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# EVALUATING THE DETERMINISTIC SEIRUS MODEL FOR DISEASE CONTROL IN AN AGE-STRUCTURED POPULATION (Research Paper Sample)

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This paper focuses on the development and analysis of the endemic model for disease control in an aged-structured population in Kenya. Upon the model framework development, the model equations were transformed into proportions with the rate of change of the different compartments forming the model, thereby reducing the model equations from twelve to ten homogenous ordinary differential equations. The model exhibits two equilibria, the endemic state, and the disease-free equilibrium state while successfully achieving a Reproductive Number R_0=0. The deterministic endemic SEIRUS model is analyzed for the existence and stability of the disease-free equilibrium state. Numerical simulations were carried to complement the analytical results in investigating the effect treatment rate and the net transmission rate on recovery for both juvenile and adult sub-population in an age-structured population.

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EVALUATING THE DETERMINISTIC SEIRUS MODEL FOR DISEASE CONTROL IN AN AGE-STRUCTURED POPULATION
ABSTRACT
This paper focuses on the development and analysis of the endemic model for disease control in an aged-structured population in Kenya. Upon the model framework development, the model equations were transformed into proportions with rate of change of the different compartments forming the model, thereby reducing the model equations from twelve to ten homogenous ordinary differential equations. The model exhibits two equilibria, the endemic state and the disease-free equilibrium state while successfully achieving a Reproductive Number R0=0. The deterministic endemic SEIRUS model is analyzed for the existence and stability of the disease-free equilibrium state. Numerical simulations were carried to complement the analytical results in investigating the effect treatment rate and the net transmission rate on recovery for both juvenile and adult sub-population in an age-structured population.
Keyword: Susceptible, Exposed, Latent, Infectious, Removed, Recovery, Undetectable.
1 INTRODUCTION
This study aims to develop and evaluate the new deterministic endemic age-structured SEIRUS compartmental model of the HIV/AIDS dynamics. As a result, a two-age-structured population framework for a deterministic endemic model is constructed for the development of an endemic deterministic model with Undetectable=Untrasmittable viral load compartment.
Previous studies (Oduwole, H. K. and Kimbir, A. R. (2018) and Mugisha, J.Y.T. and Luboobi, L.S. (2003)) have focused on the epidemiology of HIV/AIDS using various models like SIR, SEIR, SIRS, SICA models which are formulated for epidemic case of disease control. These models however, do not take into account the endemic nature/state of the diseases therefore making unrealistic to effectively control the further spread and eventual eradication of the disease. This has led to the setbacks of various interventions for the eradication of HIV/AIDS which is also a focal point of the global Sustainable Development Goals (SGDs).
However, in this paper the endemic nature of the disease with a new endemic deterministic model is investigated with an inclusion of the Undetectable=Untrasmitable (U=U) viral load compartment of the age-structured population.
2 THE MODEL VARIABLES AND PARAMETERS
The model variables and parameters for the investigation of the stability analysis of the equilibrium state for the new deterministic endemic model which is a motivation from [1] is given by;
Variable

Description

S1t

Number of susceptible juveniles at time t

S2t

Number of susceptible adult at time t

E1t

Number of exposed juvenile at time t

E2t

Number of exposed adults at time t

I1t

Number of infected juveniles at time t

I2t

Number of infected adults at time t

R1t

Number of infected juveniles receiving HAART at time t

R2t

Number of infected adults receiving HAART at time t

U1t

Number of recovered juveniles satisfying U=U case at time t

U2t

Number of recovered adults satisfying U=U case at time t

A1t

Number of AIDS cases in the juvenile sub-population at time t

A2t

Number of AIDS cases in the adult sub-population at time t

Parameter

Description

ν1

The rate at which HIV infected juveniles becomes AIDS patient.

ν2

The rate at which HIV infected adults becomes AIDS patient.

λ

Birth rate of the adult sub-population

μ1

Natural death rate of the juvenile sub-population

μ2

Natural death rate of the adult sub-population

α0

Maximum death rate due to AIDS. αi≤α0, i=1, 2

α1

Death rate of infected juvenile sub-population due to AIDS

α2

Death rate of infected adult sub-population due to AIDS

φ1

Disease induced death rate of infected juveniles not receiving HAART

φ2

Disease induced death rate of infected adults not receiving HAART

ϖ1

Disease induced death rate of infected juveniles receiving HAART

ϖ2

Disease induced death rate of infected adults receiving HAART

τ1

Disease induced death rate of recovered juvenile not receiving HAART

τ2

Disease induced death rate of removed adults not receiving HAART

T

Maximum lifespan after infection T=10 years

k

Efficacy of HAART (0≤k≤1)

c

Average number of sexual partners of adult members of class I2

c'

Average number of sexual partners of adult members of class R2

β

Probability of transmission by adult members of class I2

β'

Probability of transmission by adult members of class R2

ρ1

Probability of secondary infection by recovered juveniles population in U=U

ρ2

Probability of secondary infection by recovered adults population in U=U

σ1

Proportion of infected juveniles receiving HAART per unit time (Treatment rate)

σ2

Proportion of infected adults receiving HAART per unit time (Treatment rate)

π1

Proportion of juvenile population from susceptible to exposed/latent class

π2

Proportion of adult population from susceptible to exposed/latent class

ε1

Proportion of removed juveniles still receiving treatment and being moved to susceptible class

ε2

Proportion of removed adults still receiving treatment and being moved to susceptible class

1-ξ

Proportion of healthy newborns from infected mothers

ξ

Proportion of infected newborns from infected mothers

B1(t)

Incidence rate in the juvenile sub-population. B1t=0 (no sexual contact)

B2(t)

Incidence rate or force of infection in the adult sub-population

j

Maturation rate from Juveniles to adults (where j=s denotes the susceptible population, j=e denotes the exposed/latent population, j=i denotes the infected population, j=r denotes the removed population and j=u denotes the undetectable=untransmitable population)

m

Fixed ratio of adults to juveniles, m= N2N1

1 MODEL ASSUMPTIONS
The following assumptions would help in the derivation of the model:
1 There is no emigration from the total population and there is no immigration into the population.
2 The susceptible population are first exposed to a latent class where they can infected or not.
lation when the probability of secondary transmission is low (ρ1=0.012, ρ2=

...
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