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115

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).

ϑ−1

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 λ∆t+o(∆t)

Let the sexual contact rate between a matually sexual S person and an I person be

ωwhere ω≥0.

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

1−λn(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}=nS3λ∆t+nI3βλ∆t+o(∆t)

Pr{S1(t+ ∆t)≥n+ 2/X(t) = n}=o(∆t)

Pr{S1(t+ ∆t) = n−1/S1(t) = n}=np1S1+nS1µ1∆t+o(∆t)

Pr{S1(t+ ∆t)≤n−2/S1(t) = n}=o(∆t)

Pr{S1(t+ ∆t) = n/S1(t) = n}= 1 −nS3λ∆t−nI3βλ∆t

−np1S1−nS1µ1∆t−o(∆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)

+ [(n−1)S3λ+ (n−1)I3βλ]S1n−1(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(ηS−µS)t−1) −(µSe(ηS−µS)t−ηS)Z

(ηSe(ηS−µS)t−µS)−ηSZ(e(ηS−µS)t−1) ¶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)] = i1−A(t)

1−B(t)=ie(ηS−µS)t(2.4a)

and

δ2

S1=i(1−A(t))(A(t)+B(t))

(1−B(t))2

=i(ηS+µS

ηS−µS)e(ηS−µS)t[e(ηS−µS)t−1].(2.4b)

by taking the limits as µS→ηS( 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)aλ∆t+o(∆t)

Pr{X(t+ ∆t)≥n+ 2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n−1/X(t) = n}=nI1µ1∆t+nI1γ∆t+o(∆t)

Pr{X(t+ ∆t)≤n−2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n/X(t) = n}= 1 −nI3(1 −β)aλ∆t−nI1γ∆t

−nI1µ1∆t−o(∆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)

+ [(n−1)I3(1 −β)aλ+]I1n−1(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(ηI−µI)t)−(ηI−µIe(ηI−µI)t

µI−ηIe(ηI−µI)t−ηIZ(1 −e(ηI−µI)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)] = 1−B(t)

1−C(t)=e(ηI−µI)t(2.9)

121

and

δ2

I1=(1−B(t))(B(t)+C(t))

(1−C(t))2

= (ηI+µI

ηI−µI)e(ηI−µI)t[e(ηI−µI)t−1].(2.10)

by taking the limits as µI→ηI( 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}=nI1γ∆t+o(∆t)

Pr{X(t+ ∆t)≥n+ 2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n−1/X(t) = n}=nA1µ1∆t+o(∆t)

Pr{X(t+ ∆t)≤n−2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n/X(t) = n}= 1 −nI1γ∆t−nA1µ1∆t−o(∆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]A1n−1(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∗

2∆t+o(∆t)

Pr{S3(t+ ∆t)≥n+ 2/S3(t) = n}=o(∆t)

Pr{S3(t+ ∆t) = n−1/S3(t) = n}=nS3(ωδ +µ3)∆t+o(∆t)

Pr{S3(t+ ∆t)≤n−2/S3(t) = n}=o(∆t)

Pr{S3(t+ ∆t) = n/S3(t) = n}= 1 −p2S∗

2∆t−nS3(ωδ +µ3)∆t

−o(∆t)

Now

λn(t) = p2S∗

2∆t

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∗

2S3n−1(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+(Z−1)e−S3(ωδ+µ3)t¶i

{exp{−(p2S∗

2

s3(ωδ+µ3))(Z−1)(e−S3(ωδ+µ3)t−1)}}

(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 −e−S3(ωδ+µ3)t) + ie−S3(ωδ+µ3)t(3.4)

and

δ2(S3(t)) = ie−S3(ωδ+µ3)t[1 −e−S3(ωδ+µ3)t] + p2S∗

2

s3(ωδ +µ3)[1 −e−S3(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}= +nI3ωδ∆t+o(∆t)

Pr{I3(t+ ∆t)≥n+ 2/I3(t) = n}=o(∆t)

Pr{I3(t+ ∆t) = n−1/I3(t) = n}=nI3µ3∆t+nI3γ∆t+o(∆t)

Pr{I3(t+ ∆t)≤n−2/I3(t) = n}=o(∆t)

Pr{I3(t+ ∆t) = n/I3(t) = n}= 1 −nI3ωδ∆t−nI3γ∆t−nI3µ3∆t−o(∆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)

+ [(n−1)I3ωδ+]In−1(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(η−ν)t

(ηZ −ν) + η(1 −Z)e(η−ν)t¶(3.8)

We let

α(t) = ν1−e(η−ν)t

ν−ηe(η−ν)t

and

ω(t) = η

να(t)

124

Hence equation (4.8) becomes

GI3(Z, t) = µα(t) + [1 −α(t)−ω(t)]Z

1−ω(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)] = 1−α(t)

1−ω(t)

=e(η−ν)t(3.10)

and

δ2

I3=(1−α(t))(α(t)+ω(t))

(1−ω(t))2

= (η+ν

η−ν)e(η−ν)t[e(η−ν)t−1].(3.11)

by taking the limits as ν→η( 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}=nI3γ∆t+o(∆t)

Pr{A3(t+ ∆t)≥n+ 2/A3(t) = n}=o(∆t)

Pr{A3(t+ ∆t) = n−1/A3(t) = n}=nA3µ3∆t+o(∆t)

Pr{A3(t+ ∆t)≤n−2/A3(t) = n}=o(∆t)

Pr{A3(t+ ∆t) = n/A3(t) = n}= 1 −nI3γ∆t−nA3µ3∆t−o(∆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(n−1)(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 n−1 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}=α∆t+nS3λ∆t+nI3βλ∆t+o(∆t)

Pr{X(t+ ∆t)≥n+ 2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n−1/X(t) = n}=nSkµk∆t+nI3ωδ∆t+o(∆t)

Pr{X(t+ ∆t)≤n−2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n/X(t) = n}= 1 −nS3λ∆t−α∆t−nI3βλ∆t

−nSkµk∆t−nI3ωδ∆t−o(∆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+ (n−1)S3λ+ (n−1)I3βλ]Sn−1(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) = (η−ν)α/η [(νe(η−ν)t−ν)−Z(νe(η−ν)t−η)]m

[(ηe(η−ν)t−ν)−ηZ(e(η−ν)t−1)]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(η−ν)t+αe(η−ν)t−1

(η−ν)(4.4)

and

δ2

S=m(η+ν

η−ν)e(η−ν)t[e(η−ν)t−1] + αe(η−ν)t−1

(η−ν).(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}=α∆t+nI3(1 −β)aλ∆t+nI3ωδ∆t+o(∆t)

Pr{X(t+ ∆t)≥n+ 2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n−1/X(t) = n}=nIkµk∆t+nI3γ∆t+o(∆t)

Pr{X(t+ ∆t)≤n−2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n/X(t) = n}= 1 −nI3(1 −β)aλ∆t−α∆t

−nI3γ∆t−nSkµk∆t−nI3ωδ∆t

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)

+ [(n−1)I3(1 −β)aλ+ (n−1)I3ωδ +α]In−1(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) = (κ−ρ)α/ρ[κe(ρ−κ)t−1) −Z(κe(ρ−κ)t−1)]

[(ρe(ρ−κ)t−κ−ρZ(e(ρ−κ)t−1)]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(ρ−κ)t+αe(ρ−κ)t−1

(ρ−κ)(4.9)

and

δ2

S= (ρ+κ

ρ−κ)e(ρ−κ)t[e(ρ−κ)t−1] + αe(ρ−κ)t−1

(ρ−κ).(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}=α∆t+nIγ∆t+o(∆t)

Pr{X(t+ ∆t)≥n+ 2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n−1/X(t) = n}=nAµk∆t+o(∆t)

Pr{X(t+ ∆t)≤n−2/X(t) = n}=o(∆t)

Pr{X(t+ ∆t) = n/X(t) = n}= 1 −nAµk∆t−α∆t

−nIγ∆t−o(∆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+ (n−1)Iγ]An−1(t)

+Aµk(n+ 1)An+1(t),

for n ≥1 (4.11)

A0

0(t) = −αA0(t) + AµkA1(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 −Aµ

Iγe(Iγ −Aµ)t−Aµ ¶α/Iγ ·1−Z Iγ(e(I γ −Aµ)t−1)

Iγe(Iγ −Aµ)t−Aµ ¸−α/I γ

(4.13)

This is a negative binomial distribution, with

p=µIγ −Aµ

Iγe(Iγ −Aµ)t−Aµ ¶

and

r=α/Iγ

It is of some interest to consider the limiting form of equation (4.13) when Iγ < 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

(Aµ −Iγ)

This is related to the stable distribution of population which immigration can just

maintain against the excess of Aµ over Iγ.

The variance of the population size for large tis

αAµ

(Iγ −Aµ)2

When Aµ = 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 ·1−Z(1 −e−Iγt )¸−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

Aµ (1−e−Aµt)(Z−1)¾

When t→ ∞, it gives

G(Z) = e{α

Aµ(Z−1)}

When Iγ = 0,Aµ = 0 (that is, when there is only immigration ), the distribution

assumes a Poisson process with parameter αt.

G(Z, t) = eαt(Z−1)

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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