitrary number of groups is straightforward. 4 Different models for the same epidemic Suppose we have an age of infection model for which we know the function A(τ) representing the total infectivity of individuals with age of infection τ. That is, we assume A(τ) = A 1 (τ) = A 2 (τ). If we assume that mixing is homogeneous, we are led to a model (1). If we measure (or estimate) the initial exponential growth rate r, then, as we have seen earlier, (12) R = A(τ) dτ e rτ A(τ) dτ. Now suppose, however, that there is in fact some heterogeneity in the model, specifically that there are two subgroups of sizes and with N = + and with contact rates a 1 and a 2, respectively, which mix in an arbitrary way. Then R is the largest root of (6) with A 1 (τ) = A 2 (τ), or (13) det p 11a 1 A(τ) dτ λ p 12 a 1 A(τ) dτ =. p 21 a 2 A(τ) dτ p 22 a 2 A(τ) dτ λ The equation for the initial exponential growth rate is (1) with A 1 (τ) = A 2 (τ), or (14) det p 11a 1 e rτ A(τ) dτ 1 p 12 a 1 e rτ A(τ) dτ =. p 21 a 2 e rτ A(τ) dτ p 22 a 2 e rτ A(τ) dτ 1

8 8 FRED BRAUER Comparing (13) and (14), we see that each of R / A(τ) dτ and 1/ e rτ A(τ) dτ is the largest root of the equation Thus, x 2 (a 1 p 11 + a 2 p 22 )x + a 1 a 2 (p 11 p 22 p 12 p 21 ) =. R A(τ) dτ = 1 e rτ A(τ) dτ, which implies the same relation (12) as for the homogeneous mixing model (1). Thus, if we assume heterogeneous mixing, we obtain the same estimate of the reproduction number from observation of the initial exponential growth rate, and this conclusion remains valid for an arbitrary number of groups with different contact rates. This result does not generalize to the the case A 1 (τ) A 2 (τ). With homogeneous mixing, knowledge of the basic reproduction number translates into knowledge of the final size of the epidemic. However, with heterogeneous mixing, even in the simplest case of proportionate mixing, the size of the epidemic is not determined uniquely by the basic reproduction number. 5 Some numerical examples If the mixing is proportionate [12], so that p 11 = p 21 = p 1 = a 1 q 1 a 2 q 2, p 12 = p 22 = p 2 =, a 1 q 1 + a 2 q 2 a 1 q 1 + a 2 q 2 and assuming that initial infections come from outside the population so that S 1 () = and S 2 () =, the final size relation for (5) with proportionate mixing is [3], ln S 1 ( ) = a 1 ln S 2 ( ) = a 2 A(τ) dτ A(τ) dτ ( [p 1 1 S ) ( 1( ) + p 2 1 S )] 2( ) ( [p 1 1 S ) ( 1( ) + p 2 1 S )] 2( ). ) ) In addition, a 2 ln S 1() S 1 ( ) = a 1 ln S 2() S 2 ( ).

9 HETEROGENEOUS MIXING IN EPIDEMIC MODELS 9 Numerical simulations indicate, for example, that with homogeneous mixing. a =.375, A(τ) dτ = 4, R = 1.5, N = 1, S() = 999, we have S( ) = For heterogeneous mixing with proportionate mixing, we choose a 1 and a 2 to give p 1 a 1 + p 2 a 2 =.375 so that R = 1.5. With the choice we obtain a 1 = , a 2 = , S 1 ( ) = 3.61, S 2 ( ) = 71.24, S 1 ( ) + S 2 ( ) = 74.85, a very different epidemic size. This example emphasizes that the reproduction number of an epidemic model is not sufficient to determine the size of the epidemic if there is heterogeneity in the model, even if the mixing is proportionate. Numerical simulations suggest that the minimum number of individuals who escape infection, or the maximum final size of the epidemic is achieved for homogeneous mixing. To study this, we assume that the parameters, and A(τ) dτ remain fixed and attempt to minimize S 1 ( )+S 2 ( ) as a function of a 1 and a 2 (with a 1 and a 2 constrained to keep p 1 a 1 + p 2 a 2 = k fixed and p 1 and p 2 as specified by proportionate mixing). Homogeneous mixing corresponds to a 1 = a 2. The constraint relating a 1 and a 2 implies that when a 1 = a 2 = k, we have Also, when a 1 = a 2, da 2 = 2a 1 k N1 ; da 1 k 2a 2 da 2 da 1 =. p 1 =, p 2 =, + + S 1 = S 2, dp 1 da 1 = kn.

10 1 FRED BRAUER If we differentiate the two equations of (15) with respect to a 1, we can calculate that d[s 1 ( ) + S 2 ( )] = da 1 when a 1 = a 2. We believe that a 1 = a 2 is the only critical point of S 1 ( ) + S 2 ( ), although we have not been able to verify this analytically. Since a 1 = and a 2 = correspond to homogeneous mixing in one subgroup and no infection in the other, these states have larger values of S 1 ( ) + S 2 ( ) than if a 1 = a 2, which corresponds to homogeneous mixing in the entire population. If a 1 = a 2 is the only critical point of S 1 ( ) + S 2 ( ), this critical point must be a minimum. We conjecture that this result is also valid if we allow arbitrary mixing, that is, we conjecture that for a given value of the basic reproduction number the maximum epidemic size for any mixing is obtained with homogeneous mixing. 6 Determination of the final size of an epidemic For an epidemic model with homogeneous mixing, if the infectivity period distribution is known, measurement of the initial exponential growth rate is sufficient information to determine the basic reproduction number, and then the final size relation gives the epidemic final size. For an epidemic model with heterogeneous mixing, the final size of the epidemic may be quite different from that of an epidemic with homogeneous mixing. This raises the question of what additional information that may be measured at the start of a disease outbreak would suffice to determine the epidemic final size if the mixing is heterogeneous. We consider a model (5) with A(τ) = A 1 (τ) = A 2 (τ), and we assume that A(τ) and the mixing matrix [ ] p11 p M = 12 p 21 p 22 are known. The next generation matrix is K = a 1p 11 A(τ) dτ a 1 p 12 A(τ) dτ, a 2 p 21 A(τ) dτ a 2 p 22 A(τ) dτ and R is the largest (positive) eigenvalue of this matrix. There is a corresponding eigenvector with positive components [ ] u1 u =. u 2

11 HETEROGENEOUS MIXING IN EPIDEMIC MODELS 11 Since the components of this eigenvector give the proportions of infectious cases in the two groups initially, it is reasonable to hope to be able to determine this eigenvector from early outbreak data. The general final size relation is [5] ln S 1() S 1 ( ) = a 1 ln S 2() S 2 ( ) = a 2 A(τ) dτ A(τ) dτ ( [p 11 1 S ) ( 1( ) + p 22 1 S )] 2( ) ( [p 21 1 S ) ( 1( ) + p 22 1 S )] 2( ). These equations may be solved for S 1 ( ) and S 2 ( ) if the contact rates a 1 and a 2 can be determined form the available information. The condition that the vector u with components (u 1, u 2 ) is an eigenvector of the next generation matrix corresponding to the eigenvector R is [ a 1 p 11 A(τ) dτu 1 + p 12 [ a 2 p 21 A(τ) dτu 1 + p 22 ] A(τ) dτ u 2 = R u 1 ] A(τ) dτ u 2 = R u 2. and since it is assumed that the function A(τ), the vector u, and the mixing matrix (p ij ) are known these equations determine a 1 and a 2. In vector notation, if we define the column vector a = [ a1 a 2 ] and the row vectors M j = [ p j1 p j2 ], we have a j = R M j u A(τ) dτ u j. When these values are substituted into the final size system, S 1 ( ) and S 2 ( ) may be determined. This argument extends easily to models with an arbitrary number of activity groups.

12 12 FRED BRAUER 7 Discussion Our principal result is that the relation between the basic reproduction number and the initial exponential growth rate in an epidemic is valid for models with arbitrary heterogeneous mixing, provided only that the infectivity of an individual is the same for all individuals and does not depend on activity levels. Thus in the estimation of the basic reproduction number from data giving initial exponential growth rates it is not necessary to know the contact structure. However, with heterogeneous mixing knowledge of the basic reproduction number is not sufficient to calculate the final size of the epidemic. There is evidence that for a given basic reproduction number homogeneous mixing maximizes epidemic size. This result appears to be counter to the idea that epidemics are often driven by superspreaders, individuals with many more infective contacts than average. However, our result says only that for a given reproduction number homogeneous mixing maximizes the severity of the epidemic. Superspreaders added to a homogeneously mixing population increase the reproduction number. One central idea in preparing for an epidemic is to hope for the best but to prepare for the worst. If one uses the observed initial exponential growth rate to estimate the basic reproduction number for a developing epidemic, the assumption of homogeneous mixing would correspond to the worst case. This could be useful for a first estimate. However, in judging how to apportion treatment it would be essential to take heterogeneity into account, and this may be done if in estimating the initial exponential growth rate one also measures the proportions of infectives developing initially in the different activity groups. REFERENCES 1. F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Biosc. 198 (25), F. Brauer, Age of infection models and the final size relation, Math. Biosc. & Eng. 5 (28), F. Brauer, Epidemic models with heterogeneous mixing and treatment, Bull. Math. Biol. 7 (28), F. Brauer and G. Chowell, On epidemic growth rates and the estimation of the basic reproduction number, Notas de Modelacion y metodos numericos. Modelacion Computacional de Sistemas Biologicos (M. A. Morales Vazquez, S. Botello Rionda, eds.), CIMAT, Guanajato, Mexico, to appear. 5. F. Brauer and J. Watmough, Age of infection epidemic models with heterogeneous mixing, J. Biol. Dynamics 3 (29),

13 HETEROGENEOUS MIXING IN EPIDEMIC MODELS B. Davoudi, B. Pourbohloul, J. C. Miller, R. Meza, L. A. Meyers, and D. J. D. Earn, Early real-time estimation of infectious disease reproduction number, arxiv (29). 7. O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester, O. Diekmann. H. Metz, and J. A. P. Heesterbeek, The legacy of Kermack and McKendrick, Epidemic Models: Their Structure and Relation to Data (D. Mollison, ed.) Cambrideg University Press, 1995, C. Fraser, C. A. Donnelly, S. Cauchemez, W. P. Hanage, M. D. Van Kerkhove, T. D. Hollingsworth, J. Griff, R. F. Baggaley, H. E. Jenkins, E. J. Lyons, T. Jombart, W. R. Hinsley, N. C. Grassly, F. Balloux, A. C. Ghani, and N. M. Ferguson, Pandemic potential of a strain of influenza A (H1N1): early findings, Science 324 (29), W. O. Kermack and A. G McKendrick, A contribution to the mathematical theory of epidemics, Proc. Royal Soc. London 115 (1927), J. J. Levin and D. F, Shea, On the asymptotic behavior of the bounded solutions of some integral equations, I, II, III, J. Math. Anal. Appl. 37 (1972), 42 82, , A. Nold, Heterogeneity in disease transmission modeling, Math. Biosci. 52 (198), M. G. Roberts and J. A. P. Heesterbeek, Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection, J. Math. Biol. 55 (27), P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 18 (22), J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers, Proc. Royal Soc. B 274 (27), Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada address: brauer@math.ubc.ca

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