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MATHEMATICAL MODELING FOR HUMAN IMMUNODEFICIENCY VIRUS (HIV) TRANSMISSION USING GENERATING FUNCTION APPROACH

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Abstract

This study is concerned with the mathematical modeling for human immun-odeficiency virus (HIV) transmission epidemics. The mathematical models are specified by stochastic differential equations that are solved by use of Generating Functions (GF). Models based on Mother to child transmission (MTCT) (age group 0-5 years), Heterosexual transmission (age group 15 and more years) and combined case (incorporating all groups and the two modes of transmission) were developed and the expectations and variances of Susceptible (S) persons, Infected (I) persons and AIDS cases were found. The S 1 (t) Sus-ceptible model produces a constant expectation and increasing variance. It was shown that Mother to Child transimission and Heterosexual models are special cases of the Combined model.
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Kragujevac J. Sci. 27 (2005) 115–130.
MATHEMATICAL MODELING FOR HUMAN
IMMUNODEFICIENCY VIRUS (HIV) TRANSMISSION
USING GENERATING FUNCTION APPROACH
Rachel Waema1and Olorunsola E. Olowofeso2
1Mathematics Department, University of Nairobi, P.O.Box 30197, Nairobi, Kenya
(email address: rwaema@yahoo.com)
2Industrial Mathematics and Computer Science Department, Federal University of
Technology, P.M.B. 704, Akure, Ondo State, Nigeria
(email address: olowofeso@yahoo.com)
(Received November 15, 2004)
ABSTRACT. This study is concerned with the mathematical modeling for human immun-
odeficiency virus (HIV) transmission epidemics. The mathematical models are specified
by stochastic differential equations that are solved by use of Generating Functions (GF).
Models based on Mother to child transmission (MTCT) (age group 0-5 years), Heterosexual
transmission (age group 15 and more years) and combined case (incorporating all groups
and the two modes of transmission) were developed and the expectations and variances of
Susceptible (S) persons, Infected (I) persons and AIDS cases were found. The S1(t) Sus-
ceptible model produces a constant expectation and increasing variance. It was shown that
Mother to Child transimission and Heterosexual models are special cases of the Combined
model.
1. INTRODUCTION
Generating functions have been applied extensively in population studies, espe-
cially in branching processes, human reproduction process, birth and death pro-
cess etc. In this study, generating function (GF) technique was used in modeling
HIV/AIDS transmission. In the literature, this approach has not been used exten-
sively by researchers to study epidemic processes. There is need to extend the appli-
cation of generating functions to HIV transmission models in modern day work. Jewel
116
(1990) studied compartmental and empirical modeling approaches. In recent time,
most of the researchers have focused on deterministic models and various approaches
for studing epidemiology of infectious diseases, AIDS inclusive, have also been devel-
oped. In this study we proceed to study the deterministic models, then develop a
stochastic differential equations from the deterministic models for the spread of the
HIV/AIDS virus in a heterosexual population and then solve the equations by using
the approach of probability generating functions. We are motivated by the following
considerations.
(i) Many biological factors such as incubation periods and social factors affecting
HIV/AIDS spread are subjected to considerable random variation so that the spread
of the AIDS virus is in essence a stochastic process.
(ii) Stochastic models provide more information than deterministic models; for exam-
ple, besides the expected values, one may also compute the variances and covariances
and assess effects of various factors on these variances and covariances.
(iii) Under certain special conditions, the deterministic approach is equivalent to
working with the expected values of the stochastic models. In this sense, the deter-
ministic approach is a special case of the stochastic models if one is only interested
in the expected values.
The section two of the work focuses on Mother-to-Child Transmission Models while
section three examined the Heterosexual Model. In section four the combined model
was discussed and section five gave the concluding remarks.
1.1. ASSUMPTIONS AND NOTATIONS FOR THE MODEL
Let
m1—Survival rate of children between ages 0 5 years
m2—Survival rate of children in ages 5 15 years
m3—Survival rate of young adults and obove
µ— the death(death unrelated to HIV/AIDS) or emigration rate (migrate out of the
population because of fear of HIV/AIDS),where k= 1,2,3 (the different age groups
have different per capita mortality rates).
ϑ1
i—Average Incubation period in stage i
117
λ—birth rate for sexually mature persons per person per time.
α— the immigration rate for the sexually mature persons be αper time, this is
independent of the population
t—Present time
x1—starting time
x2—future time ( in years )
a(t)– The expected rate of new AIDS incidences at time t.
h(t)–The expected number of new incidences of HIV infection at time t.
Y(t)–Random variable corresponding to the number of newly diagnosed
AIDS incidences at time t
Thus the probability that a birth will occur in the heterosexual population during
the time interval (t, t + ∆t) is λt+o(∆t)
Let the sexual contact rate between a matually sexual S person and an I person be
ωwhere ω0.
S(t): denote the number of persons in group S at time t
I(t): denote the number of persons in group I at time t
A(t): denote the number of persons in group AIDS case at time t
It is reasonable to assume that at the beginning of the epidemic, at t= 0, that S(0)
is large, that I(0) is fairly small, and that A(0) = 0. At time t, let N(t) represent
the size of the population. Therefore the total population consists of
N(t) = S(t) + I(t) + A(t)
(a) If the population size is n(n > 0) at time t, during the small interval of time
(t, t + ∆t), the probability that “birth”(an increase to the population) will occur
is λn(t)∆t+o(∆t). The probability of no “birth”occuring in that small interval is
1λn(t)∆t+o(∆t) and the probability of more than one“birth”occurring is o(∆t).
“birth”occuring in (t, t + ∆t) are independent of time since the last occurence.
(b)The probability that “death”will occur in a small interval of time (t, t + ∆t) is
µn(t)∆t+o(∆t),the probability of no “death”occuring is 1 µn(t)∆t+o(∆t) and the
probability that more that one “death”occurs is o(∆t). “death”occuring in (t, t + ∆t)
are independent of time since the last occurence.
118
(c) n= 0 is an absorbing state of the process.
(d) For the same population size, the “birth”and “death”occur independently of each
other.
Given a sexual contact between an S person and an I person during (t, t + ∆t) ,
we let δbe the probability that this I person will transmit the AIDS virus to the S
person. This event converts the S person to an I person. ωδ =qωmδmωfδfwhere
ωmδmis the probability that an I male transmit the AIDS virus to an S female and
ωfδfis the probability that an I female transmit the AIDS virus to an S male.
Let the rate at which an infected mother does not transmitting the HIV virus to
the newborn be β
Let the transition rate from infective to AIDS case be γ.
The changes of the population for Susceptible,Infected and AIDS cases assume Birth
and Death process.
2. MOTHER-TO-CHILD TRANSMISSION MODELS
The purpose of this Section is to develop the Mother-to-child Transmission (MTCT)
model. The study population consists of the pre-school age group (0-5 years), these
are the children born of infected and susceptible mothers in group three (15 and more
years). The population is divided into those children born free of HIV virus but can
contract the virus from their mothers through breast milk (susceptibles), those who
contact the virus from their infected mothers (infectives), and the former infectives
who develop full blown symptoms (AIDS cases).
2.1. S1(t) SUSCEPTIBLE MODEL
The change in population size during the time interval (t, t + ∆t) is governed by the
119
following conditional probabilities;
Pr{S1(t+ ∆t) = n+ 1/S1(t) = n}=nS3λt+nI3βλt+o(∆t)
Pr{S1(t+ ∆t)n+ 2/X(t) = n}=o(∆t)
Pr{S1(t+ ∆t) = n1/S1(t) = n}=np1S1+nS1µ1t+o(∆t)
Pr{S1(t+ ∆t)n2/S1(t) = n}=o(∆t)
Pr{S1(t+ ∆t) = n/S1(t) = n}= 1 nS3λtnI3βλt
np1S1nS1µ1to(∆t)
Now
λn(t) = nS3λ+nI3βλ
µn(t) = np1S1+nS1µ1
then from the given rules we have the following Kolmogorov forward differential equa-
tions:
S10
n(t) = [nS3λ+nS1µ1+nI3βλ +np1S1]S1n(t)
+ [(n1)S3λ+ (n1)I3βλ]S1n1(t)
+ [(n+ 1)p1S1+ (n+ 1)S1µ1]S1n+1(t),
for n 1 (2.1)
S0
0(t) = [p1S1+S1µ1]S1(t),for n = 0 (2.2)
where the primes indicate differentiation with respect to t. Using GF technique to
solve the differential equation gives:
GS1(Z, t) = µµS(e(ηSµS)t1) (µSe(ηSµS)tηS)Z
(ηSe(ηSµS)tµS)ηSZ(e(ηSµS)t1) i
(2.3)
This is the PGF of the differential equation (2.1)
By expanding the PGF we shall obtain the probability distribution S1(t).
Differentiating the PGF in (2.3) with respect to Z, we find the expectation and
variance of S1(t):
E[S1(t)] = i1A(t)
1B(t)=ie(ηSµS)t(2.4a)
and
δ2
S1=i(1A(t))(A(t)+B(t))
(1B(t))2
=i(ηS+µS
ηSµS)e(ηSµS)t[e(ηSµS)t1].(2.4b)
by taking the limits as µSηS( where ηSis birth rate for both infected and
Susceptible mothers) we find that
E[S1(t)] = i(2.5a)
and
δ2
S1= 2ηSt(2.5b)
120
Thus when ηS=µSthe population size has a constant expectation but an increasing
variance. Where
ηS= (S3λ+I3βλ)
and
µS= (p1+µ1)S1
2.2. I1(t)(INFECTION) MODEL
The change in population size during the time interval (t, t + ∆t) is governed by the
following conditional probabilities;
Pr{X(t+ ∆t) = n+ 1/X(t) = n}=nI3(1 β)aλt+o(∆t)
Pr{X(t+ ∆t)n+ 2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n1/X(t) = n}=nI1µ1t+nI1γt+o(∆t)
Pr{X(t+ ∆t)n2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n/X(t) = n}= 1 nI3(1 β)aλtnI1γt
nI1µ1to(∆t)
Now
λn(t) = nI3(1 β)aλ
µn(t) = nI1γ+nI1µ1
Then from the given rules we have the following Kolmogorov forward differential
equations:
I0
n(t) = [nI1µ1+nI3(1 β)aλ+nI1γ]In(t)
+ [(n1)I3(1 β)aλ+]I1n1(t)
+ [(n+ 1)I1γ+ (n+ 1)I1µ1]I1n+1(t),
for n 1 (2.6)
I0
0(t) = [I1µ1+I1γ]I1(t),for n = 0 (2.7)
Where the primes indicate differentiation with respect to t. GF was used to solve the
differential equations and the results are shown below:
GI1(Z, t) = µµI(1 e(ηIµI)t)(ηIµIe(ηIµI)t
µIηIe(ηIµI)tηIZ(1 e(ηIµI)t)(2.8)
This is the PGF of the differential equation (2.6)
. Now by simply expanding the PGF we obtain the probability distribution I1(t).
Differentiating the PGF in (2.8) with respect to Z, we find the expectation and
variance of I1(t):
E[I1(t)] = 1B(t)
1C(t)=e(ηIµI)t(2.9)
121
and
δ2
I1=(1B(t))(B(t)+C(t))
(1C(t))2
= (ηI+µI
ηIµI)e(ηIµI)t[e(ηIµI)t1].(2.10)
by taking the limits as µIηI( where ηIis birth rate for both infected and Suscep-
tible mothers) we find that
E[I1(t)] = 1
and
δ2
I1= 2ηIt
Thus when the ηI=µI, the population size has a constant expectation but an
increasing variance. Where ηI=I3(1 β)aλand µI=I1(µI+γ)
2.3. A1(t) (AIDS CASE) MODEL
The change in population size during the time interval (t, t + ∆t) is governed by the
following conditional probabilities;
Pr{X(t+ ∆t) = n+ 1/X(t) = n}=nI1γt+o(∆t)
Pr{X(t+ ∆t)n+ 2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n1/X(t) = n}=nA1µ1t+o(∆t)
Pr{X(t+ ∆t)n2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n/X(t) = n}= 1 nI1γtnA1µ1to(∆t)
Now
λn(t) = nI1γ
µn(t) = nA1µ1
Then from the given rules we have the following Kolmogorov forward differential
equations:
A10
n(t) = [nI1γ+nA1µ1]A1n(t)
+ [(nI1γ]A1n1(t)
+ (n+ 1)A1µ1A1n+1(t),
for n 1 (2.11)
A0
0(t) = [A1µ1]A1(t),for n = 0 (2.12)
Where the primes indicate differentiation with respect to t. By using the GF technique
to solve the differential equation gives:
GA1(Z, t) = 1
122
3. HETEROSEXUAL MODELS
In this section,we consider a population consisting of the adults(15 and more years).
Since the age group 2 consists of HIV free population and it is the survivors of this
subgroup over the developmental period (5,15) that generate age group 3, hence we
include the survivors in the Susceptible model.
3.1. S3(t) MODEL
The change in population size during the time interval (t, t + ∆t) is governed by the
following conditional probabilities;
Pr{S3(t+ ∆t) = n+ 1/S3(t) = n}=p2S
2t+o(∆t)
Pr{S3(t+ ∆t)n+ 2/S3(t) = n}=o(∆t)
Pr{S3(t+ ∆t) = n1/S3(t) = n}=nS3(ωδ +µ3)∆t+o(∆t)
Pr{S3(t+ ∆t)n2/S3(t) = n}=o(∆t)
Pr{S3(t+ ∆t) = n/S3(t) = n}= 1 p2S
2tnS3(ωδ +µ3)∆t
o(∆t)
Now
λn(t) = p2S
2t
µn(t) = nS3(ωδ +µ3)∆t
Then from the given rules we get the following Kolmogorov forward differential equa-
tions:
S30
n(t) = [p2S
2+nS3(ωδ +µ3)]S3n(t)
+p2S
2S3n1(t)
+ (n+ 1)S3(ωδ +µ3)S3n+1(t),
for n 1 (3.1)
S0
0(t) = p2S
2S30(t) + S3(ωδ +µ3)S31(t),for n = 0 (3.2)
where the primes indicate differentiation with respect to t. Using Gf technique we
get:
GS3(Z, t) = µ1+(Z1)eS3(ωδ+µ3)ti
{exp{−(p2S
2
s3(ωδ+µ3))(Z1)(eS3(ωδ+µ3)t1)}}
(3.3)
Now it is a simple matter of expanding the PGF to obtain the probability distribution
S3(t).
123
Differentiating the PGF in equation (3.3) with respect to Z, we find the expectation
and variance of S3(t):
E[S3(t)] = p2S
2
s3(ωδ +µ3)(1 eS3(ωδ+µ3)t) + ieS3(ωδ+µ3)t(3.4)
and
δ2(S3(t)) = ieS3(ωδ+µ3)t[1 eS3(ωδ+µ3)t] + p2S
2
s3(ωδ +µ3)[1 eS3(ωδ +µ3)t] (3.5)
3.2. I3(t) (INFECTED) MODEL
The change in population size during the time interval (t, t + ∆t) is governed by the
following conditional probabilities;
Pr{I3(t+ ∆t) = n+ 1/I3(t) = n}= +nI3ωδt+o(∆t)
Pr{I3(t+ ∆t)n+ 2/I3(t) = n}=o(∆t)
Pr{I3(t+ ∆t) = n1/I3(t) = n}=nI3µ3t+nI3γt+o(∆t)
Pr{I3(t+ ∆t)n2/I3(t) = n}=o(∆t)
Pr{I3(t+ ∆t) = n/I3(t) = n}= 1 nI3ωδtnI3γtnI3µ3to(∆t)
Now
λn(t) = nI3ωδ
µn(t) = nI3(γδ +µ3)∆t
Then from the given rules we get the difference equations:
I0
n(t) = [nI3µ3+nI3ωδ +nI3γ]In(t)
+ [(n1)I3ωδ+]In1(t)
+ [(n+ 1)I3(γ+µ3)]In+1(t),
for n 1 (3.6)
I0
0(t) = [I3µ3+I3γ]I3(t),for n = 0 (3.7)
where the primes indicate differentiation with respect to t. With the application of
GF technique we have:
GI3(Z, t) = µ(ηZ ν) + ν(1 Z)e(ην)t
(ηZ ν) + η(1 Z)e(ην)t(3.8)
We let
α(t) = ν1e(ην)t
νηe(ην)t
and
ω(t) = η
να(t)
124
Hence equation (4.8) becomes
GI3(Z, t) = µα(t) + [1 α(t)ω(t)]Z
1ω(t)Z(3.9)
This is the PGF of the differential equation (3.1)
Now it is a simple matter of expanding the PGF to obtain the probability distribution
I3(t).
Differentiating the PGF in (3.9) with respect to Z, we find the expectation and
variance of I3(t):
E[I3(t)] = 1α(t)
1ω(t)
=e(ην)t(3.10)
and
δ2
I3=(1α(t))(α(t)+ω(t))
(1ω(t))2
= (η+ν
ην)e(ην)t[e(ην)t1].(3.11)
by taking the limits as νη( where ηis birth rate for both infected and non infected
mothers) we find that
E[I3(t)] = 1
and
δ2
I3= 2ηt
Thus when the birth rate is equal to the death rate, the population size has a constant
expectation but an increasing variance.
3.3. A3(t) (AIDS CASE) MODEL
The change in population size during the time interval (t, t + ∆t) is governed by
the following conditional probabilities;
Pr{A3(t+ ∆t) = n+ 1/A3(t) = n}=nI3γt+o(∆t)
Pr{A3(t+ ∆t)n+ 2/A3(t) = n}=o(∆t)
Pr{A3(t+ ∆t) = n1/A3(t) = n}=nA3µ3t+o(∆t)
Pr{A3(t+ ∆t)n2/A3(t) = n}=o(∆t)
Pr{A3(t+ ∆t) = n/A3(t) = n}= 1 nI3γtnA3µ3to(∆t)
Now
λn(t) = nI3γ
µn(t) = nA3µ3
125
Then from the given rules we get the Kolmogorov forward diffential equations:
A0
1n(t) = [nI3γ+nA3µ3]A1n(t)
+ [(nI3γ]A1(n1)(t)
+ (n+ 1)A3µ3A1(n+1)(t),
for n 1 (3.12)
A0
0(t) = [A3µ3]A3(t),for n = 0 (3.13)
Where the primes indicate differentiation with respect to t. solving these equations
by GF technique we have:
GA3(Z, t) = 1
4. COMBINED MODELS
In this section,we consider a model which combines both the two modes of trans-
mission ( that is, Heterosexual transmission and the Mother-to-child transmission
(MTCT) and the age groups. The population is subdivided into Susceptibles, Infec-
tives and AIDS cases. We assume that there is homogeneous mixing among S persons
and I persons.
4.1. S(t) MODEL
The probability that there are nindividuals in the Susceptible population during the
time interval (t, t + ∆t) is equal to the probability;
(i)That there are nindividuals by time tand nothing happens during the time interval
(t, t + ∆t)
(ii)That there are n1 individuals by time tand 1 is added by immigration or birth
during the time interval (t, t + ∆t)
(iii) That there are n+ 1 individuals by time tand 1 dies, contracts the HIV virus or
migrates from the population during the time interval (t, t + ∆t)
The change in population size during the time interval (t, t + ∆t) is governed by the
following conditional probabilities;
Pr{X(t+ ∆t) = n+ 1/X(t) = n}=αt+nS3λt+nI3βλt+o(∆t)
Pr{X(t+ ∆t)n+ 2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n1/X(t) = n}=nSkµkt+nI3ωδt+o(∆t)
Pr{X(t+ ∆t)n2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n/X(t) = n}= 1 nS3λtαtnI3βλt
nSkµktnI3ωδto(∆t)
126
Let the probability distribution of the population size at time tbe denoted by
Sn(t) = Pr{S(t) = n/S(0) = m},m < n and m= 0,1, .......
We seek to find this distribution by deriving a system of differential equations from
the assumptions above. Now
λn(t) = nS3λ+α+nI3βλ
µn(t) = nSkµk+nI3ωδ
Then from the given rules we have the following Kolmogorov forward diffrential equa-
tions:
S0
n(t) = [nS3λ+α+nSkµk+nI3βλ +nI3ωδ]Sn(t)
+ [α+ (n1)S3λ+ (n1)I3βλ]Sn1(t)
+ [(n+ 1)Skµk+ (n+ 1)I3ωδ]Sn+1(t),
for n 1 (4.1)
S0
0(t) = αS0(t)+[Skµk+I3ωδ]S1(t),for n = 0 (4.2)
where the primes indicate differentiation with respect to t.
Using the Generating Function technique we arrive at the generating function, ex-
pectation and Variance of the Susceptible persons.
G(Z, t) = (ην)α/η [(νe(ην)tν)Z(νe(ην)tη)]m
[(ηe(ην)tν)ηZ(e(ην)t1)]α/η+m(4.3)
Differentiating the PGF in (4.3) with respect to Z, we find the expectation and
variance of S(t):
E[S(t)] = me(ην)t+αe(ην)t1
(ην)(4.4)
and
δ2
S=m(η+ν
ην)e(ην)t[e(ην)t1] + αe(ην)t1
(ην).(4.5)
Where η= ((S3λ+I3βλ) and ν= (Skµk+I3ωδ).
4.2. I(t) MODEL
The change in population size during the time interval (t, t + ∆t) is governed by the
following conditional probabilities;
Pr{X(t+ ∆t) = n+ 1/X(t) = n}=αt+nI3(1 β)aλt+nI3ωδt+o(∆t)
Pr{X(t+ ∆t)n+ 2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n1/X(t) = n}=nIkµkt+nI3γt+o(∆t)
Pr{X(t+ ∆t)n2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n/X(t) = n}= 1 nI3(1 β)aλtαt
nI3γtnSkµktnI3ωδt
127
Now
λn(t) = α+nI3(1 β)aλ
µn(t) = nIkµk+nI3γ
Then from the given rules we get the following Kolmogorov forward diffrential equa-
tions:
I0
n(t) = [nIkµk+α+nI3(1 β)aλ+nI3ωδ +nI3γ]In(t)
+ [(n1)I3(1 β)aλ+ (n1)I3ωδ +α]In1(t)
+ [(n+ 1)I3γ+ (n+ 1)Ikµk]In+1(t),
for n 1 (4.6)
I0
0(t) = αI0(t) + [Ikµk+I3γ]S1(t),for n = 0 (4.7)
Applying the Generating function technique we have:
G(Z, t) = (κρ)α/ρ[κe(ρκ)t1) Z(κe(ρκ)t1)]
[(ρe(ρκ)tκρZ(e(ρκ)t1)]1+α/ρ (4.8)
Differentiating the PGF in (4.8) with respect to Z, we find the expectation and
variance of S(t):
E[S(t)] = e(ρκ)t+αe(ρκ)t1
(ρκ)(4.9)
and
δ2
S= (ρ+κ
ρκ)e(ρκ)t[e(ρκ)t1] + αe(ρκ)t1
(ρκ).(4.10)
where ρ= ((1 β)aλ+ωδ) and κ= (µk+γ).
4.3. A(t) MODEL
The change in population size during the time interval (t, t + ∆t) is governed by the
following conditional probabilities;
Pr{X(t+ ∆t) = n+ 1/X(t) = n}=αt+nIγt+o(∆t)
Pr{X(t+ ∆t)n+ 2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n1/X(t) = n}=nAµkt+o(∆t)
Pr{X(t+ ∆t)n2/X(t) = n}=o(∆t)
Pr{X(t+ ∆t) = n/X(t) = n}= 1 nAµktαt
nIγto(∆t)
Now
γn(t) = nI γ +α
µn(t) = nAµk
128
Then from the given rules we get the following Kolmogorov forward differential equa-
tions: A0
n(t) = [nAµk+α+nIγ]An(t)
+ [α+ (n1)Iγ]An1(t)
+k(n+ 1)An+1(t),
for n 1 (4.11)
A0
0(t) = αA0(t) + kA1(t),for n = 0 (4.12)
Where the primes indicate differentiation with respect to tWith the application of
GF technique we arrive at:
G(Z, t) = µIγ
Iγe( )t α/Iγ ·1Z Iγ(e(I γ )t1)
Iγe( )t¸α/I γ
(4.13)
This is a negative binomial distribution, with
p=µIγ
Iγe( )t
and
r=α/Iγ
It is of some interest to consider the limiting form of equation (4.13) when < Aµ
and the time ttends to infinity. The limiting generating function is
G(Z, t) = (1 Iγ/Aµ)α/Iγ (1 IγZ/Aµ)α/Iγ
and so the mean population size for large tis
α
(Iγ)
This is related to the stable distribution of population which immigration can just
maintain against the excess of over Iγ.
The variance of the population size for large tis
αAµ
(Iγ )2
When = 0, ( that is, when there are only births and immigration and new
infections) it is clear from equation (4.1) that the distribution will still be negative
binomial for every finite value of t.
G(Z, t) = Iγα/Iγ ·1Z(1 eIγt )¸α/I γ
129
On the hand, when Iγ = 0,(that is, when there is immigration , emigration and
HIV infection)where emigration and HIV infection depends on the population, the
distribution assumes a Poisson process.
G(Z, t) = e½α
(1eAµt)(Z1)¾
When t→ ∞, it gives
G(Z) = e{α
(Z1)}
When Iγ = 0,= 0 (that is, when there is only immigration ), the distribution
assumes a Poisson process with parameter αt.
G(Z, t) = eαt(Z1)
5. CONCLUDING REMARKS
In this paper, we developed HIV/AIDS epidemic models by using Generating
functions (GF). We came up with a conceptual framework which summarizes all the
concepts of HIV/AIDS transmission models. Stochastic models based on Mother to
child transmission(MTCT), Heterosexual transmission and Combined models were
developed. By using the stochastic models formulated, we have also demonstrated
how various factors affect the expectations of susceptible and infective persons. It
is shown from the combined model that Mother to Child transmission and Hetero-
sexual models are special cases of the Combined model. However, in the process of
achieving our goals, some problems were encountered; based on the initial condition,
it was found that when the initial condition is assumed to be zero (0), in the case of
AIDS case, most of the models showed that the Generating function is one (1), this
area need further investigation.
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Chapter
HIV is a scourge and transmitted mainly in four ways, viz., through sexual intercourse and blood transfusion, among injecting drug users, and from mother to child (MTC) (Conner et al. 1994). However, there are individual differences in the ability to transmit and acquire HIV, which remain unexplained (Abbink et al. 2003; Adriaanse et al. 1995; Agrawal et al. 2013; Ahmed et al. 1992; Aitkhaled et al. 1998; Alary et al. 2010). HIV transmission, as a result of sexual intercourse, accounts for about three quarters of all HIV infections worldwide (WHO 2018). HIV is therefore a sexually transmitted disease (STD); the majority of the world’s infections have been acquired through intercourse between men and women (heterosexual transmission) (Kretzschmar et al. 2001). A number of studies pertaining to HIV transmission at global level showed that the likelihood of HIV was found to be statistically in an increasing trend in poor and developing countries, with the number of sexual partners and anal receptive intercourse among gay population (WHO 2015; Agarwala et al. 2002). As with some other sexually transmitted diseases (STDs), HIV infection can also be transmitted through blood, for example, medical transfusion of infected blood or blood products, or from the use of contaminated injection and equipments by intravenous drug users (Rollins et al. 2007). The transmission of HIV from mother to child includes transmission during pregnancy, during delivery, and through breastfeeding (Mustapha et al. 2010). Infected children under 15 years (≤ 15 years) have contracted the virus by transmission from their mothers; their number reflects the prevalence in women of childbearing age (Ayaya et al. 2013). The schematic HIV mother to child transmission (MTCT) flow diagram and their risk factors are presented in Fig. 1.1. Different modes of HIV transmission were previously studied by Akaike et al. (1971), Akpa et al. (2010), Alary et al. (2010), Alimenti et al. (1991), Anderson et al. (1998), Anderson et al. (1986), Andrea et al. (2008), Andrijich et al. (2001), Anon et al. (1999), Arias et al. (1997), Arya et al. (1998), Avert et al. (2008), Ayaya et al. (2013), and Bai and Yin (1998).
Chapter
We consider the problem of developing joint models for a periodically observed marker of underlying disease progression and its relationship to either onset of disease or occurrence of a disease-related endpoint. We use the framework of relative risk regression models with time-dependent covariates to specify the relationship between marker and disease onset and use a mixed linear model to describe the evolution of the marker process. The construction of partial likelihood is discussed and a two step procedure is described for estimation of parameters in the model. For a special case a heuristic development of a large sample distribution theory for the proposed estimators is presented which suggests variance estimators. The method is illustrated by applying it to an analysis of periodic measurements of T4 and T8 cells and time from seroconversion to AIDS diagnosis.
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