- Email: [email protected]
A general theory of early growth?
JID:PLREV AID:782 /DIS
[m3SC+; v1.235; Prn:22/08/2016; 12:32] P.1 (1-3)
Available online at www.sciencedirect.com
ScienceDirect Physics of Life Reviews ••• (••••) •••–••• www.elsevier.com/locate/plrev
Comment
A general theory of early growth? Comment on: “Mathematical models to characterize early epidemic growth: A review” by Gerardo Chowell et al. Thomas House 1 School of Mathematics, University of Manchester, Manchester M13 9PL, UK Received 12 August 2016; accepted 16 August 2016
Communicated by J. Fontanari
Chowell et al. [1] consider the early growth behaviour of various epidemic models that range from phenomenological approaches driven by data to mechanistic descriptions of complex interactions between individuals. This is particularly timely given the recent Ebola epidemic, although non-exponential early growth may be more common (but less immediately evident) than we realise. My comment is that a more formal description of the problem may enable a unified framework for consideration of early epidemic growth, which may be helpful for further empirical work. In particular, suppose we take the renewalequation approach of e.g. [2,3] in which we consider each individual to be labelled by a ‘type’ a ∈ A. We then keep track of the density of susceptible individuals with type a at time t , denoted s(t, a), and infectious individuals with type a at time t who were infected at time t − τ , denoted i(t, τ, a). A simple model of an epidemic is then ∞ ∂s = −s(t, a) i(t, τ, α)λ(τ )c(a, α) dτ dα , ∂t (t,a) (1) α∈A τ =0 ∂s , i(t, τ, a) = − ∂t (t−τ,a) where λ(τ ) is the mean infectiousness of an individual infected a time τ ago, and c(a, α) measures the intensity of infectious contact between individuals of types a and α. The ingredients of this relatively minimal model can then be tuned to give a large variety of different early growth regimes. First, consider an example that exhibits exponential growth [2]. If we let A = [0, amax ], c(a, α) = f (a)g(α) and assume negligible susceptible depletion then we obtain amax ∞
y(t, a) =
y(t − τ, α)λ(τ )f (a)g(α) dτ dα ,
α=0 τ =0
DOI of original article: http://dx.doi.org/10.1016/j.plrev.2016.07.005. E-mail address: [email protected]. 1 Work supported by the UK Engineering and Physical Sciences Research Council. http://dx.doi.org/10.1016/j.plrev.2016.08.006 1571-0645/© 2016 Elsevier B.V. All rights reserved.
(2)
JID:PLREV AID:782 /DIS
[m3SC+; v1.235; Prn:22/08/2016; 12:32] P.2 (1-3)
T. House / Physics of Life Reviews ••• (••••) •••–•••
2
where y = i dτ . We can then substitute in the Ansatz y(t, a) = f (a)exp(rt) to give amax
1 = L[λ](r)
g(α)f (α) dα ,
(3)
α=0
where L stands for Laplace transformation, giving a formula for the rate of exponential growth r. Secondly, consider a spatial wave [2,3]. If A = R, we have constant initial population density and c(a, α) = exp(−(a − α)2 ) then we can look for wave solutions of the form w(u) where u = a − vt and v is the velocity of the wave. Substituting into (1) and linearising gives ∞ ∞ w(u) =
w(u − ζ )λ(τ )e−(ζ −vτ ) dτ dζ . 2
(4)
ζ =−∞ τ =0
We use the Ansatz w(u) = exp(u), which gives ∞ ∞ 1 = Q() , for Q() =
e−ζ λ(τ )e−(ζ −vτ ) dτ dζ . 2
(5)
ζ =−∞ τ =0
In addition to these, we have to impose Q () = 0 (see [2]). Making the (mathematically convenient) choice λ(τ ) = π −1/2 e δ(τ − T ) gives the velocity v = 1/T and hence sub-exponential, linear growth of infectives: y(t) ∝ vt. Finally, consider the time taken to reach the early asymptotic behaviour. This was shown to be longer than the time taken to achieve significant susceptible depletion for an ordinary differential equation model with strongly assortative mixing between age groups in [4]. In more generality suppose A = {1, 2, . . . , n} and consider early behaviour so that the renewal equation takes the form ∞ y(t) =
λ(τ )Cy(t − τ ) dτ .
(6)
τ =0
If we suppose the mixing matrix C has eigenvalues {μa } and associated eigenvectors {za } then we can decompose the vector of infective proportions by type y(t) = a νa (t)za and obtain the result ∞ νa (t) =
νa (t − τ )μa λ(τ ) dτ .
(7)
τ =0
Using an Ansatz νa = na exp(ra t) we end up with early behaviour of prevalence y(t) = na era t , where 1 = μa L[λ](ra ) .
(8)
a
If r1 , say, has a very much higher magnitude than other rates then we expect exponential growth with rate r1 to establish itself rapidly, however if this is not the case then early growth may look like a mixture of exponentials for some time. The fact that three rather different regimes of early growth – exponential, linear, and mixture – can be derived in relatively short order from a renewal system of the form (1) therefore suggests that these equations may be quite a general way to categorise types of early epidemic behaviour. Such a theoretical approach might have improved predictive power compared to purely phenomenological approaches, but not require the relatively large number of parameters of mechanistic models. In summary, Chowell et al. are to be congratulated for a review paper that can inform epidemiological practice at the same time as stimulating such theoretical considerations.
JID:PLREV AID:782 /DIS
[m3SC+; v1.235; Prn:22/08/2016; 12:32] P.3 (1-3)
T. House / Physics of Life Reviews ••• (••••) •••–•••
3
References [1] Chowell G, Sattenspiel L, Bansal S, Viboud C. Mathematical models to characterize early epidemic growth: a review. Phys Life Rev 2016. http://dx.doi.org/10.1016/j.plrev.2016.07.005 [in this issue]. [2] Diekmann O, Heesterbeek JAP. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. New York: John Wiley and Sons; 2000. [3] Rass L, Radcliffe J. Spatial Deterministic Epidemics. Math Surv Monogr. American Mathematical Society; 2003. [4] Rhodes CA, House T. The rate of convergence to early asymptotic behaviour in age-structured epidemic models. Theor Popul Biol 2013;85:58–62.
[m3SC+; v1.235; Prn:22/08/2016; 12:32] P.1 (1-3)
Available online at www.sciencedirect.com
ScienceDirect Physics of Life Reviews ••• (••••) •••–••• www.elsevier.com/locate/plrev
Comment
A general theory of early growth? Comment on: “Mathematical models to characterize early epidemic growth: A review” by Gerardo Chowell et al. Thomas House 1 School of Mathematics, University of Manchester, Manchester M13 9PL, UK Received 12 August 2016; accepted 16 August 2016
Communicated by J. Fontanari
Chowell et al. [1] consider the early growth behaviour of various epidemic models that range from phenomenological approaches driven by data to mechanistic descriptions of complex interactions between individuals. This is particularly timely given the recent Ebola epidemic, although non-exponential early growth may be more common (but less immediately evident) than we realise. My comment is that a more formal description of the problem may enable a unified framework for consideration of early epidemic growth, which may be helpful for further empirical work. In particular, suppose we take the renewalequation approach of e.g. [2,3] in which we consider each individual to be labelled by a ‘type’ a ∈ A. We then keep track of the density of susceptible individuals with type a at time t , denoted s(t, a), and infectious individuals with type a at time t who were infected at time t − τ , denoted i(t, τ, a). A simple model of an epidemic is then ∞ ∂s = −s(t, a) i(t, τ, α)λ(τ )c(a, α) dτ dα , ∂t (t,a) (1) α∈A τ =0 ∂s , i(t, τ, a) = − ∂t (t−τ,a) where λ(τ ) is the mean infectiousness of an individual infected a time τ ago, and c(a, α) measures the intensity of infectious contact between individuals of types a and α. The ingredients of this relatively minimal model can then be tuned to give a large variety of different early growth regimes. First, consider an example that exhibits exponential growth [2]. If we let A = [0, amax ], c(a, α) = f (a)g(α) and assume negligible susceptible depletion then we obtain amax ∞
y(t, a) =
y(t − τ, α)λ(τ )f (a)g(α) dτ dα ,
α=0 τ =0
DOI of original article: http://dx.doi.org/10.1016/j.plrev.2016.07.005. E-mail address: [email protected]. 1 Work supported by the UK Engineering and Physical Sciences Research Council. http://dx.doi.org/10.1016/j.plrev.2016.08.006 1571-0645/© 2016 Elsevier B.V. All rights reserved.
(2)
JID:PLREV AID:782 /DIS
[m3SC+; v1.235; Prn:22/08/2016; 12:32] P.2 (1-3)
T. House / Physics of Life Reviews ••• (••••) •••–•••
2
where y = i dτ . We can then substitute in the Ansatz y(t, a) = f (a)exp(rt) to give amax
1 = L[λ](r)
g(α)f (α) dα ,
(3)
α=0
where L stands for Laplace transformation, giving a formula for the rate of exponential growth r. Secondly, consider a spatial wave [2,3]. If A = R, we have constant initial population density and c(a, α) = exp(−(a − α)2 ) then we can look for wave solutions of the form w(u) where u = a − vt and v is the velocity of the wave. Substituting into (1) and linearising gives ∞ ∞ w(u) =
w(u − ζ )λ(τ )e−(ζ −vτ ) dτ dζ . 2
(4)
ζ =−∞ τ =0
We use the Ansatz w(u) = exp(u), which gives ∞ ∞ 1 = Q() , for Q() =
e−ζ λ(τ )e−(ζ −vτ ) dτ dζ . 2
(5)
ζ =−∞ τ =0
In addition to these, we have to impose Q () = 0 (see [2]). Making the (mathematically convenient) choice λ(τ ) = π −1/2 e δ(τ − T ) gives the velocity v = 1/T and hence sub-exponential, linear growth of infectives: y(t) ∝ vt. Finally, consider the time taken to reach the early asymptotic behaviour. This was shown to be longer than the time taken to achieve significant susceptible depletion for an ordinary differential equation model with strongly assortative mixing between age groups in [4]. In more generality suppose A = {1, 2, . . . , n} and consider early behaviour so that the renewal equation takes the form ∞ y(t) =
λ(τ )Cy(t − τ ) dτ .
(6)
τ =0
If we suppose the mixing matrix C has eigenvalues {μa } and associated eigenvectors {za } then we can decompose the vector of infective proportions by type y(t) = a νa (t)za and obtain the result ∞ νa (t) =
νa (t − τ )μa λ(τ ) dτ .
(7)
τ =0
Using an Ansatz νa = na exp(ra t) we end up with early behaviour of prevalence y(t) = na era t , where 1 = μa L[λ](ra ) .
(8)
a
If r1 , say, has a very much higher magnitude than other rates then we expect exponential growth with rate r1 to establish itself rapidly, however if this is not the case then early growth may look like a mixture of exponentials for some time. The fact that three rather different regimes of early growth – exponential, linear, and mixture – can be derived in relatively short order from a renewal system of the form (1) therefore suggests that these equations may be quite a general way to categorise types of early epidemic behaviour. Such a theoretical approach might have improved predictive power compared to purely phenomenological approaches, but not require the relatively large number of parameters of mechanistic models. In summary, Chowell et al. are to be congratulated for a review paper that can inform epidemiological practice at the same time as stimulating such theoretical considerations.
JID:PLREV AID:782 /DIS
[m3SC+; v1.235; Prn:22/08/2016; 12:32] P.3 (1-3)
T. House / Physics of Life Reviews ••• (••••) •••–•••
3
References [1] Chowell G, Sattenspiel L, Bansal S, Viboud C. Mathematical models to characterize early epidemic growth: a review. Phys Life Rev 2016. http://dx.doi.org/10.1016/j.plrev.2016.07.005 [in this issue]. [2] Diekmann O, Heesterbeek JAP. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. New York: John Wiley and Sons; 2000. [3] Rass L, Radcliffe J. Spatial Deterministic Epidemics. Math Surv Monogr. American Mathematical Society; 2003. [4] Rhodes CA, House T. The rate of convergence to early asymptotic behaviour in age-structured epidemic models. Theor Popul Biol 2013;85:58–62.