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Tuberculosis models with fast and slow dynamics: the role of close and casual contacts

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Diseases with chronic stage in a population with varying size
Maia Martcheva
a,*
, Carlos Castillo-Chavez
b
a
Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA
b
Department of Biological Statistics and Computational Biology, Cornell University, Ithaca, NY 14853, USA
Received 6 July 2002; received in revised form 13 September 2002; accepted 24 September 2002
Abstract
An epidemiological model of hepatitis C with a chronic infectious stage and variable population size isintroduced. A non-structured baseline ODE model which supports exponential solutions is discussed. Thenormalized version where the unknown functions are the proportions of the susceptible, infected, andchronic individuals in the total population is analyzed. It is shown that sustained oscillations are notpossible and the endemic proportions either approach the disease-free or an endemic equilibrium. Theexpanded model incorporates the chronic age of the individuals. Partial analysis of this age-structuredmodel is carried out. The global asymptotic stability of the infection-free state is established as well as localasymptotic stability of the endemic non-uniform steady state distribution under some additional condi-tions. A numerical method for the chronic-age-structured model is introduced. It is shown that this nu-merical scheme is consistent and convergent of ﬁrst order. Simulations based on the numerical methodsuggest that in the structured case the endemic equilibrium may be unstable and sustained oscillations arepossible. Closer look at the reproduction number reveals that treatment strategies directed towardsspeeding up the transition from acute to chronic stage in eﬀect contribute to the eradication of the disease.
2002 Elsevier Science Inc. All rights reserved.
Keywords:
Chronic stage; Variable infectivity; Disease-age structure; Hepatitis C; Variable population size; Diﬀerencescheme; Numerical methods; Sustained oscillations
1. Introduction
The impact of a chronic stage on the disease transmission and behavior in an exponentiallygrowing or decaying population is the focus of this paper. The framework is applied to the case of
Mathematical Biosciences 182 (2003) 1–25
www.elsevier.com/locate/mbs
*
Corresponding author. Present address: Department of Biological Statistics and Computational Biology, 434Warren Hall, Cornell University, Ithaca, NY 14853-7801, USA. Tel.: +1-718 260 3294; fax: +1-718 260 3660.
E-mail addresses:
mayam@duke.poly.edu (M. Martcheva), cc32@cornell.edu (C. Castillo-Chavez).0025-5564/02/$ - see front matter
2002 Elsevier Science Inc. All rights reserved.PII: S0025-5564(02)00184-0
hepatitis C, a disease characterized by a long chronic stage. Hepatitis C, priorly referred to as
non-A, non-B
hepatitis, is a viral infection of the liver which was ﬁrst recognized as a separatedisease in 1975. The causative agent was identiﬁed in 1989. The pathogen responsible for hepatitisC virus (HCV) is quite diﬀerent from the causative agents for hepatitis A and B. It is an envelopedRNA virus in the ﬂaviviridae family [1]. Researchers have classiﬁed more than 100 strains of thevirus. Hepatitis C has been identiﬁed as the most common cause of post-transfusion hepatitisworldwide. It accounts for
90% of this disease in Japan, USA and Western Europe. HCV isresponsible for both acute and chronic cases of hepatitis C.Data from the World Health Organization suggest that up to 3% of the world population hasbeen infected with HCV. It is expected that more than 170 million people are chronic carriers andat risk of developing liver cirrhosis and/or liver cancer. In the United States, the annual number of acute HCV infections has declined during the past decade from 180000 to 35000, but an ap-proximately 3.9 million Americans (1.8% of the population of the US) are currently infected withHCV, and an estimated 8000–10000 deaths each year are attributed to HCV-associated chronicliver disease [2].Hepatitis C is transmitted mainly through blood transfusions. However, in 1990 an eﬀectivescreening test was developed and the transmission has begun to decrease, at least in the developedcountries. HCV can also be contracted via contacts with inadequately sterilized or unsterilizedequipment. Needle-sharing among drug-users is also a risk factor. Two major risk groups are thehealth care personnel (needle-stick injuries) as well as patients undergoing hemodialysis. Evidenceof sexual and perinatal transmission exist, although these ways of transmission seem to occur lessfrequently and their relevancy and eﬀectiveness are often questioned. Ear and body piercing, cir-cumcision, tattooing and other percutaneous procedures may also be a source of HCV infections.The majority of patients with acute hepatitis C develop a chronic infection which is charac-terized by detection of HCV RNA for a period of at least six months after a newly acquired in-fection. Data suggest that at least 85% of patients with acute HCV infection eventually progress tothe chronic stage. Chronic hepatitis C can lead to cirrhosis and end-stage liver disease (HCC).Cirrhosis can develop either within 1–2 years or after 2–3 decades following exposure. Some longterm studies suggest that cirrhosis develops in 20–30% of patients. A fraction (1–5%) of cirrhoticindividuals develop cancer of the liver in about a decade after developing cirrhosis.The disease has extremely slow average rate of progression, ranging from 2 to 4 or even moredecades. Using data collected in Japan, investigators estimated that, following acute infection,chronic hepatitis could be identiﬁed 13
:
7
10
:
9 years later, chronic active hepatitis 18
:
4
11
:
2years later, cirrhosis 20
:
6
10
:
1 years later, and HCC, 28
:
3
11
:
5 years later [3].The most common symptoms of acute hepatitis C are fatigue and jaundice. However, the vastmajority of cases (up to 90%), including those with chronic disease, are asymptomatic. This makesthe diagnosis of hepatitis C very diﬃcult and is the reason why the HCV epidemic is often called
the silent epidemic
[4].The enzyme immunosorbent assay is used for the detection of speciﬁc antibodies which conﬁrmthepresence ofHCV. Thepresence of HCVRNAin serumindicates thepresence of activeinfectionand a potential for transmission of the infection and/or the development of chronic liver disease.Hepatitis C can be treated with interferon. The treatment is eﬀective in about 20% of the cases.Ribavirin in combination with interferon is eﬀective as an antiviral agent against HCV in ap-proximately 40% of previously untreated patients. The combination therapy has been approved by
2
M. Martcheva, C. Castillo-Chavez / Mathematical Biosciences 182 (2003) 1–25
FDA in 1998. However, it also appears that the cost of such combined treatment is high – about$20000 for 48-week treatment. No vaccine is available for hepatitis C. The high mutability of hepatitis C genome [5] complicates its development. There is no evidence that the successfultreatmentofHCVgivesanykindofpartialortemporaryimmunity.Hencethemodelsdevelopedfallwithin the class of models that treated or recovered individuals move back to the susceptible class.Several studies exist treating epidemics in populations with active demographic patterns, par-ticularly in the ODE case, [6 – 10]. The epidemic contact process or the force of infection is typi-
cally assumed to be modeled by proportionate mixing. Derrick and van den Driessche [11] discussa general SIRS disease transmission model in which the population varies and the rate at whichsusceptibles become infective is given by a rather general non-linear function. Another model withgeneral contact function is treated in [12] where the eﬀective contact rate is an arbitrary functionof the total population size. The rate of transmission used here allows for the possibility of generation of secondary infections via contacts with two types of infectious individuals: those withacute and those with chronic infections.Very little analysis of diseases with chronic stage has been performed. In fact, the only twoworks known to the authors are [13,14]. A model of Cytomegalovirus infection with disease-agestructured chronic stage is considered in [13]. A model structured by age-since-infection has alsobeen discussed in relation to HIV in [15,16]. Reade et al. consider an ODE model for infections
with acute and chronic phases with feline calicivirus [14]. Their work is mostly numerical andconcentrates on the impact of vaccination on the acute and chronic phases.A mathematical model of hepatitis C has been considered in [17]. The authors use a backcal-culation approach to reconstruct the history of HCV infection. Their parameters are estimatedwith data from France. The model is structured by age and sex since there is evidence thatprogression to cirrhosis depends strongly on these two characteristics. The model traces thehepatitis C epidemic in France back to the 1940s and makes projections until the year 2025.This paper is structured as follows. Section 2 presents and analyzes the baseline ODE model of hepatitis C infection. Section 3 extends the model of Section 2 to the case where the chronic classis structured by age-since-infection. The basic reproduction number is computed and used to showthat the disease-free equilibrium is locally stable and to establish the existence of a unique endemicstate. Section 4 discusses the global stability of the disease-free equilibrium and the local stabilityof the unique endemic equilibrium. In Section 5 we present a numerical method and discuss theresults of our simulation. Section 6 summarizes our ﬁndings and conclusions. The Appendix Acontains the proof of Theorem 5.2.
2. A simple ODE model
The asymptomatic nature of hepatitis C and its slow progression make it diﬃcult to charac-terize the natural history of the disease. The following assumptions are used in the construction of the models.(1) Only the acute and chronic stages are diﬀerentiated. Patients with either acute or chronic in-fections are capable of transmitting the disease.(2) All infected individuals develop an acute hepatitis C ﬁrst.
M. Martcheva, C. Castillo-Chavez / Mathematical Biosciences 182 (2003) 1–25
3
(3) Some individuals with an acute infection progress towards the chronic state and later on de-velop cirrhosis.(4) Since the disease-induced death rate is relatively low, it is ignored.(5) The acute stage of infection is short and often asymptomatic and there is no possibility fortreatment during this state.Since the disease progression is slow the model should also incorporate demographic changes inthe population. It is assumed that the total population is growing exponentially ([18], Fig. 1.8).We divide the population in three subclasses:
S
––susceptible;
I
––infected with acute hepatitis C;
V
––infected with chronic hepatitis C (with or without cirrhosis);
N
¼
S
þ
I
þ
V
(total number of people).The model introduces the following rates of transition from one class to another (parameters):
b
––birth/recruitment rate;
l
––death rate;
a
––treatment/recovery rate for the chronic state;
k
––rateof progression to chronic stage;
c
––eﬀective contact rate of individuals with acute hepatitis C;
d
–– eﬀective contact rate of individuals with chronic hepatitis C. All parameters are assumed positive.The model is homogeneous (of degree one).
S
0
¼
bN
ð
c
I
þ
d
V
Þ
S N
l
S
þ
a
V I
0
¼ ð
c
I
þ
d
V
Þ
S N
ð
l
þ
k
Þ
I V
0
¼
kI
ð
l
þ
a
Þ
V
8<:
:
ð
2
:
1
Þ
The demographic equation for the dynamics of the total population size is given by:
N
0
¼
bN
l
N
:
We set
r
¼
b
l
. We obtain
N
0
¼
rN
and therefore,
N
¼
N
0
e
rt
. Hence,
r
gives the growth rate of the population. If
r
>
0, that is
b
>
l
, the population grows exponentially, if
r
<
0, that is
b
<
l
,the population decreases exponentially. The case
r
¼
0 or
b
¼
l
implies that the population isstationary. These thresholds are often interpreted in terms of the demographic reproductionnumber. In particular, in [19] a
net reproduction number of the population
is thus deﬁned as
R
¼
b
l
:
Clearly, if
R
>
1 the population is exponentially increasing, if
R
<
1 the population is expo-nentially decreasing, and if
R
¼
1, the population is stationary. Since the model (2.1) is homo-geneous of degree one, we consider also the equations for the normalized quantities. Setting
s
¼
S
=
N
,
i
¼
I
=
N
,
v
¼
V
=
N
leads to the following equivalent non-homogeneous system
s
0
¼
b
ð
1
s
Þ ð
c
i
þ
d
v
Þ
s
þ
a
vi
0
¼ ð
c
i
þ
d
v
Þ
s
ð
b
þ
k
Þ
iv
0
¼
ki
ð
b
þ
a
Þ
v
8<:
ð
2
:
2
Þ
where
s
þ
i
þ
v
¼
1
:
ð
2
:
3
Þ
In a sequence of papers Hadeler [20 – 22] develops a theory of ODEs with homogeneous non-
linearities. These models, in general, do not have time independent solutions. Instead, they supportsolutions of the form
S
¼
e
k
t
S
,
I
¼
e
k
t
I
,
V
¼
e
k
t
V
. The normalized model (2.2), on the otherhand, is non-linear, non-homogeneous and it has steady states in the classical sense. Hadeler callsthe exponential solutions of (2.1) stable (unstable), if the corresponding steady states of (2.2) are
4
M. Martcheva, C. Castillo-Chavez / Mathematical Biosciences 182 (2003) 1–25
stable (unstable) in the classical sense. Under these conditions investigating model (2.1) is to a largeextent reduced to investigating model (2.2).The disease-free state of (2.2) is (1,0,0) and the epidemiological reproduction number for theproportions is
R
0
¼
c
b
þ
k
þ
k
d
ð
b
þ
k
Þð
b
þ
a
Þ
:
ð
2
:
4
Þ
The ﬁrst term
c
=
ð
b
þ
k
Þ
can be interpreted as the contribution to the reproduction number due tosecondary infections generated by an infective with acute hepatitis C. Naturally, it increases as theeﬀective contact rate,
c
, of individuals with acute hepatitis C increases. The second term
k
d
=
ð
b
þ
k
Þð
b
þ
a
Þ
is the contribution to the reproduction number due to secondary infections generated byan infective with chronic hepatitis C. It increases with the increase of eﬀective contact rate of chronic individuals,
d
, as well as with the rate of progression to chronic stage,
k
. Overall, thereproduction number
R
0
has a more complicated response to variations of the rate of progressionto chronic stage,
k
. In particular, it decreases when
b
ð
d
c
Þ
ac
<
0 and increases when theopposite inequality is valid. However, the probability of transmitting the disease from an indi-vidual with acute infection is larger than that from an individual with chronic infection, that is,
d
<
c
. Therefore, we expect that for realistic values of the parameters
R
0
will decrease as the rateprogression to chronic stage increases.
Lemma 2.1.
If
R
0
<
1
then the disease-free equilibrium (1,0,0) is globally stable. If
R
0
>
1
thedisease-free equilibrium is unstable and there is a unique endemic equilibrium
ð
s
;
i
;
v
Þ
which islocally stable
.
Proof.
The Jacobian of (2.2) is
J
ð
1
;
0
;
0
Þ ¼
b
c
d
þ
a
0
c
ð
b
þ
k
Þ
d
0
k
ð
b
þ
a
Þ
0@1A
:
This is a block-triangular matrix. It can be seen that one of the eigenvalues is
b
. The other twohave negative real part if and only if
c
ð
b
þ
k
Þ ð
b
þ
a
Þ
<
0 and
½
c
ð
b
þ
k
Þ½ð
b
þ
a
Þ
k
d
>
0. The second inequality is equivalent to
R
0
<
1 (in fact, this can be used for the compu-tation of
R
0
). The condition
R
0
<
1 guarantees both inequalities. If
R
0
>
1 the second inequalityis not satisﬁed and the disease-free equilibrium is unstable. In this case an endemic equilibriumexists and is given by
ð
s
;
i
;
v
Þ
, where
s
¼
1
R
0
;
i
¼ ð
b
þ
a
Þ
b
þ
a
þ
k
1
1
R
0
;
v
¼
k b
þ
a
þ
k
1
1
R
0
:
ð
2
:
5
Þ
Since
s
<
1,
i
<
1,
v
<
1 the quantities in (2.5) are well deﬁned. To see the global stability of thedisease-free equilibrium, we integrate the second and third equation in (2.2) to obtain the fol-lowing pair of inequalities:
M. Martcheva, C. Castillo-Chavez / Mathematical Biosciences 182 (2003) 1–25
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