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AIMS’ Journals
Volume X, Number 0X, XX 200X pp. X–XX
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN
EPIDEMIOLOGY
Jacques Demongeot
Jacques.Demongeot@agim.eu
FRE 3405, AGIM (AGeing Imaging Modeling), CNRS-UJF-EPHE-UPMF
Faculty of Medicine of Grenoble, 38700 La Tronche, France
Mohamad Ghassani and Mustapha Rachdi
Mohamad.Ghassani@agim.eu; Mustapha.Rachdi@agim.eu
FRE 3405, AGIM (AGeing Imaging Modeling), CNRS-UJF-EPHE-UPMF
Universit´e Pierre Mend`es France, UFR SHS, BP.47, 38040 Grenoble Cedex 09
Faculty of Medicine of Grenoble, 38700 La Tronche, France
Idir Ouassou and Carla Taramasco
iouassou@yahoo.fr; Carla.Taramasco@polytechnique.edu
FRE 3405, AGIM (AGeing Imaging Modeling), CNRS-UJF-EPHE-UPMF
Faculty of Medicine of Grenoble, 38700 La Tronche, France
(Communicated by the associate editor name)
Abstract. The aim of this paper is first to find interactions between com-
partments of hosts in the Ross-Macdonald Malaria transmission system. So,
to make clearer this association we introduce the concordance measure and then
the Kendall’s tau and Spearman’s rho. Moreover, since the population com-
partments are dependent, we compute their conditional distribution function
using the Archimedean copula. Secondly, we get the vector population par-
tition into several dependent parts conditionally to the fecundity and to the
transmission parameters and we show that we can divide the vector population
by using p-th quantiles and test the independence between the subpopulations
of susceptibles and infecteds. Third, we calculate the p-th quantiles with the
Poisson distribution. Fourth, we introduce the proportional risk model of Cox
in the Ross-Macdonald model with the copula approach to find the relationship
between survival functions of compartments.
1. Introduction. Advances in epidemics modelling have been done recently by in-
troducing demographic aspects (i.e. consideration of host populations whose global
size changes during the epidemic and the endemic history) as well as spatial aspects
about vector or infectious agents spread.
As examples of application, malaria endemics in South Mali or recurrent seasonal
influenza epidemics with irregular pandemics, are modelled by the same type of
mathematical framework, coming from the Ross-Macdonald tradition. We will fo-
cus in this paper to some improvements of this classical model, in terms of:
2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.
Key words and phrases. Archimedean copula, quantile, Malaria, transmission system, measure
of concordance.
The first author is supported by NSF grant xx-xxxx.
1
2 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
• mechanism of contacts, supposed to be assimilated to shocks, like in stochastic
chemistry.
• demography of non constant population, with a fecundity and mortality pa-
rameters specific to considered populations of hosts and vectors.
• differential infectious risk caused by a given population of hosts or vectors,
in comparison with other populations less or more subjected to infectious risk.
These improvements come from the fact that the Ross-Macdonald model has, in
the case of the malaria, the following interaction graph (cf.[9, 10, 13, 15, 21, 22, 25,
27, 28, 32, 37]):
S
1
Susceptible Hosts
I
1
Infective Hosts
S
2
Susceptible Vectors
E
2
Infected non Infective
Vectors
I
2
Infective Vectors
h
k
j
Figure 1. Interaction graph of the Ross-Macdonald model
The equations of the Ross-Macdonald model are:
dS
1
dt
=
−β
2
S
1
I
2
N
H
+ rI
1
dI
1
dt
=
β
2
S
1
I
2
N
H
− rI
1
dS
2
dt
= ω + f S
2
−
β
1
S
2
I
1
N
V
− δS
2
(1)
dI
2
dt
= fI
2
+ KE
2
− δI
2
dE
2
dt
= fE
2
+
β
1
S
2
I
1
N
V
− KE
2
− δE
2
where f (resp. δ) is the fecundity (mortality) rate of the vector population
(susceptible, infected and infective vectors being supposed to have the same fecun-
dity and mortality), β
1
(resp. β
2
) is the host (resp. vector) contagion parameter,
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 3
N
H
(resp. N
V
) is the host (resp. vector) population size, the ratio m = N
V
/N
H
is
the vector/host ratio, K (resp. r) is the vector (resp. host) speed of passage from
the infected/not infective (resp. infected) state to the infective (resp. susceptible)
state. If f = δ − µ (the fecundity compensating partly the mortality), the value of
R
0
, the mean number of secondary infected vectors for one infective host, is equal
to:
R
0
= β
1
β
2
K/[N
H
N
V
µr(K + µ)]
If R
0
> 1, assuming that ω = 1, N
H
= 1/µ and m(0) =
N
V
(0)
N
H
= 1, the stationary
state (0, 0, 0, 0, 0) is unstable and the endemic stable stationary state is stable and
reached after a transient epidemic wave for the values:
i
∗
1
= I
∗
1
/N
H
, i
∗
2
= I
∗
2
/N
V
, e
∗
2
= E
∗
2
/N
V
,
with i
∗
1
= (R
0
− 1)/(R
0
+ β
1
/µ), i
∗
2
= i
∗
1
r/[mβ
2
(1 − i
∗
1
)], e
∗
2
= i
∗
1
µr/[Kmβ
2
(1 − i
∗
1
)].
If µ is small with respect to K, then K/(K + µ) ≈ e
−µ/K
, where 1/K is
the mean sojourn time in the compartment E
2
(sporogonic cycle duration) and
R
0
= [β
1
β
2
/N
V
N
H
µr]e
−µ/K
.
In this paper, we are interested in analyzing interactions between vector compart-
ments in system (1), by using a copula approach. In fact, partitioning into several
compartments the studied population must be done in the model in a rigorous
way. For this aim, we focus on estimating the conditional quantiles of several com-
partments of the vector population. This can be made by estimating the marginal
distribution of the sizes of susceptible and infected not infectious vector populations
which may done through an Archimedean copula approach. It must be said that
copulas functions permit to model compartments associations and then highlight
their dependence structure which allows a better stratification of the population.
So, to make clearer these associations we introduce the concordance measure and
then the Kendall’s tau and Spearman’s rho. Afterward, we show that we can divide
the population by using the p-th percentiles, where p = 95%. Remark that it is
known that the copula functions are used essentially in finance and this is the first
time we use the information it brings to explain an epidemiological phenomena. We
believe that this study could open a window on several future developments.
Recall that, in statistics, a copula is used as a general way of formulating a
multivariate distribution in such a way that various general types of dependence
can be represented. The approach to formulating a multivariate distribution using
a copula is based on the idea that a simple transformation can be made on each
marginal variable in such a way that each such transformed variable has a uniform
distribution. Once this is done, the dependence structure can be expressed as a
multivariate distribution of the obtained uniform variables, and a copula is precisely
the multivariate distribution of these marginally uniform random variables. There
are many families of copulas which differ in the detail of the dependence they
represent. A parameterized family of distribution functions will typically depend
on several parameters which relate to the strength and form of the dependence.
A typical use of copulas consists in choosing such a family in order to define the
adequate multivariate distribution fitting the empirical distribution observed from
a sample of data, and then to derive the copula corresponding to this multivariate
distribution.
4 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
A copula is a multivariate joint distribution defined on the n-dimensional unit
cube [0, 1]
n
such that every marginal distribution be uniform on the interval [0, 1].
Let consider a copula function C for an n-dimensional random variable X = (X
1
, ...,
X
n
) defined on a probability space (Ω, Σ, P) with a joint distribution function F
X
such that for any x
1
, ..., x
n
∈ R, (cf. [3, 6, 18, 19, 20, 24, 33, 36]), we have:
F
X
(x
1
, ..., x
n
) = C(F
1
(x
1
), ..., F
n
(x
n
))
where F
i
is the marginal distribution function of X
i
for i = 1, . . . , n.
This paper is organized as follows. In Section 2, we give the definition of the
Archimedean copula and provide the two examples of the Gumbel and Clayton
copulas, we define and discuss the Kendall’s tau, the Spearman’s rho, and then, we
propose a measure of concordance. At the end of this Section, we give the formulas
of conditional distribution functions using Archimedean copulas. In Section 3, we
discuss the interaction between the vector compartments, susceptible and infected
not infectious, of the system (1), by providing a link between the Kendall’s tau
and the regression parameter of copulas. After, we will conduct a simulation study
to highlight the interaction between the distribution functions of these two com-
partments. Eventually, we will discuss the conditional quantiles from copulas. In
Section 4, we study an application in which we calculate the p-quantiles with any
distribution function and after with the Poisson distribution. In the last Section,
we introduce the proportional risk model of Cox in the Ross-Macdonald model (1)
with the copula approach to quantify the relationships existing between the survival
functions of the considered compartments.
2. Preliminaries.
2.1. Archimedean copula. The most important family of copulas is the Archime-
dean one [13]. This latter is defined as follows :
C(u
1
, ..., u
n
) =
φ
−1
(φ(u
1
) + · · · + φ(u
n
)) if φ(u
1
) + ... + φ(u
n
) ≤ 0
0 otherwise
(2)
where the generator of the copula φ is a twice continuously differentiable function
which satisfies:
φ(1) = 0, φ
(1)
(u) < 0 and φ
(2)
(u) > 0 for all u ∈ [0, 1]
n
where φ
(i)
denotes the ith order derivative of φ.
There are some popular Archimedean copulas:
• Clayton copula:
C(u, v) = (u
−α
+ v
−α
− 1)
−1
α
where α > 0 and φ(t) = t
−α
− 1
• Gumbel copula:
C(u, v) = exp{−[(− ln u)
α
+ (− ln v)
α
]
1
α
} where α ≥ 1 and φ(t) = (− ln t)
α
Let consider that U and V are two random variables defined on the probability
space (Ω, Σ, P) and uniformly distributed on [0, 1], and let X be the random variable
C(U, V ) valued in [0, 1].
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 5
The real α corresponds to a regression parameter.
In the case of Archimedean copula, the distribution function of X is given by:
F
X
(t) = t −
φ(t)
φ
(1)
(t)
In the case of the Gumbel copula, we have φ(t) = (− ln t)
α
, so φ
(1)
(t) =
−α
t
(− ln t)
α−1
.
Then:
F
X
(t) = t −
(− ln t)
α
−α
t
(− ln t)
α−1
= t +
t
α
(− ln t)
α−α+1
= t −
t ln t
α
and the probability density of X is given by:
f
X
(t) = 1 −
1
α
−
ln t
α
(3)
Figure 2. Distribution functions of Clayton bivariate copula
(resp. for α = 2, α = 5 and α = 10)
Figure 3. Distribution functions of Gumbel bivariate copula
(resp. for α = 2, α = 5 and α = 10)
2.2. Concordance, Kendall’s tau, Spearman’s rho, Copula. In order to de-
fine the concordance measure, we begin by defining the concordance (cf. [24]): two
observations (x
1
, y
1
) and (x
2
, y
2
) of a pair (X, Y ) of continuous random variables
are said to be concordant if both values of one pair are greater than the correspond-
ing values of the other pair, that is if x
1
< x
2
and y
1
< y
2
or if x
1
> x
2
and y
1
> y
2
;
and they are said to be discordant if for one pair one value is greater and the other
is smaller than the corresponding value of the other pair, that is if x
1
< x
2
and
y
1
> y
2
or x
1
> x
2
and y
1
< y
2
. The simple version of the measure of association
known as Kendall’s tau is defined in terms of concordance as follows (cf. [24]):
6 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
let (x
1
, y
1
), (x
2
, y
2
), ..., (x
n
, y
n
) denote a random sample of n observations from a
continuous random vector (X, Y ). The Kendall’s tau for the latter random sample
is defined by:
τ =
number of concordant pairs − number of discordant pairs
total number of pairs
(4)
Equivalently, τ may be written and interpreted as the difference between the
empirical probabilities of concordance and discordance for a pair of observations
(x
i
, y
i
) and (x
j
, y
j
), chosen randomly from the sample. The multivariate version of
Kendall’s tau will be defined similarly.
Let (X
1
, Y
1
) and (X
2
, Y
2
) be independent and identically distributed random vec-
tors, each with joint distribution function H. So the probability that (X
1
, Y
1
) and
(X
2
, Y
2
) are concordant is equal to P ((X
1
− X
2
)(Y
1
− Y
2
) > 0) and the probability
that these two vectors are discordant is equal to P ((X
1
− X
2
)(Y
1
− Y
2
) < 0), and
we have :
P ((X
1
− X
2
)(Y
1
− Y
2
) > 0) =
number of concordant pairs
total number of pairs
P ((X
1
− X
2
)(Y
1
− Y
2
) < 0) =
number of discordant pairs
total number of pairs
Then the multivariate generalization of Kendall’s tau is defined as the difference
between the probability of concordance and the probability of discordance:
τ = P ((X
1
− X
2
)(Y
1
− Y
2
) > 0) − P ((X
1
− X
2
)(Y
1
− Y
2
) < 0)
2.2.1. Kendall’s tau and copula relation.
Theorem 2.1. Let X and Y be continuous random variables whose copula is C.
Then the multivariate version of Kendall’s tau for X and Y , denoted by τ
X,Y
or
τ
C
, is given by:
τ
C
= 4
Z
1
0
Z
1
0
C(u, v)c(u, v)dudv − 1, where c(u, v) = ∂
2
C(u, v)/∂u∂v (5)
The proof is in Appendix A.
For computational purpose, there are alternative expressions for τ
C
. The integral
which appears in (5) can be interpreted as the expected value of the function C(U, V )
where U and V are random variables uniformly distributed on [0, 1] for which the
joint distribution function is C, i.e.,
τ
C
= 4 E (C(U, V )) − 1 = 4
Z
1
0
tf
C
(t)dt − 1 (6)
where f
C
(t) denotes the density function of C.
2.2.2. Spearman’s rho. The multivariate version of Spearman’s rho is based on the
concordance and discordance values. In order to obtain the population value of
this measure (cf.[24]), let (X
1
, Y
1
), (X
2
, Y
2
) and (X
3
, Y
3
) be three independent ran-
dom vectors whose components are continuous random variables with a common
joint distribution function H (whose marginal distribution are F and G) and let
C be a copula function. The multivariate version of Spearman’s rho is three time
the difference between the probabilities of concordance and discordance of the two
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 7
vector pairs (X
1
, X
2
) and (Y
1
, Y
3
), i.e., two vector pairs with the same margins,
but one vector has distribution function H, while the components of the other are
independent:
ρ = 3 [P ((X
1
− X
2
)(Y
1
− Y
3
) > 0) − P ((X
1
− X
2
)(Y
1
− Y
3
) < 0)]
The pair (X
1
, X
3
) could be used equally as well.
On the other hand, notice that while the joint distribution function of (X
1
, Y
1
) is
H(x, y), the joint distribution function of (X
2
, Y
3
) is F (x)G(y) (because X
2
and
Y
3
are independent), thus the copula of X
2
and Y
3
is the independence copula
(C(u
1
, u
2
) = u
1
u
2
).
Theorem 2.2. Let X and Y be continuous random variables whose copula function
is C. Then the multivariate version of Spearman’s rho for X and Y , denoted by
ρ
X,Y
or ρ
C
, is given by:
ρ
C
= 12
Z Z
I
2
u
1
u
2
dC(u
1
, u
2
) − 3
The proof is in Appendix B.
2.2.3. Measure of Concordance.
Definition 2.3. A measure of association, denoted M , between two continuous
random variables X and Y whose copula function is C, is said to be a measure of
concordance, if it satisfies the following properties (in which we denote M by M
X,Y
or M
C
when convenient):
• M
X,Y
is defined for every pair (X, Y ) of continuous random variables
• M
X,Y
∈ [−1, 1] with M
X,X
= 1 and M
X,−X
= −1
• M
X,Y
= M
Y,X
• If X and Y are independent, then M
X,Y
= 0
• M
−X,Y
= M
X,−Y
= −M
X,Y
• if C
1
and C
2
are copulas with C
1
≤ C
2
then M
C
1
≤ M
C
2
• if {(X
n
, Y
n
)}
n∈N
∗
is a sequence of pairs of continuous random variables with
copula functions C
n
and if C
n
converges pointwise to C then lim
n→+∞
M
C
n
=
M
C
Theorem 2.4. The multivariate versions of Kendall’s tau and Spearman’s rho are
two measures of concordance.
The proof is in Appendix C.
2.3. Multidimensional conditional distribution function using copulas.
Let X
1
, X
2
, ..., X
k
be random variables defined on a probability space (Ω, Σ, P),
where the joint probability density function f
k
of the vector X = (X
1
, X
2
, ..., X
k
)
is assumed to exist. By using the Archimedean copula construction, the joint prob-
ability density function may be written as follows:
f
k
(x
1
, ..., x
k
) =
∂
k
∂x
1
...∂x
k
φ
−1
{φ [F
1
(x
1
)] + · · · + φ [F
k
(x
k
)]}
= φ
−1(k)
{φ [F
1
(x
1
)] + · · · + φ [F
k
(x
k
)]}
k
Y
j=1
φ
(1)
[F
j
(x
j
)] F
(1)
j
(x
j
)
(7)
8 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
for all (x
1
, ..., x
k
) ∈ R
k
, where F
j
denotes the marginal distribution function of
X
j
.
Thus, the conditional density of X
k
given X
1
, ..., X
k−1
is given by:
f
k
(x
k
|x
1
, ..., x
k−1
) =
f
k
(x
1
, ..., x
k
)
f
k−1
(x
1
, ..., x
k−1
)
= φ
(1)
[F
k
(x
k
)] F
(1)
(x
k
)
φ
−1(k)
{φ [F
1
(x
1
)] + · · · + φ [F
k
(x
k
)]}
φ
−1(k−1)
{φ [F
1
(x
1
)] + · · · + φ [F
k−1
(x
k−1
)]}
(8)
Further, the conditional distribution function of X
k
given X
1
, ..., X
k−1
is also
given by:
F
k
(x
k
|x
1
, ..., x
k−1
) =
Z
x
k
−∞
f
k
(x|x
1
, ..., x
k−1
)dx
=
φ
−1(k−1)
{φ [F
1
(x
1
)] + · · · + φ [F
k
(x
k
)]}
φ
−1(k−1)
{φ [F
1
(x
1
)] + · · · + φ [F
k−1
(x
k−1
)]}
=
φ
−1(k−1)
{c
k−1
+ φ [F
k
(x
k
)]}
φ
−1(k−1)
(c
k−1
)
(9)
where c
k
= φ[F
1
(x
1
)] + · · · + φ[F
k
(x
k
)].
3. A case where the population sizes, S
1
and I
2
, are Poissonian random
variables. A simple way to prove the Poissonian character of the distribution of
the sizes S
1
and I
2
, considered as random variables, is to take into account only the
two compartments of random size S and I (for the sake of simplicity, we will omit in
the following the indices), assuming that there is at least one event (contact, birth,
death or recovering) in (t, t + dt), and by denoting by f, ν, µ and ρ respectively the
fecundity, contagion, mortality and recovering rates, we can write:
P
S(t + dt) = k, I(t + dt) = j
=
P
S(t) = k, I(t) = j
h
1 −
νkj − f k + µk − ρj
dt
i
+
"
ν(k + 1)(j − 1)P
S(t) = k + 1, I(t) = j − 1
+ (f(k − 1) + ρ(j + 1))
P
S(t) = k − 1, I(t) = j + 1
+ µ(k + 1)P
S(t) = k + 1, I(t) = j − 1
#
dt
(10)
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 9
and by multiplying equation (10) by k and summing over k and j, we obtain:
X
k,j≥0
k
h
P
S(t + dt) = k, I(t + dt) = j
− P
S(t) = k, I(t) = j
i
dt
=
d
h
P
k,j≥0
kP
S(t) = k, I(t) = j
i
dt
=
X
k,j≥0
"
− νk
2
jP
S(t) = k, I(t) = j
+ νk(k + 1)(j − 1)P
S(t) = k + 1, I(t) = j − 1
+ k(f(k − 1) + ρ(j + 1))P
S(t) = k − 1, I(t) = j + 1
+ (fk
2
+ ρkj)
P
S(t) = k, I(t) = j
− µk
2
P
S(t) = k, I(t) = j
+ µk(k + 1)P
S(t) = k + 1, I(t) = j − 1
#
(11)
Hence:
d
h
P
k,j≥0
kP
S(t) = k, I(t) = j
i
dt
≈
−
X
k,j≥0
h
νkjP
S(t) = k, I(t) = j
+ (fk + ρj)P
S(t) = k, I(t) = j
− µkP
S(t) = k, I(t) = j
i
(12)
and we get from (12) the following expectation equation:
dE(S)
dt
= fE(S) − νE(SI)dt − µE(S) + ρE(I) (13)
If the random variables S and I can be considered as uncorrelated, we can obtain
from (13) a deterministic differential equation ruling E(S) and E(I):
dE(S)
dt
= E(S)[f − µ − νE(I)] + ρE(I) (14)
Then, we get the Ross macroscopic equation for the deterministic variable
¯
S
(resp.
¯
I) which represent the size of the susceptible (resp. infected) population:
d
¯
S
dt
= −ν
¯
S
¯
I + (f − µ)
¯
S + ρ
¯
I (15)
By replacing k by k
2
in equation (12), we obtain more, the differential system
ruling the not centred moments of order 2, E(S
2
(t)), then we get the differential
system ruling the variance: V (S(t)) = E(S
2
(t)) − E
2
(S(t)), and we can draw the
confidence cylinder (or viability tube) around the expected trajectory. By replacing
k by s
k
in the case where f = µ (the host population is constant) and where I can
be considered as independent of S (which can be tested statistically and is observed
for example if the contagions rate ν is sufficiently small with respect to the whole
population sizes of hosts and vectors), we get the differential system ruling the
generating function ψ(s) of the distribution of S:
10 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
dψ
dt
≈ ρψ − ν
dψ
ds
this equation having a stationary solution: ψ = exp((ρ/ν)(s − 1)). The gener-
ating function ψ corresponds to a Poisson distribution for S, with expectation and
variance equal to ρ/ν, the last one being in general not negligible, which explains
the differences observed between the random and the deterministic models. The
same argument can be used for proving the Poissonian character of the distribution
of I:
As equation (15) similar to the equation relative to S
2
in system (1), it is possible to
obtain all other equations of system (1) from a microscopic random contact mech-
anism and by using the same type of arguments, to show that all variable involved
(S
1
, I
1
, S
2
, I
2
, E
2
) are Poissonian. Then, the copula approach can serve to show
the independency between the pair of variables (S
1
, I
2
) and (S
2
, I
1
), hypothesis
necessary to derive the system (1) from the microscopic equations.
In the following, the statistical sampling of S and I will be numerically obtained
from Poissonian distributions, whose parameter is fixed by their steady state value
in the deterministic version of the epidemic model (equation (1)). Near this steady
state solution, we can consider that the non correlation between S and I can be
postulated if the host and vector populations are sufficiently numerous and the
contagion rate is sufficiently small. This circumstance is checkable by the statistical
test associated to the Kendall’s tau, whose value is about zero near the steady state
and whose absolute value is near 1 during the transient phase of the trajectories
[34].
4. Quantiles regression based on the system of transmission of Malaria.
Recall that the regression function is the most widely used tool for describing multi-
variate relationships. Then, copulas functions can help to understand the full joint
distribution and thus be used to address some important applications, which we
tackle to explain in the sequel. In this part, we will use the Gumbel copula.
4.1. Interaction between compartments of hosts. Let X be the random vari-
able C(U, V ) defined on the probability space (Ω, Σ, P),where F
X
is the distribution
function of X. From (3) we obtain (cf. [33]):
E(X) =
Z
1
0
tf
X
(t)dt
=
1 −
1
α
Z
1
0
tdt −
1
α
Z
1
0
t ln tdt
=
1
2
−
1
2α
−
1
α
−1
4
=
2α − 1
4α
(16)
Then, from the equation (6) the Kendall’s tau is expressed in the following man-
ner:
τ
C
= 4
2α − 1
4α
− 1
=
α − 1
α
(17)
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 11
where C denotes the Gumbel copula. Notice that the parameter α measures a
degree of dependence. Recall that the higher it is, the strong is dependence between
the two studied variables.
4.1.1. Relationship between the Kendall’s tau and the equilibrium of the system of
transmission of Malaria. Let (S
21
, E
21
), (S
22
, E
22
), ..., (S
2n
, E
2n
) be calculated from
n successive independent observations of samples of m individuals extracted from
the part of the vector population made of the two compartments, susceptible and
infected non infectious: S
2i
(resp. E
2i
) is the number of susceptible (resp. infected
non infectious) vectors, counted in the ith sample observed in mosquitoes population
transmitting malaria [4, 12, 16, 23, 26, 35, 38].
From the equation (4), if the number of concordant pairs is equal to the number
of discordant pairs, so we will have the Kendall’s tau equal to 0 and the regression
parameter equal to 1 (τ
C
= 0, α = 1), then in this case the two compartments will
be non concordant (notion similar to uncorrelated), property observed in case of
stochastic independence, and for example when the system is in equilibrium (but
also in case of deterministic chaos, cf. [5]). However, if the number of concordant
pairs is equal to the total number of pairs, so we will have τ
C
= 1 and then we will
have the perfect concordance and so the system is in transient state. The sampling
used to calculate τ
C
can also be obtained by simulating Individual Based Models
(IBM) in which social networks allow to simulate all the possibilities of contact
between hosts and vector, by using the same simulation techniques as in stochastic
molecular kinetics [1, 2, 7, 17, 29, 30, 31]. An example of such a social network has
been simulated for explaining the contagion process of a social disease, the obesity
[11]. This social network has been simulated on Figure 4, in which the homophilic
rule is based on the fact that individuals tend to interact with those who resemble
them in terms of social behaviour.
In the system (1), when the contagion parameter β
1
or the kinetic parameter K
between the two compartments S
2
and E
2
increases, then the number of concordant
pairs increases, because in both cases the size of S
2
diminishes and the size of E
2
is growing, thus the Kendall’s tau increases, so we could expect that the system is
still in a transient state.
Let consider a simplified version of the system (1), given by:
dS
1
dt
= −β
2
S
1
I
2
+ rI
1
dI
1
dt
= β
2
S
1
I
2
− rI
1
dS
2
dt
= ω − β
1
S
2
I
1
− µS
2
(18)
dE
2
dt
= β
1
S
2
I
1
− (µ + K)E
2
dI
2
dt
= KE
2
− µI
2
If we calculate its Jacobian matrix J, we get :
12 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
a
b
c
d
e
Figure 4. Simulation of social graphs representing a contagion
network, with initial conditions (a) and asymptotic state in case of
an homophilic graph (b), random graph (c), scale free graph (d)
and small world graph (e).
J =
−β
2
I
2
r 0 0 −β
2
S
1
β
2
I
2
−r 0 0 β
2
S
1
0 −β
1
S
2
−β
1
I
1
− µ 0 0
0 β
1
S
2
β
1
I
1
−(µ + K) 0
0 0 0 K −µ
0 is a trivial eigenvalue and the rest of the spectrum is given by the eigenvalues
of the submatrix A:
A =
−r 0 0 rI
1
/I
2
−β
1
S
2
−β
1
I
1
− µ 0 0
β
1
S
2
β
1
I
1
−(µ + K) 0
0 0 K −µ
If we neglect the mortality and the recovering parameter r, the eigenvalues can
be calculated at the stationary state by making explicit the characteristic polyno-
mial: λ(K − λ)(λ− β
1
(I
∗
1
−S
∗
2
)) = λ(K −λ)(λ −β
1
(I
∗
1
−ω/(β
1
I
∗
1
))). If β
1
increases,
we have seen in Introduction that R
0
increases, then I
∗
1
tends to 1 if K >> β
1
− ω,
the rate of convergence to the stationary state being given by log(K/(β
1
− ω)),
which causes a long transient. We observe the same behaviour, if β
1
>> K − ω.
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 13
For example, let us take a sample of tail 1000 in each compartment. Figures 5, 6,
7 and 8 show that the interaction between some pairs of random variables coming
from the set S
1
, I
1
, S
2
, I
2
, E
2
with the Gumbel and Clayton copulas. In each figure
we have three graph, in the first graph on the left we assume that there is neither
a contagion nor a fertility for the two compartments so that the number of concor-
dant pairs is equal to the number of discordant pairs: it is the case of independence
between the two compartments, then the regression parameter is equal to 1 and the
Kendall’s tau is equal to 0, so in this case the equilibrium of the system is reached.
In the graph that is in the middle, we assume that there is a dependency between
the two compartments, with a regression parameter α equal to 3, so the Kendall’s
tau increases and we have an unbalanced system. In the graph on the right, we
increased the part of dependency between the two compartments, by taking the
regression parameter equal to 5, thus the Kendall’s tau increases a little more and
we have a system increasingly unbalanced. To summarize, when τ
C
= 1, the system
is totally unbalanced and when τ
C
= 0, the system is totally balanced.
In the graphs that are in the middle and right of Fig. 5 to 8, it is clear that we are
not in the case of complete independence between variables of the studied couples,
so for checking if there are sub-populations where the dependency is more or less
important, we will use in the next section the quantiles to divide the population
into several parts, where variables of the studied couples inside each part are more
or less dependent, regarding their Kendall’s tau.
Figure 5. Interaction between the distribution functions of S
2
and E
2
using the Gumbel copula with the parameter of regression
equal to 1, 3 and 5 respectively
Figure 6. Interaction between the distribution functions of E
2
and I
2
using the Gumbel copula with the parameter of regression
equal to 1, 3 and 5 respectively
14 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
Figure 7. Interaction between the distribution functions of S
2
and E
2
using the Clayton copula with the parameter of regression
equal to 1, 3 and 5 respectively
Figure 8. Interaction between the distribution functions of E
2
and I
2
using the Clayton copula with the parameter of regression
equal to 1, 3 and 5 respectively
4.2. Quantile regression using the bivariate Gumbel copula. In general, it
is known that the calculation of the regression function is tedious. As an alternative,
copulas are well-suited to the concept of quantile regression. Instead of examining
the mean of a conditional distribution, one looks at the median or some other quan-
tiles (for instance, percentiles) of this distribution. For p ∈ [0, 1], the p-th quantile
is defined as the solution t
p
of the equation: p = F
X
k
(t
p
|X
1
= x
1
, ..., X
k−1
= x
k−1
),
that we will denote, for the sake of simplicity, in the following by:
p = F
k
(t
p
|x
1
, ..., x
k−1
)
Let F
S
2
and F
E
2
be the distribution functions of the sizes of the two vector
compartments, susceptible and infected respectively. The bivariate Gumbel copula
for these two compartments is defined as follows:
C (F
S
2
, F
E
2
) = exp{−[(− ln F
S
2
)
α
+ (− ln F
E
2
)
α
]
1
α
} (19)
where α is the parameter of regression. So, we can write the equation (19) as
follows:
[− ln C (F
S
2
, F
E
2
)]
α
= (− ln F
S
2
)
α
+ (− ln F
E
2
)
α
(20)
For simplicity, we denote C (F
S
2
, F
E
2
) by C. Now, we take the partial derivative
with respect to F
S
2
of both sides of the equation (20) to get:
(− ln C)
α−1
C
∂C
∂F
S
2
=
(− ln F
S
2
)
α−1
F
S
2
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 15
Figure 9. Interaction between the distribution functions of S
2
and E
2
using the Gumbel copula with α = 3, using the quantile
regression (of E
2
on S
2
) curves.
Then, we extract the first partial derivative denoted C
1
= ∂C/∂F
S
2
as follows:
C
1
:= C
1
(F
S
2
, F
E
2
)
=
∂C(F
S
2
, F
E
2
)
∂F
S
2
=
∂C
∂F
S
2
=
ln F
S
2
ln C
α−1
C
F
S
2
(21)
By symmetry, we get also the second partial derivative as follows:
C
2
:=
∂C
∂F
E
2
=
ln F
E
2
ln C
α−1
C
F
E
2
(22)
Using the conditional distribution function from (9) and (21), we obtain the p-th
quantile, e
2,p
(s) as follows:
p = F
E
2
(e
2,p
(s)|S
2
= s)
=
φ
−1
{c
1
(s, e
2,p
(s)) + φ [F
E
2
(e
2,p
(s))]}
φ
−1
(c
1
(s, e
2,p
(s)))
=
φ
−1
{φ [F
S
2
(s)] + φ [F
E
2
(e
2,p
(s))]}
φ
−1
[φ(F
S
2
(s))]
= C
1
[F
S
2
(s), F
E
2
(e
2,p
(s))]
(23)
where C
1
is the first partial derivative of the Archimedean copula, φ is the gen-
erator of the Gumbel copula (φ(t) = (− ln t)
α
), and F
E
2
(e
2,p
(s)) is the distribution
function of the infected compartment where e
2,p
(s) denotes the p-th conditional
16 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
quantile conditioned by S
2
= s.
Notice that, in the case of the Gumbel-Hougaard copula, one can use Equation (21)
to get the distribution function F
E
2
(e
2,p
):
p =
ln(F
S
2
)
ln C(F
S
2
, F
E
2
(e
2,p
))
α−1
C(F
S
2
, F
E
2
(e
2,p
))
F
S
2
(24)
Finding the p-th quantile e
2,p
permits to divide the population into several parts
where the individuals in each part are dependent. In Figure 9, there are 500 indi-
viduals in each compartment and then there are 1000 individuals in the population.
Taking into account the interaction between the two distribution functions F
S
2
and
F
E
2
of the two compartments susceptible and infected respectively, we can divide
the population, with the conditional quantiles e
2,p
’s, taken for different p’s, for in-
stance for p ∈ {0.25, 0.5, 0.75}. For tracing the separation lines of the Figure 9, we
must calculate the former conditional quantiles related to the two compartments
susceptible and infected. Then, by computing these quantiles, we can analyze each
part to find the dependence between the two vector compartments.
4.3. Quantiles from the distribution functions. The purpose of this Section is
to calculate the p-th percentile, as in equation (24), from the distribution functions
of the susceptible and infective host populations sizes F
S
1
and F
I
1
, by writing:
F
S
1
(s) = 1 − R
γ
(s) (25)
F
I
1
(k) = 1 − R
ν
(k) (26)
where R
γ
(s) and R
ν
(k) are the rests at order k of the distribution functions F
S
1
and F
I
1
respectively.
Then:
ln [F
S
1
(s)] = ln [1 − R
γ
(s)]
ln [F
I
1
(k)] = ln [1 − R
ν
(k)]
Consequently, using the Gumbel copula in (19), we will have :
C
F
S
1
(s), F
I
1
(k)
= exp
n
− [[− ln (1 − R
γ
(s))]
α
+ [− ln (1 − R
ν
(k))]
α
]
1
α
o
(27)
or with the Development Limited at Order 2 DL
2
(+∞) we will have when k
tends to infinity:
ln (1 − R
γ
(s)) = −R
γ
(s) −
1
2
R
2
γ
(s) + o(R
2
γ
(s))
and
ln (1 − R
ν
(k)) = −R
ν
(k) −
1
2
R
2
ν
(k) + o(R
2
µ
(k))
Then :
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 17
C (F
S
1
(s), F
I
1
(k)) = exp
−
R
γ
(s) +
1
2
R
2
γ
(s) + o(R
2
γ
(s))
α
+
R
ν
(k) +
1
2
R
2
ν
(k) + o
R
2
ν
(k)
α
1
α
)
= exp
n
−
h
R
α
γ
(s)
1 +
α
2
R
γ
(s) + o(R
γ
(s))
+R
α
ν
(k)
1 +
α
2
R
ν
(k) + o(R
ν
(k))
i
1
α
= exp
(
−
R
α
γ
(s) + R
α
ν
(k)
1
α
.
"
1 +
α
2
R
α+1
γ
(s) + R
α+1
ν
(k)
R
α
γ
(s) + R
α
ν
(k)
+
R
α
γ
(s)
R
α
γ
(s) + R
α
ν
(k)
o
R
α
γ
(s)
+
R
α
ν
(k)
R
α
γ
(s) + R
α
ν
(k)
o(R
α
ν
(k))
1
α
)
= exp
(
−
h
R
α
γ
(s) + R
α
ν
(k)
i
1
α
.
"
1 +
1
2
R
α+1
γ
(s) + R
α+1
ν
(k)
R
α
γ
(s) + R
α
ν
(k)
+ o (R
γ
(s) + R
ν
(k))
i
)
= exp
(
−
h
R
α
γ
(s) + R
α
ν
(k)
i
1
α
)
.
1 −
1
2
R
α+1
γ
(s) + R
α+1
ν
(k)
R
α
γ
(s) + R
α
ν
(k)
1−
1
α
−
R
α
γ
(s) + R
α
ν
(k)
1
α
o (R
γ
(s) + R
ν
(k))
#
We denote by: L
α
(s, k) = R
α
γ
(s) + R
α
ν
(k). For simplicity, we denote in the
following L
α
(s, k) by L
α
.
Because o (R
γ
(s) + R
ν
(k)) tends to 0 when s → ∞ and k → ∞ then:
C (F
S
1
(s), F
I
1
(k)) ≈
1 −
1
2
L
α+1
L
1−α
α
α
.e
−L
1
α
α
(28)
Thus, the p-th percentile of the conditional distribution of the random variable
I
1
given the random variable S
1
is obtained from a formula similar to (24) by:
p ≈
[ln (1 − R
γ
(s))]
α−1
h
−L
1
α
α
+ ln
1 −
1
2
L
α+1
L
1−α
α
α
i
α−1
×
h
1 −
1
2
L
α+1
L
1−α
α
α
i
.e
−L
1
α
α
1 − R
γ
(s)
(29)
where L
α
= R
α
γ
(s) + R
α
ν
(k
ν
(p)), and k
ν
(p) = inf {m ∈ N, R
ν
(m) ≤ 1 − p}.
Because ln (1 − R
γ
(s)) ≈ −R
γ
(s) and ln
1 −
1
2
L
α+1
L
1−α
α
α
≈ −
1
2
L
α+1
L
1−α
α
α
,
then:
[ln (1 − R
γ
(s))]
α−1
≈ (−R
γ
(s))
α−1
(30)
18 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
and
−L
1
α
α
+ ln
1 −
1
2
L
α+1
L
1−α
α
α
α−1
≈
−L
1
α
α
−
1
2
L
α+1
L
1−α
α
α
α−1
≈ (−L
α
)
α−1
α
1 +
1
2
L
α+1
L
α
α−1
≈ (−L
α
)
α−1
α
1 +
α − 1
2
L
α+1
L
α
(31)
Therefore :
p ≈ (−1)
α+
1
α
(R
γ
(s))
α−1
1 − R
γ
(s)
×
1 −
1
2
L
α+1
L
1−α
α
α
e
−L
1
α
α
L
α−1
α
α
1 +
α−1
2
L
α+1
L
α
(32)
In the case of the independence between I
1
and S
1
in the Gumbel copula, α = 1,
and we have:
p ≈
1 −
1
2
L
2
e
−L
1
1 − R
γ
(s)
This formula is general and can be applied to any distribution function. In the
next section we will apply it on the Poisson distribution.
4.4. Quantile regression with Poisson distributions. In this Section we will
assume that F
S
1
and F
I
1
follow Poisson distributions, whose parameters γ and ν are
respectively the expected values of the susceptible and infective host populations
sizes.
The distribution functions F
S
1
(s) and F
I
1
(k) of the Poisson distributions are
defined as:
F
S
1
(s) =
s
X
i=0
γ
i
i!
e
−γ
= 1 − R
γ
(s)
(33)
with: R
γ
(s) =
P
+∞
i=s+1
γ
i
i!
e
−γ
F
I
1
(k) =
k
X
i=0
ν
i
i!
e
−ν
= 1 − R
ν
(k)
(34)
with: R
ν
(k) =
P
+∞
i=k+1
ν
i
i!
e
−ν
Then the p-th quantile is as in equation (32) with L
α
= R
α
γ
(s) + R
α
ν
(k(p)), and
k(p) = inf
n
m ∈ N,
P
∞
i=m+1
ν
i
i!
e
−ν
≤ 1 − p
o
.
5. The proportional risk model and the copula approach. Let us consider
now the Cox model with proportional risk [6] and suppose that the risk function
would be given by h(t, z) = e
ρz
b(t), where ρ is a regression parameter and b(t) the
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 19
baseline risk function. Then, by denoting q = e
ρz
, the survival function T (t, q) (i.e.,
the probability to survive until the age t with a risk q) is given by:
T (t, q) = exp
−
Z
t
0
h(s, z)ds
= B(t)
q
(35)
where B(t) = exp[−
R
t
0
b(s)ds].
In the Macdonald model (1), for calculating the survival function T
2
(t, q) of the
subpopulation I
2
, it is possible to identify z = log (β
1
β
2
/N
v
N
H
µr), ρ = −K/µ,
t = 1/K, b(s) = cste = µ, B(1/K) = e
−µ/K
and R
0
= [β
1
β
2
/N
V
N
H
µr]e
−µ/K
=
exp[log(β
1
β
2
/N
V
N
H
µr)] . exp[−µ/K] ≈ T
2
(1/K, (β
1
β
2
/N
V
N
H
µr)
−K/µ
), if β
1
β
2
/N
V
N
H
µr
is close to 1.
If there exist n age classes into the vector subpopulation E
2
whose sojourn times
U
i
(i = 1, ..., n) are independent random variables related to the survival function
T
i
, we have:
P(U
i
> t
i
, i = 1, ..., n|q) =
n
Y
i=1
T
i
(t
i
, q) =
n
Y
i=1
B
i
(t
i
)
q
(36)
If z is a random variable, then q = e
ρz
is also a random variable and we define
the mean survival function as T (t) = E
q
[B(t)
q
].
If we consider now the Laplace transform defined by: E
q
[e
−vq
] = exp(−v
p
) = L(v),
where p is a parameter depending on the probability distribution of q, we can write:
P(U
i
> t
i
, i = 1, ..., n) = E
q
[
n
Y
i=1
B
i
(t
i
)
q
]
= E
q
[exp(q
n
X
i=1
ln B
i
(t
i
))]
= exp
−
n
X
i=1
[− ln B
i
(t
i
)]
p
!
= exp
−
"
n
X
i=1
(− ln T
i
(t
i
))
1
p
#
p
!
= C(S
1
, ..., S
n
)
(37)
where C is an Archimedean copula, precisely the Gumbel Copula.
6. Conclusion. In this paper, we have given some definitions of Archimedean cop-
ulas used to define conditional quantiles for analysing interactions between compart-
ments of vectors and hosts in a system of transmission of malaria. By using the
bivariate Gumbel copula, we have calculated explicitely conditional quantiles and
applied it when the compartments sizes are supposed to be random and Poissonian.
Two of the direct applications of this work are:
• If the calculation of the regression parameter α of the Gumbel copula gives
the conclusion that α is close to 1, then the sizes of the concerned populations
(e.g., susceptible and infected) are not concordant or uncorrelated, which is in
favour of a total population size non constant, hypothesis rarely done in the
20 DEMONGEOT GHASSANI RACHDI OUASSOU AND TARAMASCO
epidemiologic studies, especially in malaria spread modelling [5, 23], which
leads to incorporate a refined demographic part in the epidemic model.
• The calculation of conditional quantiles with the general formula from the
empiric distribution functions of the observed population sizes allows to re-
construct their density, hence allows to test a posteriori their Poissonian char-
acter, which only authorizes the use of the specific simplified formulae. The
Poisson hypothesis has the interest to link the epidemic interaction mecha-
nism to the stochastic molecular kinetics [1, 31], which is a way to model the
contagious contacts.
7. Appendix.
7.1. Appendix A (after [24]).
τ = P {(X
1
− X
2
)(Y
1
− Y
2
) > 0} − P {(X
1
− X
2
)(Y
1
− Y
2
) < 0}
= 2 × P {(X
1
− X
2
)(Y
1
− Y
2
) > 0} − 1
= 2 × P {(X
1
> X
2
; Y
1
> Y
2
) ∪ (X
1
< X
2
; Y
1
< Y
2
)} − 1
= 2 × [P {(X
1
> X
2
; Y
1
> Y
2
)} + P {(X
1
< X
2
; Y
1
< Y
2
)}] − 1
= 4 × P {(X
1
> X
2
; Y
1
> Y
2
)} − 1
= 4
Z
x
Z
y
P {X
2
≤ x ; Y
2
≤ y | X
1
= x ; Y
1
= y} f
XY
(x, y)dxdy − 1
= 4
Z
x
Z
y
F
XY
(x, y)f
XY
(x, y)dxdy − 1
= 4
Z
x
Z
y
C (F
X
(x), F
Y
(y)) f
XY
(x, y)dxdy − 1
(38)
In making the change of variables u = F
X
(x) and v = F
Y
(y), we get:
τ = 4
Z
1
0
Z
1
0
C(u, v)c(u, v)dudv − 1
where c(u, v) =
∂
2
C(u,v)
∂u∂v
7.2. Appendix B (after [24]).
ρ = 3 [P ((X
1
− X
2
)(Y
1
− Y
3
) > 0) − P ((X
1
− X
2
)(Y
1
− Y
3
) < 0)]
So according to Appendix A, we can write:
ρ = 3
4
Z
x
Z
y
C (F
X
(x), F
Y
(y)) dC (F
X
(x), F
Y
(y)) − 1
In making the change of variables u
1
= F
X
(x) and u
2
= F
Y
(y), we get:
ρ = 12
Z
1
0
Z
1
0
C(u
1
, u
2
)dC(u
1
, u
2
) − 3
where ∂C(u
1
, u
2
) = ∂
2
C(u, v)/∂u
1
∂u
2
du
1
du
2
Since X
2
and Y
3
are independent, then C(u
1
, u
2
) = u
1
u
2
So
ρ = 12
Z
1
0
Z
1
0
u
1
u
2
dC(u
1
, u
2
) − 3
ARCHIMEDEAN COPULA AND CONTAGION MODELING IN EPIDEMIOLOGY 21
7.3. Appendix C (after [24]). For both tau and rho, the first six properties in
Definition (2.3) are obvious from the properties of the Kendall’s tau and the Spear-
man’s rho. For the seventh property, we note that the Lipschitz condition implies
that any family of copulas is equicontinuous, thus the convergence of C
n
to C is
uniform.
Lipschitz condition:
Let C
0
be a copula. If for every (u
1
, u
2
), (v
1
, v
2
) in DomC
0
| C
0
(u
2
, v
2
) − C
0
(u
1
, v
1
) |≤| u
2
− u
1
| + | v
2
− v
1
|
then C
0
is uniformly continuous on its domain.
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Received xxxx 20xx; revised xxxx 20xx.
E-mail address: Jacques.Demongeot@agim.eu
E-mail address: Mohamad.Ghassani@agim.eu
E-mail address: Mustapha.Rachdi@agim.eu
E-mail address: iouassou@yahoo.fr
E-mail address: Carla.Taramasco@polytechnique.edu
