Download Analytic calculation of finite-population reproductive numbers for

Analytic calculation of finite-population reproductive numbers for direct- and vector-transmitted
diseases with homogeneous mixing
Lindsay Keegan and Jonathan Dushoff
McMaster University
Introduction
Results
Vector-borne Transmission
The basic reproductive number, R0 , measures the expected number of new infections caused by a single
infectious individual in an otherwise totally susceptible population. R0 provides a foundation for understanding when interventions can eliminate disease.
The generating function for the expected number of
bites from one infectious vector is
Vector-to-vector transmission, Z(H):
From equation (2), we know that the number of
hosts infected by a single infectious vector is:
In a study of malaria reproductive numbers, Smith et al. (PLoS Biol 5:e42) pointed out that R0
calculations implicitly assume infinite population sizes, and are hard to interpret in real populations when
R0 approaches or exceeds the population size. They introduced the idea of measuring the typical number
of new infections per infectious individual for malaria invading a finite population and showed the number
of vectors infected per vector is not necessarily the same as the number of hosts infected per host. They
used simulations to estimate the vector-to-vector (Z) and host-to-host (R) reproductive numbers and show
how these numbers change when vector biting is heterogeneous.
1 − Pvh
φv1 (x) =
,
1 − Pvh x
(2)
Those I1 infected hosts go on to infect τhv vectors.
Thus, Z(H) is:
Hτvh
I1 =
.
H + τvh
H
Hτvh τhv
=
R0
Z(H) =
H + τvh
H + τvh
(3)
Using generating functions, we find that the distinct
number of hosts infected by m vectors is:
m
H
Im = H − H
.
(4)
H + τvh
Methods
Direct Transmission, R(N )
Assumptions
Calculation Framework
Direct-transmission:
The probability that any individual host escapes
infection is (1 − H1 )b . Thus, the expected number of
1 b
new infections is H(1 − (1 − H ) ).
• Finite population of size N
• Each infected host produces on average τ new
infections
– Some may fall on the same host
– Realized number of infections will be
smaller
Vector-borne transmission:
• Host population is finite with size H
• Vector population is effectively infinite
• A single infected host produces a geometrically distributed number of new infections
with mean τvh
We use probability distributions over numbers
of potentially infectious events p(a), P
and correa
sponding generating functions, φ(x) =
a p(a)x .
If φb (x) corresponds to the distribution of infectious
bites on the host population p(b), then the expected
number of infections, I, is:
X
b
I=
H(1 − (1 − 1/H) )p(b)
(1a)
b
= H(1 − φb (1 − 1/H))
τvh H
=
H + τvh
(1b)
(1c)
(6)
where Pvh is the probability corresponding to τvh .
The expected number of infections from one infectious vector is I1 = Hφv1 (1 − 1/H), substituting:
Here we calculate new, simple analytic formulas for these ”finite-population reproductive numbers”.
We show that Z is approximately the same as R0 over a wide range of parameters, while R diverges from
R0 earlier than expected. Both Z and R are reduced for a given R0 when the efficiency transmission from
vector to humans relatively is increased relative to transmission from humans to vectors.
We describe transmission using the transmission fac- To calculate R(N ), R(H), and Z(H), we trace intors: τhv , the average number of vector infections fections through one cycle of transmission for both
caused by a single infectious host and τvh , the av- hosts and vectors in the case of vector-borne diseases.
erage number of host infections caused by a single
infectious vector. In the infinite case, R0 = τhv τvh
and we call the ratio ρ = τhv /τvh . For a directly
transmitted disease, there is only one “transmission
factor”, τ , which is precisely R0 .
Hτvh
I1 =
.
H + τvh
Host-to-host transmission, R(H):
Starting from a single infected host, the number
of infected vectors is distributed geometrically with
mean Phv = τhv /(τhv + 1). Thus, the distribution of
infected mosquitoes is:
X
m
p(m) =
(1 − Phv )Phv
(8)
m
Since we assume that hosts produce infections the
same way that vectors produce infectious bites,
fromwe get I1 = N τ /(N + τ ), since R0 is exactly
τ:
Nτ
N R0
R(N ) =
=
N +τ
N + R0
(7)
(5)
We get the number of hosts infected P
by m vectors
from equation (4) . We find R(H) = m ph (m)Im .
Using generating functions, we find:
R0 H
τhv τm H
=
R(H) =
H + τvh τhv + τvh
R0 + H + τvh
(9)
Discussion
These simple analytic formulas shed light on the relationship between R0 and the finite-population reproductive numbers. Under the assumption that host population, and not vector population is limiting, we find
that:
• The host-based finite-population reproductive number, but not the vector-based one, is always less
than the host population
• For a given value of R0 , higher levels of τvh lead to lower finite-population reproductive numbers.
• R(H) diverges from R0 when R0 is around 1/4 of the population size H