Journal of Theoretical Biology 356 (2014) 133–143
Contents lists available at ScienceDirect
Journal of Theoretical Biology
journal homepage: www.elsevier.com/locate/yjtbi
Incorporating heterogeneity into the transmission dynamics
of a waterborne disease model
O.C. Collins 1, K.S. Govinder n
School of Mathematics, Statistics and Computer Science, University of KwaZulu–Natal, Private Bag X54001, Durban 4000, South Africa
H I G H L I G H T S
Devised new model for the evolution of water-borne diseases incorporating heterogeneity.
Incorporated multiple water sources for the first time.
Fitted model to real data from Haiti.
art ic l e i nf o
a b s t r a c t
Article history:
Received 21 October 2013
Received in revised form
4 April 2014
Accepted 16 April 2014
Available online 24 April 2014
We formulate a mathematical model that captures the essential dynamics of waterborne disease
transmission to study the effects of heterogeneity on the spread of the disease. The effects of
heterogeneity on some important mathematical features of the model such as the basic reproduction
number, type reproduction number and final outbreak size are analysed accordingly. We conduct a realworld application of this model by using it to investigate the heterogeneity in transmission in the recent
cholera outbreak in Haiti. By evaluating the measure of heterogeneity between the administrative
departments in Haiti, we discover a significant difference in the dynamics of the cholera outbreak
between the departments.
& 2014 Elsevier Ltd. All rights reserved.
Keywords:
Cholera
Basic reproduction number
Disease dynamics
1. Introduction
Waterborne diseases can be transmitted via person–water–person contact. This means that an infected individual will first shed
pathogens into the water source and susceptible individuals can then
contact the disease when they drink contaminated water. In reality,
the transmission rate and the shedding rate vary from one individual
to another, hence leading to heterogeneity in the transmission of
waterborne diseases. Even though, in some of the theoretical studies
on the dynamics and control intervention strategies (Tien and Earn,
2010; Zhou et al., 2012; Mwasa and Tchuenche, 2011; Liao and Wang,
2011; Capasso and Paveri-Fontana, 1979; Pourabbas et al., 2001;
Codeco, 2001; Ghosh et al., 2004; Hartley et al., 2006; King et al.,
2008; Eisenberg et al., 2003; Mukandavire et al., 2011a, 2011b) this is
not taken into account, heterogeneity is crucial to understand the
dynamics of waterborne disease and how best to reduce the spread
of the infection. Since most of the factors affecting the spread of
n
Corresponding author.
E-mail addresses: [email protected] (O.C. Collins),
[email protected] (K.S. Govinder).
1
Permanent home address: Department of Mathematics, University of Nigeria,
Nsukka, Nigeria.
http://dx.doi.org/10.1016/j.jtbi.2014.04.022
0022-5193/& 2014 Elsevier Ltd. All rights reserved.
waterborne diseases vary within and across a population, it is
expected that most of the important mathematical features of
waterborne disease models such as the basic reproduction number,
the type reproduction number and the final outbreak size will also
vary. Understanding the behaviour of each of these mathematical
features is very important in defining better control intervention
strategies that will reduce the spread of the disease. It is our interest
in this study to explore the effects of heterogeneity on each of the
mathematical feature of waterborne disease model which is necessary for defining better control strategies that will reduce the spread
of the disease.
Waterborne disease can be transmitted through contaminated
environmental water sources such as lake, river as well as through
contaminated household water sources like pond, private water
reservoir, etc. (Huq et al., 2005). It is important to know that some
individuals can be exposed to more than one contaminated water
source thus adding more heterogeneity in disease transmission.
To take this into account, it is necessary to consider a situation
whereby individuals are exposed to multiple contaminated water
sources.
Consider a community where individuals are exposed to multiple contaminated water sources. Despite the fact that individuals
are exposed to contaminated water sources, studies have shown
134
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
that some groups of individuals (especially children) are more
vulnerable to infection. Some of the reasons for this differences
might be due to hygienic practices of the individuals (like boiling
water before drinking, washing hands after going to the toilet,
proper washing of dishes and food before eating) and the level of
the immune system of the individuals. Understanding the
dynamics of waterborne diseases for such a community is complicated as homogeneous models cannot explain such situations.
As a result we resort to a multi-group model where a number of
environmental, biological and socio-economic factors are used to
categorise a group or sub population.
Tuite et al. (2011) constructed a mathematical model of cholera
epidemic dynamics for the ten departments in Haiti that is based
on both population and distance (a “gravity” model) between the
departments. They used the model to predict the sequence and
timing of regional cholera epidemics in Haiti and explore the
potential effects of disease-control strategies. Bertuzzo et al. (2011)
formulated a mathematical model which describes the epidemiological dynamics and pathogen transport and use it to determine
the timing and the magnitude of the epidemic in the ten Haitian
departments. Mukandavire et al. (2013) formulated a system of
coupled stochastic differential equations and use it to estimate the
reproductive numbers and vaccination coverage for the cholera
outbreak in Haiti. Robertson et al. (2013) extended the Tien and
Earn (2010) model to an n-patch waterborne disease model in
networks with a common water source to investigate the effect of
heterogeneity in dual transmission pathways on the spread of the
disease. Other works being done on spatially explicit models of
waterborne diseases include those by Bertuzzo et al. (2010), Mari
et al. (2011) and Gatto et al. (2012).
There is no doubt that the above studies have contributed
immensely towards understanding the dynamics and control of
waterborne diseases particularly for the recent cholera outbreak in
Haiti. To the best of our knowledge, the heterogeneity in the
transmission dynamics of waterborne disease has not yet been
explored neither have it been investigated for the recent cholera
outbreak in Haiti. The objectives of this paper is to develop and
analyse a mathematical model in order to improve the understanding of the transmission dynamics of waterborne disease. We
do this by investigating the heterogeneity in the transmission
dynamics of the model and consequently use the model to
investigate the heterogeneity in the recent cholera outbreak
in Haiti.
The remaining part of this work is organized as follows: the
model we are going to discuss is formulated in Section 2 and its
qualitative analyses are carried out in Section 3. In Section 4,
we apply our model to investigate heterogeneity in the recent
cholera outbreak in Haiti. We conclude the paper by discussing our
results in Section 5.
We partition N, the total human population of a community at
risk for waterborne disease infections, into n groups or homogeneous sub populations of size Nj such that each group is made
up of susceptible Sj(t), infected Ij(t) and recovered Rj(t), individuals.
The compartment W k measures pathogen concentration in water
reservoir k. In this study, we assume that there is no person to
person transmission and only consider transmission through
contact with contaminated water, as it is often considered to be
the main driver of waterborne disease outbreaks (Mukandavire et
al., 2011b; Sanches et al., 2011). Susceptible individuals Sj(t)
become infected through contact with the contaminated water
sources Wk at rate bjk. Infected individuals Ij(t) can contaminate the
water source k by shedding pathogen into it at rate θjk. The Ij(t) can
recover naturally at rate γj. Pathogens in the contaminated water
source k grow naturally at rate αk and decay at rate ξk. We assume
that sk ¼ ðαk ξk Þ o 0 is the net decay rate of pathogens in the
kth water reservoir. Natural death occurs in all the groups at rate μ.
Note that j ¼ 1; 2; …; n and k ¼ 1; 2; …; m. Putting these assumptions together, we obtain the model
2. Model formulation
S_ j ðtÞ ¼ μN j ðtÞ Sj ðtÞ ∑ bjk W k ðtÞ μSj ðtÞ;
To formulate the model, we consider a total human population
N where individuals are exposed to m multiple contaminated
water sources. We partition the population into n distinct sub
populations or groups based on the activity level. These groups
when combined together form the total population model in
which secondary infections can be generated both within a given
group and between groups. The secondary infections within a
group occur when an individual from a group sheds pathogens
into water sources with which susceptible individuals from the
same group subsequently come into contact. However, if the
susceptible individuals that come in contact with the pathogens
shed from an individual are from different groups, we say that
secondary infections between groups have occurred.
I_ j ðtÞ ¼ Sj ðtÞ ∑ bjk W k ðtÞ ðμ þ γ j ÞI j ðtÞ;
m
S_ 1 ðtÞ ¼ μN 1 ðtÞ S1 ðtÞ ∑ b1j W k ðtÞ μS1 ðtÞ;
k¼1
m
I_1 ðtÞ ¼ S1 ðtÞ ∑ b1k W k ðtÞ ðμ þ γ 1 ÞI 1 ðtÞ;
k¼1
R_ 1 ðtÞ ¼ γ 1 I 1 ðtÞ μR1 ðtÞ:
m
S_ 2 ðtÞ ¼ μN 2 ðtÞ S2 ðtÞ ∑ b2j W k ðtÞ μS2 ðtÞ;
k¼1
m
I_ 2 ðtÞ ¼ S2 ðtÞ ∑ b2k W k ðtÞ ðμ þγ 1 ÞI 2 ðtÞ;
k¼1
R_ 2 ðtÞ ¼ γ 2 I 2 ðtÞ μR2 ðtÞ;
⋮¼⋮
m
S_ n ðtÞ ¼ μN n ðtÞ Sn ðtÞ ∑ bnk W k ðtÞ μSn ðtÞ;
k¼1
m
I_n ðtÞ ¼ Sn ðtÞ ∑ bnk W k ðtÞ ðμ þ γ n ÞI n ðtÞ;
k¼1
R_ n ðtÞ ¼ γ n I n ðtÞ μRn ðtÞ;
n
_ 1 ðtÞ ¼ ∑ θj1 I j ðtÞ s1 W 1 ðtÞ;
W
j¼1
n
_ 2 ðtÞ ¼ ∑ θj2 I j ðtÞ s2 W 2 ðtÞ;
W
j¼1
⋮¼⋮
n
_ m ðtÞ ¼ ∑ θjm I j ðtÞ sm W m ðtÞ:
W
ð1Þ
j¼1
A pictorial illustration of model (2) showing all the possible
transmission dynamics that resulted in heterogeneity is given in
Fig. 1. The model (1) can be written in compact form as
m
k¼1
m
k¼1
n
_ k ðtÞ ¼ ∑ θjk I j ðtÞ sk W k ðtÞ;
W
j¼1
R_ j ðtÞ ¼ γ j I j ðtÞ μRj ðtÞ;
ð2Þ
where j ¼ 1; 2; …; n and k ¼ 1; 2; …; m. Variables and parameters of
the model (2) with their meaning are given in Table 1. The force of
infection in patch j is given by the linear term ∑m
k ¼ 1 bjk W k (Guo,
2012; Lloyd and May, 1996). Since our interest is on heterogeneity
in transmission dynamics of the waterborne disease which can be
generated due to differences in contact rates and shedding rates,
we will not consider explicit movement of individuals from one
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
135
Fig. 1. A model diagram illustrating all the possible transmission that leads to heterogeneity in the dynamics of the disease. Each group of three circles represents the
susceptible, infected and recovered individuals for each of the sub populations while the rectangles represent the contaminated water sources with distinct pathogen
concentration (i.e., W 1 ; W 2 ; …; W m ). The lines joining S1 to the W 1 ; W 2 ; …; W m represent contact rates of S1 with the contaminated water sources W 1 ; W 2 ; …; W m
respectively (i.e., b11 ; b12 ; …; b1m ) while the arrows joining I1 to the W 1 ; W 2 ; …; W m represent shedding rates of I1 into the contaminated water sources W 1 ; W 2 ; …; W m
respectively (i.e., θ11 ; θ12 ; …; θ1m ). Similarly, for S1 ; I2 ; …; Sn ; In . We do not label the model diagram neither do we include other dynamics in the model such as recovery rates
and birth/death rates, in the model diagram, otherwise the diagram will be complicated.
Table 1
Variables and parameters for model (3).
3. Model analysis
Variables
Meaning
N(t)
Sj ðtÞ
Ij(t)
Rj ðtÞ
W k ðtÞ
bjk
bj
θjk
θj
γj
Total human population
Susceptible individuals in sub population j
Infected individuals in sub population j
Recovered individuals in sub population j
Measure of pathogen concentration in water reservoir k
Partial contact rate of Sj ðtÞ with W k ðtÞ
Effective contact rate of Sj ðtÞ with all the water sources
Partial shedding rate of Ij ðtÞ into W k ðtÞ
Effective shedding rate of I j ðtÞ into all the water sources
Recovery rate of Ij
Net decay rate of pathogen in water source k
Natural death rate of individuals
sk
μ
In this section, we carry out a qualitative analysis of model (3).
The initial conditions are assumed as follows:
sj ð0Þ 40;
ij ð0Þ Z 0;
wk ð0Þ Z 0;
r j ð0Þ Z 0:
ð4Þ
We can show that all solutions ðs; i; w; r Þ of model (3) are positive
and bounded for all t 4 0, where s ¼ ðs1 ðtÞ; s2 ðtÞ; …; sn ðtÞÞ,
i ¼ ði1 ðtÞ; i2 ðtÞ; …; in ðtÞÞ, w ¼ ðw1 ðtÞ; w2 ðtÞ; …; wm ðtÞÞ and r ¼ ðr 1 ðtÞ;
r 2 ðtÞ; …; r n ðtÞÞ. Thus, the feasible region of model (3) is given by
nn
m
Ω ¼ ΩnH Ωm
P Rþ Rþ ;
ð5Þ
where
m
Ωm
P ¼ fw A R þ : 0 rwk r 1g;
and
group to another in this paper, however, we will consider this in
our future work. Note that model (2) is an extension of the model
considered by Collins and Govinder (2014) to study the dynamics
of waterborne disease with multiple water sources.
A dimensionless version of model (2) is given by
m
s_ j ¼ μ sj ∑ βjk wk μsj ;
k¼1
ΩnH ¼ fðs; i; rÞ A R3n
þ : sj þ ij þ r j ¼ 1g;
are the feasible regions of the pathogen and human components
respectively of model (3). The feasible region Ω is positively
invariant under the flow induced by (3), thus it is sufficient to
study the dynamics of (3) in Ω. Note that the superscript m is used
to emphasize on the number of water sources and n is the number of
sub populations. In order to understand the dynamics of model (3),
we start with a qualitative analysis of the model with special
case n¼ 2.
m
i_j ¼ sj ∑ βjk wk ðμ þγ j Þij ;
3.1. Quantifying heterogeneity
k¼1
_ k ¼ sk
w
!
n
∑ νjk ij wk ;
j¼1
r_ j ¼ γ j ij μr j ;
ð3Þ
s
wk ¼ k W k =∑nj¼ 1 θjk Nj ,
sj ¼ Sj =N j ; ij ¼ I j =N j ; r j ¼ Rj =N j ,
n
n
bjk ∑p ¼ 1 θpk N p = k ,
νjk ¼ θjk N j =∑p ¼ 1 θpk N p .
Note
that
where
βjk ¼
s
∑nj¼ 1 νjk ¼ 1. Thus the parameter νjk can be interpreted as the
proportion of total shedding from Ij into Wk while βjk is the scaled
contact rate of Sj with the water source Wk. Note that βj ¼ ∑m
k ¼ 1 β jk
is the scaled effective contact rate of Sj with all the water sources
and νj ¼ ∑m
k ¼ 1 νjk is the scaled effective proportion of total shedding from Ij into all the water sources.
Here, we investigate the heterogeneity in the transmission
dynamics of the model (3). Since secondary infections are generated in two different ways: within each sub population and
between the sub populations, it is expected that heterogeneity
could also arise in the same manner.
3.1.1. Heterogeneity within each sub population only
Heterogeneity in disease transmission can be measured as the
variance in transmission rates or contact rates among sub populations (Robertson et al., 2013). For heterogeneity within sub
population 1, we propose the following measure of heterogeneity:
!,
H1 ¼
m
m
j¼1
j¼1
∑ wj ðν1j ν 1 Þ2 þ ∑ wj ðβ1j β 1 Þ2
m
∑ wj :;
j¼1
136
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
m
m
m
where ν 1 ¼ ∑m
j ¼ 1 wj ν1j =∑j ¼ 1 wj and β 1 ¼ ∑j ¼ 1 wj β1j =∑j ¼ 1 wj . By
considering the following transformation:
,
m
w0k ¼ wk
∑ wj :;
j¼1
k ¼ 1; 2; …; m;
ð6Þ
the measure of heterogeneity H1 can be normalized to
m
m
j¼1
j¼1
H 1 ¼ ∑ w0j ðν1j ν 1 Þ2 þ ∑ w0j ðβ1j β 1 Þ2 ;
ð7Þ
0
m
0
m
0
with ν 1 ¼ ∑m
j ¼ 1 wj ν1j and β 1 ¼ ∑j ¼ 1 wj β1j , since ∑j ¼ 1 wj ¼ 1.
Similarly, the measure of heterogeneity H2 within sub population 2 is given by
m
m
j¼1
j¼1
H 2 ¼ ∑ w0j ðν2j ν 2 Þ2 þ ∑ w0j ðβ2j β 2 Þ2 ;
ð8Þ
0
m
0
where ν 2 ¼ ∑m
j ¼ 1 wj ν2j and β 2 ¼ ∑j ¼ 1 wj β 2j .
The total heterogeneity within the two sub populations can be
defined as
H ¼ H1 þ H2 :
ð9Þ
If we fix ðν1 ; β1 Þ and H and allow ðν2 ; β2 Þ to vary, geometrically,
pffiffiffiffiffi
Eq. (12) will represent a circle with center ðν1 ; β1 Þ and radius H
while if we fix ðν1 ; β1 Þ only and allow H and ðν2 ; β2 Þ to vary, Eq. (12)
becomes a paraboloid. The numerical illustrations of these are given
in Fig. 2. In Fig. 2(a), ðν1 ; β1 Þ ¼ ðν1 ; β1 Þ is fixed as the center of the
circles while ðν2 ; β2 Þ ¼ ðν2 ; β2 Þ and H varies. The radius of each circle
represents the magnitude of heterogeneity between the two sub
populations. Thus the bigger the radius of the circle, the greater the
heterogeneity between the two sub populations.
3.1.3. Total heterogeneity
We have seen that there are two sources of heterogeneity in
the system (3) namely, heterogeneity due to variation in transmission within a sub population and heterogeneity due to variation in
transmission between the sub populations. Therefore, the total
heterogeneity H in the system (3) can be defined as the sum of all
the heterogeneity H within the sub populations and heterogeneity
H between the patches and is given by
H ¼ H þH:
ð13Þ
3.2. Homogeneous version of model (3)
3.1.2. Heterogeneity between two sub populations only
The contact rate and the shedding rate of individuals in patch
1 is certainly not the same as that of sub population 2. To estimate
this variation in transmission dynamics between the sub populations 1 and 2, we proposed the following measure of heterogeneity:
m
m
j¼1
j¼1
H12 ¼ ∑ w0j ðν1j ν2j Þ2 þ ∑ w0j ðβ1j β2j Þ2 :
ð10Þ
Noting that H21 ¼ H12 , the total variation in transmission between
the two sub populations can be written as
H ¼ H12 :
If we have a single water source, i.e., m¼1, the measure of
heterogeneity H becomes
2
2
H ¼ ðν11 ν21 Þ þ ðβ11 β21 Þ :
ð11Þ
Geometrically,
Eq. (11) represents a circle with center ðν11 ; β11 Þ and
pffiffiffiffiffi
radius H when ðν11 ; β11 Þ is fixed and ðν21 ; β21 Þ is allowed to vary.
Since our interest is in the dynamics of waterborne disease for
multiple water sources, to have a geometric view of measure of
heterogeneity H between the two sub populations, we can also
define H as
H ¼ ðν1 ν2 Þ2 þ ðβ1 β2 Þ2 :
ð12Þ
1.5
To determine the effects of heterogeneity, it is necessary to obtain
some of the mathematical features of the homogeneous version of
model (3) and thereafter compare them with that of the heterogeneous model (3). The homogeneous version of the model (3)
is obtained by considering the entire population as a homogeneous
mixing population where all the individuals have access to a
single water source. The homogeneous version of model (3) is
given by
_ ¼ μNðtÞ bSðtÞWðtÞ μSðtÞ;
SðtÞ
_ ¼ bSðtÞWðtÞ ðμ þ γÞIðtÞ;
IðtÞ
_ ðtÞ ¼ θIðtÞ sWðtÞ;
W
_ ¼ γIðtÞ μRðtÞ:
RðtÞ
Note that (14) is simply obtained from the original model (2) by
taking n¼ m¼1 and ignoring the subscripts. By rescaling (14) as
follows: s¼S/N, i¼I/N, r¼R/N, w ¼ sW=θN, β ¼ bθN=s, we obtain a
non-dimensional version of it:
s_ ¼ μ βsw μs;
i_ ¼ βsw ðμ þ γÞi;
_ ¼ sði wÞ;
w
r_ ¼ γi μr:
H=0.2
H=0.4
H=0.6
H=0.8
H=1.0
1
ð14Þ
ð15Þ
1.2
1.4
1
1.2
1
0.8
0.5
H
β2
0.8
0.6
0.6
v1,β1
0
0.4
−0.5
0
0.8
−1
−1
0.4
0.2
−0.5
0
0.5
v2
1
1.5
0.2
1
0.6
0.4
β2
0.5
0.2
0
0
v2
Fig. 2. Numerical illustration of heterogeneity between patches 1 and 2 assuming homogeneity within the sub populations: (a) for ðν1 ; β1 Þ, H fixed and (b) for ðν1 ; β1 Þ fixed.
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
The disease free equilibrium (DFE) and the basic reproduction
number of the homogeneous model (15) are given by
ðs0 ; i0 ; w0 Þ ¼ ð1; 0; 0Þ;
ð16Þ
and
R0 ¼ β=ðμþ γÞ;
ð17Þ
respectively. To establish a relationship between the heterogeneous
model (3) and the homogeneous model (15), we define the contact
rate, recovery rate and decay rate as follows:
2
2
m
β ¼ ∑ βj N j =N ¼ ∑ ∑ βjk Nj =N;
j¼1
j¼1k¼1
m
s¼ ∑
k¼1
w0k
2
sk ;
γ ¼ ∑ N j γ j =N:
j¼1
Based on this definition, we will see later that in the absence of
heterogeneity (both within and between the sub populations), most
of the mathematical features of the heterogeneous model (3) such as
the basic reproduction number coincide to that of the homogeneous
model (15).
3.3. The basic reproduction number
The basic reproduction number is a useful epidemiological
quantity that determines whether a disease will persist or not in
a population. We determine the basic reproduction number of (3)
using the next generation matrix approach of van den Driessche
and Watmough (2002). The associated next generation matrices
are given by
0
1
0 0 β11 β12 β13 … β1m
B0 0 β
β22 β23 … β2m C
B
C
21
B
C
B
0
0 …
0 C
F ¼B0 0 0
C;
B
C
…
@⋮ ⋮
A
0
0
0
μ þ γ1
B
0
B
B
B s1 ν11
B
V ¼B
B s2 ν12
B
B
⋮
@
0
sm ν1m
with
0
Rm
11
B m
B R21
B
1
¼B
FV
B 0
B
@ ⋮
0
0
…
0
0
μ þ γ2
0
0
s1 ν21
s1
s2 ν22
0
0
0
0
0
0
0
…
…
0
0
0
…
0
s2 0
0
…
0
0
…
0
⋮
sm ν2m
0
β11 =s1
Rm
22
β21 =s1
β12 =s2
0
0
0
…
0
…
…
Rm
12
⋮
0
0
β22 =s2
…
…
m
1
C
β2m =sm C
C
C;
0
C
C
A
0
j¼1
m
Rm
21 ¼ ∑ ν1j β 2j =ðμþ γ 1 Þ;
j¼1
m
Rm
12 ¼ ∑ ν2j β1j =ðμ þ γ 2 Þ;
ð18Þ
ð19Þ
j¼1
m
Rm
22 ¼ ∑ ν2j β2j =ðμ þ γ 2 Þ:
Next, we investigate the effects of heterogeneity by comparing
the basic reproduction numbers of the heterogeneous model (3)
and homogeneous model (15). Noting that the basic reproduction
numbers are parameter dependent, we modify the contact rates,
shedding rates and recovery rates as follows.
Since individuals do not have equal access to each of the
contaminated water sources, we can rewrite νij and βij as
νij ¼ νi1 aj 1 ;
j1
βij ¼ βi1 b
;
ð23Þ
where 0 oa; b o 1. If, in addition, we assume that individuals in
sub population 1 are more exposed to infections than those in sub
population 2, those in sub population 2 are more exposed than
those in sub population 3, in this order till sub population n, then
we have that
νij ¼ ν11 qi 1 aj 1 ;
j1
βij ¼ β11 pi 1 b
;
γ i ¼ γ 1 ci 1 ;
ð24Þ
where 0 o p; q o 1 and c 4 1. Note that we also assumed that
individuals in sub population n recover faster than those in sub
population n 1 while those in sub population n 1 recover faster
than those in n 2 and so on until sub-population 1. This accounts
for having c 4 1. By considering Eq. (24), Rm
0 simplifies to
R0 o Rm
0
where
Rm
11 ¼ ∑ ν1j β 1j =ðμþ γ 1 Þ;
Theorem 3.1. The DFE of model (3) is globally asymptotically stable
if Rm
0 o 1.
ð25Þ
With these modifications, we can now go ahead and carry out the
analysis as follows. Firstly, we compare the basic reproduction
number Rm
0 of model (3) with that of R0 homogeneous model
(15). Taking limits of Rm
0 and R0 as a; b; c; p; q⟶1 or as a; b; p;
q⟶0; c⟶1 we obtain that
sm
β1m =sm
where R111 ¼ ν11 β11 =ðμ þ γ 1 Þ, R112 ¼ ν21 β11 =ðμþ γ 2 Þ, R121 ¼ ν11 β21 =
ðμ þ γ 1 Þ and R122 ¼ ν21 β21 =ðμ þ γ 2 Þ. Note that if there is homogeneity
(both within and between the patches), then we can easily see that
Rm
0 simplifies to R0 .
Eq. (21) reveals that to have any chance of controlling the
m
spread of infection (i.e., Rm
0 o 1), then it is necessary that R11 o1
and Rm
o
1
hold.
Using
a
global
stability
result
by
Castilo-Chavez
22
et al. (2002) we establish the following theorem.
m
m
Rm
0 ¼ R11 þ R22 :
1
0
C
0 C
C
0 C
C
C;
0 C
C
C
A
137
ð20Þ
j¼1
The basic reproduction number is the dominant eigenvalue of the
matrix FV 1 and is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
m
m 2
m
m
ðRm
Rm
ð21Þ
0 ¼ ðR11 þ R22 þ
11 R22 Þ þ 4R12 R21 Þ=2:
Suppose the entire population share a common water source (i.e.,
m ¼1), then the basic reproduction number Rm
0 becomes
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R10 ¼ ðR111 þ R122 þ ðR111 R122 Þ2 þ 4R112 R121 Þ=2; ¼ R111 þ R122 ; ð22Þ
ð26Þ
for both cases. This suggests that heterogeneity increases the basic
reproduction number.
Note that a; b; c; p; q⟶1 means a situation when the difference
in transmission between the sub populations becomes very small,
while a; b; p; q⟶0; c⟶1 implies that the difference in transmission between the sub populations becomes very large.
Secondly, we compare the basic reproduction number of our
models for the case when there is heterogeneity only within the
sub populations i.e., H a 0 and H ¼ 0. In this case, ν1j ¼ ν2j ; β1j ¼ β2j
and the basic reproduction number Rm
0 becomes
m
m
RH
0 ¼ R11 þ R22 :
ð27Þ
Taking limits as a; b; c; p; q⟶1 or as a; b; p; q⟶0; c⟶1, we
obtain that
R0 o RH
0:
ð28Þ
Thirdly, for the case when there is heterogeneity only between the
sub populations i.e., H ¼0 and H a 0. Here we have ν 1 ¼ ν1j ,
ν 2 ¼ ν2j , β 1 ¼ β1j and β 2 ¼ β2j and Rm
0 becomes
m
m
RH
0 ¼ R11 þ R22 :
ð29Þ
138
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
Taking limits as a; b; c; p; q⟶1 or as a; b; p; q⟶0; c⟶1, we
obtain that
R0 o RH
0:
ð30Þ
The above results show that an increase in heterogeneity increases
the basic reproduction number. These are also consistent with the
results of Robertson et al. (2013) that says that the basic reproduction number is an increasing function of heterogeneity.
Suppose the sub populations are isolated such that there is no
sharing of water sources, then Rm
0 becomes
m
Rs0 ¼ maxfRm
11 ; R22 g;
ð31Þ
where R11 ¼ ν11 β11 =ðμþ γ 1 Þ and R22 ¼ ν22 β22 =ðμ þγ 2 Þ. It is obvious
that
Rs0 o Rm
0:
ð32Þ
This shows that sharing of water sources increases the basic
reproduction number compared to when sub populations are
isolated. Notice that if there is no sharing of water sources,
heterogeneity within the sub populations vanishes i.e., H¼ 0.
Therefore, sharing of water sources increases heterogeneity in
transmission of waterborne diseases.
Furthermore, we have from Eq. (22) that when the entire
1
population share common water sources, Rm
0 becomes R0 ¼
R111 þ R122 . For this case, we also notice that heterogeneity within
the sub populations vanishes i.e., H ¼0. This implies that reducing
the number of water sources that the population shared deceases
heterogeneity in transmission. Moreover,
R10 o Rm
0;
ð33Þ
showing that the greater the number of water sources shared by
the population, the greater the basic reproduction number. Therefore, an increase in the number of water sources shared by the
population leads to increased heterogeneity in transmission of
waterborne diseases. This support our earlier results that say
that heterogeneity increases the reproduction number of the
disease. Thus, heterogeneity has some influence on the dynamics
of waterborne disease.
The type reproduction number T i represents the expected
number of secondary infections produced by an infected individual in a susceptible patch i over his/her lifetime. To determine the
proper control effort needed to eradicate the spread of the
infection while targeting control at one particular patch, and
having no control over reducing the spread of the disease in other
patches, it is necessary that we consider the type reproduction
number rather than the basic reproduction number (Roberts and
Heesterbeek, 2003; Heesterbeek and Roberts, 2007). We compute
T 1 for patch 1 to be
Similarly, the type reproduction number T 2
provided that
for sub population 2 is given by
ð35Þ
provided that Rm
11 a 1.
Eq. (34) can be re-written as
m
m
m
ðT 1 Rm
11 Þð1 R22 Þ ¼ R12 R21 :
Rm
22 o T 2 :
ð37Þ
m
On the other hand, if Rm
11 4 1 and R22 4 1, then
Rm
11 4T 1 ;
Rm
22 4 T 2 :
ð38Þ
In this case, there is no chance of controlling the spread of
infection.
To determine the effect of heterogeneity on the type reproduction numbers of model (3), we consider the following cases:
suppose there is heterogeneity only within the sub populations
i.e., H a 0 and H ¼ 0, then the type reproduction numbers T 1 and
T 2 become
m
m
TH
1 ¼ R11 =ð1 R22 Þ;
m
m
TH
2 ¼ R22 =ð1 R11 Þ:
ð39Þ
Similarly, if there is heterogeneity only between the sub populations i.e., H¼ 0 and H a 0, the type reproduction numbers T 1 and
T 2 become
m
m
TH
1 ¼ R11 =ð1 R22 Þ;
m
m
TH
2 ¼ R22 =ð1 R11 Þ:
ð40Þ
Suppose there is no sharing of water sources, then the type
reproduction numbers T 1 and T 2 become
T s1 ¼ Rm
11 ;
T s2 ¼ Rm
22 :
ð41Þ
We can easily see
T s1 o T 1 ;
T s2 o T 2 :
ð42Þ
This implies that sharing of water sources increases the type
reproduction numbers. Based on our earlier results, we can say
that heterogeneity increases the type reproduction numbers.
3.5. Final outbreak size
The basic reproduction number/type reproduction number is
very important for determining whether or not an outbreak will
occur. To determine the magnitude of an outbreak, it is necessary
to compute the final outbreak size. The final outbreak size denoted
by z of the SIR models together with some other related models is
given by the relation (Ma and Earn, 2006)
ð43Þ
This relation does not hold for model (3). However, when μ ¼ 0,
Rm
0 4 1 and wj ð0Þ ¼ 0, then the final outbreak size in sub populations 1 and 2 denoted by z1 and z2 respectively can be calculated.
Proposition 3.2. If μ ¼ 0, Rm
0 4 1 and wj ð0Þ ¼ 0, then the final
outbreak size in sub populations 1 and 2 denoted by z1 and z2
respectively are given by the following equations:
m m
m m
zm
1 ¼ 1 expð R11 z1 R12 z2 Þ;
ð44Þ
m m
m m
zm
2 ¼ 1 expð R22 z2 R21 z1 Þ:
ð45Þ
ð34Þ
Rm
22 a1.
m
m
m
T 2 ¼ Rm
22 þ R12 R21 =ð1 R11 Þ;
Rm
11 oT 1 ;
z ¼ 1 expð R0 zÞ:
3.4. Type reproduction numbers
m
m
m
T 1 ¼ Rm
11 þ R12 R21 =ð1 R22 Þ;
m
Rm
11 o1 and R22 o 1, we must have that
Proof. We consider the same approach used in Tien and Earn
(2010) and Ma and Earn (2006). Consider the functions
m
m
j¼1
j¼1
F 1 ðtÞ ¼ log s1 ðtÞ þ r 1 ðtÞ ∑ ν1j β1j =γ 1 þ r 2 ðtÞ ∑ ν2j β1j =γ 2
m
∑ β1j wj ðtÞ=sj ;
ð46Þ
j¼1
ð36Þ
m
m
Since Rm
12 4 0 and R21 4 0, then we must have that T 1 4 R11 and
m
m
m
1 4 R22 or T 1 o R11 and 1 oR22 . Similarly, from (35) we obtain
m
m
m
T 2 4 Rm
22 and 1 4 R11 or T 2 o R22 and 1 o R11 . Given that a
necessary condition to control the spread of infection is that
m
m
j¼1
j¼1
F 2 ðtÞ ¼ log s2 ðtÞ þ r 2 ðtÞ ∑ ν2j β2j =γ 2 þ r 1 ðtÞ ∑ ν1j β2j =γ 1
m
∑ β2j wj ðtÞ=sj :
j¼1
ð47Þ
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
and final outbreak size zm in the entire population is
Differentiating F1 with respect to time t gives
m
m
m
_ j ðtÞ=sj ;
F_ 1 ¼ s_ 1 =s1 þ r_ 1 ðtÞ ∑ ν1j β1j =γ 1 þ r_ 2 ðtÞ ∑ ν2j β1j =γ 2 ∑ β1j w
j¼1
j¼1
j¼1
m
m
m
j¼1
j¼1
j¼1
ð52Þ
Taking limits as a; b; c; p; q⟶1 or as a; b; p; q⟶0; c⟶1, we
obtain that
m
∑ β1j ðν1j i1 ðtÞ þ ν2j i2 ðtÞ wj ðtÞÞ;
z r zH :
j¼1
¼ 0:
Hence, F1 is a constant function along solution trajectories of
model (3). Similarly, F2 is also a constant function along the
solution trajectories. Since μ¼ 0, then susceptible individuals s1 ðtÞ
and s2 ðtÞ decrease monotonically to limits s 1 and s 2 respectively
while the recovered individuals r 1 ðtÞ and r 2 ðtÞ increase monotonically to limits r 1 and r 2 respectively. By Tien and Earn (2010,
Lemma 2), ði1 ðtÞ; i2 ðtÞÞ⟶ð0; 0Þ and wj ðtÞ⟶0. Consequently,
s 1 ¼ 1 r 1 and s 2 ¼ 1 r 2 . Taking limits of (46) and (47), we obtain
ð53Þ
On the other hand, if there is heterogeneity only between the sub
populations i.e., H ¼0 and H a 0, the final out break size zm
1 in sub
population 1 and zm
2 in sub population 2 become
m H
m H
zH
1 ¼ 1 expð R11 z1 R12 z2 Þ;
ð54Þ
m H
m H
zH
2 ¼ 1 expð R22 z2 R21 z1 Þ;
ð55Þ
and final outbreak size z
m
in the entire population is
2
zH ¼ ∑ N i zH
i =N:
ð56Þ
m
m
j¼1
m
j¼1
m
Taking limits as a; b; c; p; q⟶1 or as a; b; p; q⟶0; c⟶1, we
obtain that
j¼1
j¼1
z r zH :
i¼1
lim F 1 ðtÞ ¼ log ð1 r 1 Þ þ r 1 ∑ ν1j β1j =γ 1 þ r 2 ∑ ν2j β1j =γ 2 ;
lim F 2 ðtÞ ¼ log ð1 r 2 Þ þ r 2 ∑ ν2j β2j =γ 2 þ r 1 ∑ ν1j β2j =γ 1 :
t⟶1
2
zH ¼ ∑ zH
i N i =N:
i¼1
¼ ∑ β1j wj ðtÞ þ i1 ðtÞ ∑ ν1j β1j þi2 ðtÞ ∑ ν2j β1j
t⟶1
139
At t¼ 0,
m
m
j¼1
j¼1
These results suggest that an increase in heterogeneity increases
the final outbreak size.
In addition to this, if the sub populations are isolated such that
there is no sharing of water sources, then the final out break size
relation in patch 1 and patch 2 becomes
F 1 ð0Þ ¼ log s1 ð0Þ þ r 1 ð0Þ ∑ ν1j β1j =γ 1 þ r 2 ð0Þ ∑ ν2j β1j =γ 2
m
∑ β1j wj ð0Þ=sj ;
j¼1
m
m
m
j¼1
j¼1
j¼1
ð57Þ
F 2 ð0Þ ¼ log s2 ð0Þ þ r 2 ð0Þ ∑ ν2j β2j =γ 2 þ r 1 ð0Þ ∑ ν1j β2j =γ 1 ∑ β2j wj ð0Þ=sj :
zs1 ¼ 1 expð R11 zs1 Þ;
ð58Þ
Letting s1 ð0Þ⟶1, s2 ð0Þ⟶1 then ðr 1 ð0Þ; r 2 ð0ÞÞ⟶ð0; 0Þ and
wj ð0Þ⟶0. Since F 1 ðtÞ and F 2 ðtÞ are constant along solution
trajectories,
then
limt⟶1 F 1 ðtÞ ¼ F 1 ð0Þ ¼ 0
and
limt⟶1
F 2 ðtÞ ¼ F 2 ð0Þ ¼ 0, so
zs2 ¼ 1 expð R22 zs2 Þ
ð59Þ
m
m
j¼1
m
j¼1
m
j¼1
j¼1
log ð1 r 2 Þ þ r 2 ∑ ν2j β2j =γ 2 þr 1 ∑ ν1j β2j =γ 1 ¼ 0:
Rm
11
and noting that
¼ ∑m
j ¼ 1 ν2j β2j =γ 2 and
β1j =γ 1 when μ¼0 gives the desired result. □
¼ ∑m
j ¼ 1 ν1j β 1j =γ 1 ,
Rm
∑m
21 ¼
j ¼ 1 ν2j
The final outbreak size zm in the entire population is therefore
given by Ma and Earn (2006):
2
zm ¼ ∑ zm
i N i =N:
ð48Þ
i¼1
Taking limits as a; b; c; p; q⟶1 or as a; b; p; q⟶0; c⟶1, we
obtain that
z r zm ;
ð49Þ
where z is the final outbreak size of the homogeneous model (15).
We observe that the final outbreak size zm
1 in sub population
1 is affected by the shedding from sub population 2 and vice versa.
This could be due to heterogeneity in transmission between the
two sub populations. Hence, it is necessary to determine the
effects of heterogeneity in the final outbreak size. Suppose that
there is heterogeneity only within the patches i.e., H a0 and
m
H ¼ 0, the final outbreak size zm
1 in patch 1 and z2 in patch
2 become
m H
m H
zH
1 ¼ 1 expð R11 z1 R12 z2 Þ;
m H
m H
zH
2 ¼ 1 expð R22 z2 R21 z1 Þ;
zs1 o zm
1;
zs2 o zm
2:
ð60Þ
This shows that sharing of water sources increases the final
outbreak size compared to when sub populations are isolated.
log ð1 r 1 Þ þ r 1 ∑ ν1j β1j =γ 1 þr 2 ∑ ν2j β1j =γ 2 ¼ 0;
m
Letting r 1 ⟶zm
1 , r 2 ⟶z2
m
m
m
R12 ¼ ∑j ¼ 1 ν2j β1j =γ 2 , R22
respectively where R11 ¼ ν11 β11 =γ 1 and R22 ¼ ν22 β22 =γ 2 . Clearly
3.6. The general n-sub population model with shared multiple water
sources
In this section, we extend some of the results obtained in the
2-sub population model to the general n-sub population model
(3). The next generation matrix FV 1 of the general n-sub
population model is given by
0 m
1
… Rm
β11 =s1 β12 =s2 … β1m =sm
R11 Rm
12
1n
B Rm Rm … Rm β =s
β22 =s2 … β2m =sm C
B 21
C
1
21
22
2n
B
C
B ⋮
C
⋮
…
…
B
C
B
C
m
m
R
…
R
β
=
s
β
=
s
…
β
=
s
FV 1 ¼ B Rm
C;
m
1
2
n1
n2
nm
n2
nn
B n1
C
B 0
C
0
…
0
0
0
…
0
B
C
B
C
⋮
…
@ ⋮
A
0
0
…
0
0
0
…
0
ð61Þ
Rm
jk
¼ ∑m
p ¼ 1 νkp βjp =ðμ þ γ k Þ.
Similar to the case of two sub
where
populations, the basic reproduction number Rm
0 is the dominant
eigenvalue of the matrix FV 1 . In this case, Rm
0 is the largest
positive root of the polynomial
an þ m λn þ m þan þ m 1 λn þ m 1 þ an þ m 2 λn þ m 2 þ ⋯ þ a1 λ þ a0 ¼ 0;
ð62Þ
ð50Þ
ð51Þ
where a0 ; a1 ; …; an þ m are constant functions of
and N j βjk =sk .
If the sub populations are isolated such that there is no sharing of
Rm
jk
140
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
water sources, then Rm
0 for the general case becomes
Rs0 ¼ maxfRjj g;
j ¼ 1; 2; …; n;
m
where Rjj ¼ νjj βjj =ðμ þ γ j Þ.
For heterogeneity within any patch i of the general n-sub
population model (3), we propose the following measure of
heterogeneity:
m
m
j¼1
j¼1
∑ βjk
ð63Þ
H i ¼ ∑ w0j ðνij ν i Þ2 þ ∑ w0j ðβij β i Þ2 ;
ð64Þ
0
m
0
where ν i ¼ ∑m
j ¼ 1 wj νij and β i ¼ ∑j ¼ 1 wj βij .
The total heterogeneity within all the sub populations can be
defined as
n
H ¼ ∑ Hi :
¼ 0:
Thus Fj is a constant function along solution trajectories of model
(3). Since μ ¼ 0, then susceptible individuals sj(t) decrease monotonically to limits s j while the recovered individuals rj(t) increase
monotonically to limits r j for each j ¼ 1; 2; …; n. By Tien and Earn
(2010, Lemma 2), ij ðtÞ⟶0 and wk ðtÞ⟶0 for each k ¼ 1; 2; …; m.
This implies that s j ¼ 1 r j . Since Fj is constant, setting
F j ð0Þ ¼ limt⟶1 F j ðtÞ gives
m
k¼1
m
j¼1
k¼1
ð66Þ
Note that Hpq ¼ Hqp and Hpp ¼ Hqq ¼ 0. The total heterogeneity
between all the sub populations in the system can be estimated as
follows: the heterogeneity between sub population 1 and each of
the remaining n 1 sub populations starting from sub population
2 to sub population n are H12 , H13 , H14 ,…, H1n . Similarly, the
heterogeneity between patch 2 and each of the remaining n 2
sub populations starting from sub population 3 to sub population
n are H23 , H24 , H25 ,…, H2n . Notice that at each stage the number of
measures of heterogeneity decreases by 1. We can continue in this
order to the last term which is Hn 1n . There are a total of
nðn 1Þ=2 distinct measure of heterogeneities Hpq between any
two sub populations. Hence the total heterogeneity between each
of the sub populations in the system can be calculated explicitly as
n
n
i¼2
i¼3
i¼4
k¼1
m
m
k¼1
k¼1
¼ log ð1 r j Þ þ r 1 ∑ ν1k βjk =γ 1 þ r 2 ∑ ν2k βjk =γ 2
m
þ ⋯ þr n ∑ νnk βjk =γ n :
k¼1
Hpq ¼ ∑ w0j ðνpj νqj Þ2 þ ∑ w0j ðβpj βqj Þ2 :
n
k¼1
m
þ r n ð0Þ ∑ νnk βjk =γ n ∑ βjk wk ð0Þ=sk
The variation in transmission dynamics between any two sub
populations p and q can be estimated by the following measure of
heterogeneity:
j¼1
m
log sj ð0Þ þ r 1 ð0Þ ∑ ν1k βjk =γ 1 þ r 2 ð0Þ ∑ ν2k βjk =γ 2 þ ⋯
ð65Þ
m
∑ νjk ij wk ;
j¼1
k¼1
i¼1
m
!
n
H ¼ ∑ H1i þ ∑ H2i þ ∑ H3i þ ⋯ þ
n
n
∑
i ¼ n1
Hn 2i þ ∑ Hn 1i ;
Let sj ð0Þ⟶1 and wk ð0Þ⟶0 and note that sj ð0Þ⟶1 will force
r j ð0Þ⟶0, we have
log ð1 r j Þ
m
m
m
k¼1
k¼1
!
¼ r 1 ∑ ν1k βjk =γ 1 þ r 2 ∑ ν2k βjk =γ 2 þ ⋯ þr n ∑ νnk βjk =γ n :
k¼1
By letting r j ¼
relation
zm
j
zm
j
and simplifying gives the final outbreak size
m
¼ 1 exp ∑
k¼1
!
m
Rm
jk zk
;
m
which is the desired result with Rm
jk ¼ ∑p ¼ 1 νkp βjp =γ k . Therefore,
the final outbreak size in sub population j of the general model (3)
m
is given by zm
in the entire
j . Thus, the final outbreak size z
population is
n
i¼n
ð67Þ
ð69Þ
zm ¼ ∑ zm
j N j =N:
ð70Þ
j¼1
where ∑ni¼ n Hn 1i ¼ Hn 1n .
The total heterogeneity H in the general n-sub population
system (3) can be defined as the sum of all the heterogeneity H
within the sub populations and heterogeneity H between the sub
populations and is given by
If there is no sharing of water sources, then the final out break
size relation in sub population j becomes
H ¼ H þ H:
zs ¼ ∑ zsj N j =N:
ð68Þ
The final outbreak size relation can also be derived for each of
the sub populations of the general model (3). Consider the
function
m
m
k¼1
k¼1
zsj ¼ 1 expð Rjj zsj Þ;
where Rjj ¼ νjj βjj =γ j . Moreover,
n
ð72Þ
j¼1
By a similar reasoning as in the case of 2-sub populations, we can
show that
zsj ozm
j :
F j ðtÞ ¼ log sj ðtÞ þ r 1 ðtÞ ∑ ν1k βjk =γ 1 þ r 2 ðtÞ ∑ ν2k βjk =γ 2
ð71Þ
ð73Þ
This implies that sharing of water sources also increases the final
outbreak size compared to when sub populations are isolated for
the general case.
m
þ ⋯ þ r n ðtÞ ∑ νnk βjk =γ n
k¼1
m
∑ βjk wk ðtÞ=sk :
4. Heterogeneity in dynamics of cholera outbreak in Haiti
k¼1
Differentiating F j with respect to time gives
m
m
m
k¼1
k¼1
k¼1
F_j ¼ s_ j =sj þ r_ 1 ∑ ν1k βjk =γ 1 þ r_ 2 ∑ ν2k βjk =γ 2 þ ⋯ þ r_ n ∑ νnk βjk =γ n
m
_ k =sk ;
∑ βjk w
k¼1
m
m
m
m
k¼1
k¼1
k¼1
k¼1
¼ ∑ βjk wk þ i1 ∑ ν1k βjk þ i2 ∑ ν2k βjk þ ⋯ þin ∑ νnk βjk
Here we consider our model to investigate heterogeneity in the
recent cholera outbreak in Haiti. This will help to improve our
understanding on the dynamics of the cholera outbreak in Haiti, as
well as to control and possibly predict future epidemics. Cholera
outbreak was confirmed in Haiti on October 21, 2010, by the
National Laboratory of Public Health of the Ministry of Public
Health and Population (MSPP) (Mukandavire et al., 2013). By
August 4, 2013, 669,396 cases and 8217 deaths have been reported
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
since the beginning of the epidemic (Centers for Disease Control
and Prevention, 2013). The outbreak started in Artibonite region, a
rural area north of Port-au-Prince, but spread to all the administrative departments in the country. This suggests that there are
connections/network between the individuals across the departments in Haiti. According to the Central Intelligence Agency (CIA)
(2013), 49% and 90% of Haiti population in rural areas do not have
access to improved drinking water sources and sanitation facilities
respectively. Furthermore, 15% and 76% of Haiti population living
in urban areas also do not have access to improved drinking water
source and sanitation facilities respectively.
In order to determine the measure of heterogeneity in the
recent cholera outbreak in Haiti, we need to estimate the contact
rates ðβjk Þ and shedding rates (νjk) for each department in Haiti.
To get reasonable results, we estimate the parameters as follows:
first, we take n ¼11 such that each sub population Ni in our model
represents a department in Haiti while the total population N
becomes the total population in Haiti. Since, we do not have
sufficient information about the water sources in Haiti, we assume
that everybody share a common water source. So we take m ¼ 1
and this implies that the measure of heterogeneity H within each
department vanishes. Therefore, we can only determine the
measure of heterogeneity H between any two departments in
Haiti. The population of each department in Haiti is taken from
2009 Haiti population data before the outbreak started (Centers
for Disease Control and Prevention (CDC), 2013). The birth/death
rate μ, recovery rate γj and net decay rate of pathogen in water
reservoir sj are taken from published data as shown in Table 2.
To estimate the contact rates ðβjk Þ and shedding rates (νjk) for
the cholera outbreak in the departments in Haiti, we used the
monthly data on the number of reported cholera cases to the
Ministry of Public Health and Population (MSPP) for the period
from October 30, 2010 to December 24, 2012. A pictorial representation of the log of number of reported hospitalized cholera
cases in each department in Haiti from October 30, 2010 to
December 24, 2012 is given in Fig. 3. We estimate ðβjk Þ and (νjk)
by fitting it to match the reported infections in each department
with other parameter values fixed as given in Table 2. The cholera
data are fit by using the built-in MATLAB (Mathworks, Version,
R2012b) least-squares fitting routine fminsearch in the optimization tool box. The estimated contact rates ðβjk Þ and shedding rates
(νjk) obtained from the fitting are given in Table 3 while the model
fitting is presented in Fig. 4.
By considering the estimated contact rates ðβjk Þ and shedding
rates (νjk) in Table 1 and applying our measure of heterogeneity
between any two sub populations as given in (66), we determine
the measure of heterogeneity between any two departments in
Haiti and is presented in Table 4. Since Hjk ¼ Hkj , we do not
include the lower diagonal in Table 4 to avoid repetition.
Even though we are not able to compute the explicit value of
the basic reproduction number when there is more than two sub
Table 3
Estimates of contact rates βj1 and shedding rates νj1 for the departments in Haiti
using the number of reported hospitalized cases from October 30, 2010 to
December 24, 2012. The population sizes were extracted from http://emergency.
cdc.gov/situationawareness/haiticholera/data.asp.
Table 2
Parameter values for numerical simulations with reference.
Parameter
Symbol Value
Reference
1
Birth/death rate
μ
0.02 day
Recovery rate in department j
γj
0.0793 day 1
Net decay rate of pathogen in
water
sj
0.0333 day 1
141
Robertson et al.
(2013)
Tien and Earn
(2010)
Tien and Earn
(2010)
Departments
Population size
βj1
νj1
βj1 νj1
Artibonite
Centre
Grande Anse
Nippes
Nord
Nord-Est
Nord-Quest
Quest
Port-au-Prince
Sud
Sud-Est
1,571,020
678,626
425,878
311,497
970,495
358,277
662,777
1,187,833
2,476,787
704,760
575,293
0.0714
0.0666
0.0633
0.0332
0.1093
0.0941
0.0498
0.0337
0.0144
0.0437
0.0240
0.8712
0.5916
0.0198
0.0653
0.0037
0.1122
0.0794
0.0091
0.1418
0.0051
0.0277
0.0622
0.0394
0.0013
0.0022
0.0004
0.0106
0.0040
0.0003
0.0020
0.0002
0.0006
10
Artibonite
Centre
Grande Anse
Nippes
Nord
Nord−Est
Nord−Quest
Quest
Port−Au−Prince
Sud
Sud−Est
9
Log(number of reported cases)
8
7
6
5
4
3
2
1
0
0
5
10
15
20
25
30
Time (months)
Fig. 3. Bar chart representing the number of reported hospitalized cholera cases in each department in Haiti from October 30, 2010 to December 24, 2012.
0.02
5
10
15
20
25
30
0
0
5
10
Nippes
0.02
0.01
0
0
5
10
15
20
25
30
0.02
10
15
20
25
30
Cummulative cases
Cummulative cases
Nord−Quest
5
0.02
10
15
20
25
30
Cummulative cases
Cummulative cases
Sud
5
0
5
10
0.05
0
0
5
10
15
20
25
30
Quest
0.01
0
5
10
15
20
20
25
30
25
30
25
30
Nord−Est
0.02
0
0
5
10
15
20
Time (months)
0.02
0
15
0.04
25
30
Port−au−Prince
0.02
0.01
0
0
5
10
Time (months)
0.04
0
0
Time (months)
Nord
Time (months)
0
30
0.02
Time (months)
0.04
0
25
0. 1
Time (months)
0
20
Grande Anse
0.04
Time (months)
Cummulative cases
Cummulative cases
Time (months)
15
Cummulative cases
0
0.02
Cummulative cases
0
Centre
0.04
Cummulative cases
Artibonite
0.04
Cummulative cases
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
Cummulative cases
142
15
20
Time (months)
Sud Est
0.02
0.01
0
0
5
10
Time (months)
15
20
25
30
Time (months)
Fig. 4. Cholera model fitting for the fraction of cumulative cholera cases where the bold lines represent the model fit and the stars mark the reported cases in the
departments from October 30, 2010 to December 24, 2012.
Table 4
Estimates of heterogeneities in cholera transmission between the departments in Haiti over the period of October 30, 2010 to December 24, 2012. Due to space, we use the
following abbreviations in the table. Atb stand for Artibonite, G-A is Grande Anse, N-E is Nord-Est, N-Q is Nord-Quest and P-P is Port-au-Prince.
Atb
Centre
G-A
Nippes
Nord
N-E
N-Q
Quest
P-P
Sud
S-E
Atb
Centre
G-A
Nippes
Nord
N-E
N-Q
Quest
P-P
Sud
S-E
0.000
0.078
0.000
0.725
0.327
0.000
0.651
0.278
0.003
0.000
0.754
0.347
0.002
0.010
0.00
0.577
0.231
0.010
0.006
0.012
0.000
0.627
0.263
0.004
0.0004
0.009
0.003
0.000
0.745
0.340
0.001
0.003
0.006
0.014
0.005
0.000
0.535
0.205
0.017
0.006
0.028
0.007
0.005
0.018
0.000
0.751
0.345
0.001
0.004
0.004
0.014
0.006
0.0001
0.020
0.000
0.714
0.320
0.002
0.002
0.008
0.012
0.003
0.0004
0.013
0.0008
0.000
populations, we know that the basic reproduction number is an
increasing function of contact rates βj1, shedding rates νj1 and their
product. From Table 3, we notice that the product βj1 νj1 is greatest
in Artibonite followed by Centre. This suggests that Artibonite
which is the where the outbreak start has the greatest number of
secondary infections. This discovery is consistent with that of
Mukandavire et al. (2013) that Artibonite has the greatest reproduction number.
Furthermore, the magnitude of heterogeneity between any two
departments in Haiti presented in Table 4 also shows that the
greatest difference in transmission is between Artibonite and
other departments. This is reasonable since the outbreak started
in Artibonite and this also support the findings from Mukandavire
et al. (2013) that Artibonite has the greatest reproduction number.
We can also see from the table that there is a significant
magnitude of heterogeneity between any two departments in
Haiti. The model fittings presented in Fig. 4 also support this
augment. From the figure we notice significant differences in the
behaviour of the fraction of reported cumulative cases. Therefore,
to improve our understanding of the cholera outbreak in Haiti as
well as defining better control strategies, heterogeneity in the
transmission dynamics must be put into considerations.
5. Discussion
Most of the factors affecting waterborne disease transmission
are not constant, hence leading to heterogeneity in disease
transmission. To improve our understanding on the effects of
heterogeneity on the dynamics of waterborne disease, we formulated an n-sub population model where disease can spread
both within a sub population and between sub populations. We
noted that heterogeneity can arise both within a sub population
and between sub populations. To understand the magnitude of
differences in transmission within a sub population, we define a
measure of heterogeneity within a sub population. Similarly, the
O.C. Collins, K.S. Govinder / Journal of Theoretical Biology 356 (2014) 133–143
magnitude of differences in transmission between any two sub
populations is also determined by defining a measure of heterogeneity between sub populations. The total variation in transmission existing in the whole population automatically becomes the
total sum of measures of heterogeneity.
A homogeneous version of the n-sub population model was
formulated to help understand whether heterogeneity has a
positive or negative impact on dynamics of the disease. Some
important mathematical features of the models such as the basic
reproduction number, type reproduction numbers and final outbreak size were determined and analysed accordingly. Our analysis
revealed that an increase in heterogeneity increases the basic
reproduction number. Since heterogeneity is more realistic, it
means that not considering heterogeneity implies an under
estimation of the basic reproduction number which might
lead to an under estimation of an outbreak – this is obviously
dangerous to the population. Similarly, we proved that the type
reproduction number increases as heterogeneity increases.
The final outbreak size relations were determined for each sub
population of the n-sub population model and consequently for
the entire population. Our analysis revealed that an increase in
heterogeneity increases the final outbreak size. Furthermore, we
discovered that sharing of water sources also increases the final
outbreak size compared to when sub populations are isolated.
We verified the analytical predictions by considering the
most recent Haiti cholera outbreak as a realistic case study. We
investigated the heterogeneity in transmission in the recent
cholera outbreak in Haiti. The results of our analysis revealed that
there are significant differences in the transmission dynamics of
cholera between the departments particularly between Artibonite
which is the starting point of the outbreak and the other departments. By fitting our model to Haiti data, we discovered that our
model (3) is applicable to the cholera dynamics in Haiti. It should
be noted that our model, (3), describes the evolution of the disease
in Haiti. It can thus provide insight into the future evolution of
cholera dynamics in Haiti. Since our model is a more general
heterogeneous model, we expected that it can be used to carry out
similar studies to other cholera-endemic countries (with different
parameter values).
Acknowledgments
O.C.C. acknowledges the financial support of the African Institute for Mathematical Sciences (AIMS) and the University of
KwaZulu-Natal, South Africa. K.S.G. thanks the University of
KwaZulu-Natal and the National Research Foundation of South
Africa for ongoing support. We are grateful to Dr. J. Tien, Ohio State
University and his research group, for providing us with the Haiti
data to compare our model.
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