A Mathematical Model Analysis on the Dynamics of HIV/AIDS with Age Structure and Inflow Immigrants in Ethiopia

In this work we considered a nonlinear deterministic dynamical system to study the dynamics of HIV/AIDS with age structure and different mode of transmissions in Ethiopia. We found that the diseases free equilibrium point and endemic equilibrium points exist and we perform their local stability and global stability analysis using nonlinear stability methods. We found that the basic reproduction number of the considered dynamical system depends on the considered parameters and using real data collected from different health sectors in Ethiopia we found the numerical value of the reproduction number is R_0=1.05>1. This shows that the considered disease spreads in the community. From the sensitivity index of the dynamical system we found that the most sensitive parameter is the transmission rate of unaware infective humans to aware infective θ. We also showed that the effect of all parameters on the basic reproduction number using numerical simulation.


Introduction
HIV/AIDS remains a major global health problem affecting approximately 70 million people worldwide causing significant morbidity and mortality [1].
In Ethiopia in 2018, 690 000 people were living with HIV. HIV incidence per 1000 uninfected the number of new HIV infections among the uninfected population over one year among all people of all ages was 0.24. 23 000 people were newly infected with HIV and 11 000 people died from an AIDS-related illness. There has been progress in the number of AIDS-related deaths since 2010, with a 45% decrease, from 20 000 deaths to 11 000 deaths. The number of new HIV infections has also decreased, from 29 000 to 23 000 in the same period. The 90-90-90 targets envision that, by 2020, 90% of people living with HIV will know their HIV status, 90% of people who know their HIV-positive status will be accessing treatment and 90% of people on treatment will have suppressed viral loads. In terms of all people living with HIV, reaching the 90-90-90 targets means that 81% of all people living with HIV are on treatment and 73% of all people living with HIV are virally suppressed. In 2018 in Ethiopia 79% of people living with HIV knew their status and 65% of people living with HIV were on treatment [2].
The Human Immunodeficiency Virus (HIV) infects cells of immune system such as helper T cells (specifically CD4+ T cells), macrophages, and dendritic cells. HIV compromises the human immune system and reduces the ability of the body to fight back infections and diseases. The most advanced stages of HIV infection is usually called Acquired Immunodeficiency Syndrome (AIDS). AIDS is one of the leading causes of death worldwide that is affecting virtually every nation. Even if HIV/AIDS is not permanently curable, main methods used to fight against it are preventive mechanisms (which include: abstinence, faithfulness and protection) which mainly rely on the level of behavioral change of the population, and providing Antiretroviral Therapy (ART) for those infected [3].
Mathematical models have played a major role in increasing our understanding of the dynamics of infectious diseases. Several models have been proposed to study the effects of some factors on the transmission dynamics of these infectious diseases including HIV/AIDS and to provide guidelines as to how the spread can be controlled. Among these models include those of Anderson et al. [4] who presented a preliminary study of the transmission dynamics of HIV by proposing a model to study the effects of various factors on the transmission of the disease, Stilianakis et al. [5]. who proposed and gave a detailed analysis of a dynamical model that describes the pathogenesis of HIV, and Tripathi et al. [6] who proposed a model to study the effects of screening of unaware infective on the transmission dynamics of HIV/AIDS. Several other models proposed to study dynamics of HIV/AIDS can be found in ( [7][8][9][10][11][12][13][14][15], and the references therein).
This paper presents a deterministic model for age structure, the combined effect of unaware infective immigrants, different mode of transmissions and aware infective immigrants, for predicting the epidemiological trends of HIV that exploits HIV surveillance data to model the disease evolution in Ethiopia. The results are presented graphically and discussed qualitatively in the following sections.

The Mathematical Model
Our initial model [16] is represented by five ordinary differential equations. Our extended model is represented by seven ordinary differential equations by adding two more compartments based on the following basic assumptions. For this dynamical system we considered susceptible classes as sexually mature S t and sexually immature S t , Unaware infective class I t , Aware infective class I t , Pre-AIDS class P t , AIDS class A t and Seropositive/treatment class S t . Individuals will join the susceptible mature compartment S t by immigrants. Some of these people will leave this compartment due to natural deaths and some others will go to I t compartment after getting infected. The remaining people will stay in the S t compartment itself. People of S t compartment are likely to get infected by the people of I t , I t and P t compartments only. Individuals will join the susceptible immature compartment S t by birth. Some of these people will leave this compartment due to natural deaths and some others will go to I t compartment after getting infected. The remaining people will stay in the S t compartment itself. People of S t compartment are likely to get infected by the people of I t , I t and P t compartments only. But the people of AIDS compartment A t being physically too weak to participate in sexual activities, cannot transfer infection to susceptible people. In this study we considered that, the transfer of HIV from infected people to susceptible people is by sexual intercourse, vertical transmission and transferring HIV by any other means like sharing needles; blood transfusion.
The population under this study is heterogeneous and varying with time, the whole human population is divided in to seven classes, the HIV can be transmitted by the sexual intercourse with infective peoples, vertical transmission and blood borne transmission. The full blown AIDS class is sexually inactive, the seropositive class could not transmit the disease, all the new infected people are assumed to be initially unaware of the infection and the probability of transferring the disease to susceptible population by Pre-AIDS class person is more than unaware infected and aware infected person i. e.
, . The Pre-AIDS class people grow to AIDS much faster than unaware and aware infected people i. e.
, . Based on these assumptions we construct the following flow chart which shows the movement of individuals from compartment to compartment. Based on the above basic assumptions and flow chart we do have the following corresponding dynamical system represented by seven non-linear ordinary differential equations.

Disease Free Equilibrium Point /DFE/
The disease free equilibrium point is obtained by setting the right hand side of the dynamical system (1) -(7) equal to zero with assumptions there are neither infective people nor AIDS patients, that is = = / = = 0 . And thus we obtain the disease free equilibrium point of the dynamical system is ^1 = ( ? @ , ? ! @ , 0, 0, 0, 0, 0).

Basic Reproduction Number _`
The basic reproduction number is defined as the average number of secondary infections that occur when one infective is introduced into a completely susceptible host population [17].
We can calculate the basic reproduction number, a 1 , using the next generation approach proposed by van den Driessche and Watmough [18]. According to this approach, in order to compute the basic reproduction number, it is important to distinguish new infections from all other class transitions in the population. The infected classes are , , 6bE . We can write system (1)-(7) as: where c = ( , , , , , , , / ) . ℱ is the rate of appearance of new infection in each class, f is the rate of transfer into each class by all other means, and f ; is the rate of transfer of the infectious individuals out of each class.
The associated matrices, ℱ(c) for the new infection terms, and f(c) for the remaining transition terms are respectively given by, Evaluating the partial derivatives of (9) and bearing in mind that system (1) -(7) has three infected classes, namely , 6bE , we obtain At disease free equilibrium point we have + ≈ G. Thus Similarly, the partial derivatives of (10) with respect to , 6bE at ^1 gives The spectral radius or the largest eigenvalue of the next generation matrix mr > is the required basic reproduction number and is computed as

Local Stability of the Disease Free Equilibrium Point
The disease free equilibrium point ^1 of the dynamical system (1) - (7) is locally asymptotically stable if a 1 < 1 and unstable if a 1 > 1.
Proof The Jacobean matrix of the dynamical system (1) - (7) The corresponding characteristic equation for the eigenvalue ~ is with After some calculations using Routh Hurwitz stability criterion we found that all roots of the characteristic equation have negative real part. Hence the disease free equilibrium point is locally asymptotically stable.

Global Stability of Disease-free Equilibrium Point
Theorem-4: The disease free equilibrium point ^1 is globally asymptotically stable if a 1 < 1. Proof We construct a Liapunov function by r = . + . + . + . € , + . • / and thus we get r is continuous function and has first order partial derivatives and r has minimum at ^1 = ( Take the coefficients of , , , and / are equal to zero.
We note that ˆ ≤ 0 if a 1 < 1. invariance principle then implies that ^1 is globally stable in Ω if a 1 < 1 otherwise it is unstable.

Endemic Equilibrium Point
The endemic equilibrium point is obtained by setting the right hand side of the dynamical system (1)-(7) equal to zero. Thus we get the endemic equilibrium point is ^ * = ( * , * , * , * , * , ,

Local Stability of Endemic Equilibrium Point
Theorem-5: The positive endemic equilibrium point ^ * of the system of equations (1) - (7) is locally asymptotically stable if a 1 > 1.

Proof
The Jacobian matrix of the system of equations (1) - (7) at the endemic equilibrium point is The corresponding characteristic equation is After some calculations using Routh Hurwitz stability criterion we found that all roots of the characteristic equation have negative real part, therefore the endemic equilibrium point is locally asymptotically stable.

Global Stability of Endemic Equilibrium Point
Theorem-6: The endemic equilibrium point ^ * is globally asymptotically stable if ž < Ÿ, where Therefore the endemic equilibrium point ^ * is globally asymptotically stable in Ω if ž < Ÿ.

Parameter Values for Numerical Simulation and Sensitivity Analysis
To perform numerical simulation and sensitivity analysis we collected the following parameter values from different data sources.

Numerical Simulations
The numerical analysis is obtained from the graphs of basic reproduction number with respect to the parameters obtained and given in Table 1.           Figure 6. it is graphical representation of the basic reproduction number a 1 versus rate of progress of unaware infective to AIDS, and keeping other parameters constant. This figure shows that an increase in the rate of progress of unaware infective to AIDS, between the parametric values 0 and 0.099024 makes a decrease in the reproduction number but the reproduction number is greater than one that indicates the disease persists. If the parameter value of greater than 0.099024 , then the reproduction number decreases and becomes less than one where the disease not persists. Figure 7. it is graphical representation of the basic reproduction number a 1 versus rate of transmission of Pre-AIDS to seropositive class, ' and keeping other parameters constant. This figure shows that an increase in the rate of transmission of Pre-AIDS to seropositive class between the parametric values 0 and 0.85235 makes a decrease in the reproduction number but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of ' greater than 0.85235, then the reproduction number is less than one and we can say the disease not persists.

Rate of transmission of the disease from unaware & aware infective classes
Rate of progress of Pre-AIDS to AIDS & rate of transmission of unaware infective to aware infective   and keeping other parameters constant. This figure shows that an increase in the rate of progress of Pre-AIDS to AIDS between the parametric values 0 and 0.66445 makes a decrease in the reproduction number but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of greater than 0.66445 , then the reproduction number decreases and becomes less than one where the not persists. Figure 9. it is graphical representation of the basic reproduction number a 1 versus rate of transmission of unaware infective to aware infective, ( and keeping other parameters constant. This figure shows that the rate of transmission of unaware infective to aware infective between the parametric values 0 and 0.921330 makes a decrease in the reproduction number but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of ( greater than 0.921330 , then the reproduction number decreases and becomes less than one where the not persists.
Rate of transmission of unaware & and aware infective to seropositive class   decrease in the reproduction number, but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of ' greater than 0.059024, then the reproduction number decreases and becomes less than one where the disease dies out. Figure 11. it is graphical representation of the basic reproduction number a 1 versus rate of transmission of aware infective to seropositive class, ' and keeping other parameters constant. This figure shows that an increase in the rate of transmission of aware infective to seropositive class, ' , between the parametric values 0 and 0.01822 makes a decrease in the reproduction number, a 1 , but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of ' greater than 0.01822, then the reproduction number is less than one and almost constant in value.
Rate of aware infective immigrants & rate of progress of unaware infective to Pre-AIDS   177813 makes an increase in the reproduction number but the reproduction number is less than one that indicates the disease not persists. If the rate of aware infective immigrants between 0.177 and 1.2865 makes an increase in the reproduction number with, a 1 1 and tell us the disease persists. Whereas, the rate of aware infective immigrants greater than 1.2865, makes an increase in the reproduction number, a 1 Y 1, and tell us the disease not persists. Figure 13. it is graphical representation of the basic reproduction number a 1 versus rate of progress of unaware infective to Pre-AIDS, ) and keeping other parameters constant. This figure shows an increase in the rate of progress of unaware infective to Pre-AIDS between the parametric values 0 and 0.420598 makes a decrease in the reproduction number but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of ) greater than 0.420598 , then the reproduction number is less than one and we can say the disease not persists.   that an increase in the rate of vertical transmission between the parametric values 0 and 0.409848 makes an increase in the reproduction number but the reproduction number is less than one that indicates the disease not persists. Whereas, the rate of vertical transmission greater than 0.409848, makes an increase in the reproduction number with a 1 1, and tell us the disease persists. Figure 15. it is graphical representation of the basic reproduction number a 1 versus natural mortality, and keeping other parameters constant. This figure shows an increase in natural mortality between the parametric values 0 and 1.25185 makes a decrease in the reproduction number but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of greater than 1.25185 , then the reproduction number is less than one and we can say the disease not persists. HIV/AIDS with Age Structure and Inflow Immigrants in Ethiopia

Sensitivity Analysis
The parameter values and assumptions of any model are subject to change and error. Sensitivity analysis is the investigation of these potential changes & errors and their impacts on conclusions to be drawn from the model. Here we use it to discover parameters that have a high impact on reproduction number a 1 . We calculate the normalized forward sensitivity index of a variable ) that depends differentiable on a parameter # is defined by (#) =¯-,°, -

.
After some simplifications and numerical calculation we get values of sensitivity index for the important parameters mentioned by the table below:

Results and Discussion
Results from Numerical simulation show that as the transmission rate of unaware infective humans to aware infective increases, the basic reproduction number decreases. This will result in decreasing on the transmission of HIV/AIDS. An increase in the rate of horizontal transmission of unaware infective, , makes an increase in the reproduction number, a 1 . If > 0.7349096 the reproduction number a 1 > 1 that indicates the disease persists. When < 0.7349096 the reproduction number a 1 < 1 this indicates that the disease not persists.
An increase in the rate of horizontal transmission of aware infective, , makes an increase in the reproduction number, a 1 . If > 0.546716 the reproduction number a 1 > 1 that indicates the disease persists. When < 0.546716 the reproduction number a 1 < 1 this indicates that the disease not persists. We can also observe that an increase in the rate of horizontal transmission of Pre-AIDS, , makes an increase in the reproduction number, a 1 . For > 0.750956 we can see the reproduction number a 1 > 1 that indicates the disease persists. When < 0.750956 the reproduction number a 1 < 1 and this indicates that the disease not persists. Similarly we can see that an increase in the rate of progress of unaware infective to AIDS, , between the parametric values 0 and 0.099024 makes a decrease in the reproduction number, a 1 , but the reproduction number is greater than one that indicates the disease persists. If the parameter value of greater than 0.099024 , then the reproduction number decreases and becomes less than one where the disease not persists.
We can see that an increase in the rate of progress of Pre-AIDS to AIDS, , between the parametric values 0 and 0.66445 makes a decrease in the reproduction number, a 1 , but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of greater than 0.66445 , then the reproduction number decreases and becomes less than one where the disease not persists. For an increase in the rate of transmission of unaware infective to aware infective, ( , between the parametric values 0 and 0.921330 makes a decrease in the reproduction number, a 1 , but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of ( greater than 0.921330, then the reproduction number decreases and becomes less than one where the not persists. An increase in the rate of transmission of aware infective to seropositive class,' , between the parametric values 0 and 0.01822 makes a decrease in the reproduction number, a 1 , but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of ' greater than 0.01822 , then the reproduction number is less than one and almost constant in value this indicates the disease not persists. An increase in the rate of progress of unaware infective to Pre-AIDS, ) , between the parametric values 0 and 0.420598 makes a decrease in the reproduction number, a 1 , but the reproduction number is greater than one that indicates the disease persists with a decreasing rate. If the parameter value of ) greater than 0.420598, then the reproduction number is less than one and we can say the disease not persists. We can also observe that an increase in the rate of vertical transmission, &, between the parametric values 0 and 0.409848 makes an increase in the reproduction number, a 1 , but the reproduction number is less than one that indicates the disease not persists. Whereas, the rate of vertical transmission greater than 0.409848 , makes an increase in the reproduction number, a 1 > 1, and tell us the disease persists.
From sensitive analysis we observed that the most sensitive parameter is the transmission rate of unaware infective humans to aware infective, ( and the least sensitive parameter is the recruitment into sexually immature class, . The indices having positive signs increase the value of a 1 as one increase them and those having negative signs decrease the value of a 1 , when they are increased.

Conclusion
In this study we have developed a deterministic mathematical model for Age structure and Inflow Infective Immigrants on the Dynamics of HIV/AIDS: dividing susceptible individuals in to sexually immature (i.e age below 15 years) and sexually mature (i.e age 15 years and above), aware and unaware infective, infective immigrants, Pre-AIDS individuals and treatments of infectious individuals. The stability analysis on the model shows that the disease -free equilibrium point ^1 is to be locally asymptotically stable and globally asymptotically stable when a 1 Y 1 and the positive endemic equilibrium point ^ * is shown to be locally asymptotically stable and globally asymptotically stable for ž < Ÿ . Results from Numerical simulation show that as the transmission rate of unaware infective humans to aware infective increases, the basic reproduction number decreases. This will result in decreasing on the transmission of HIV/AIDS. We evaluated the numerical value of the basic reproduction number. Consequently, a 1 = 1.0498 that shows the HIV/AIDS disease spread in the community. A sensitivity analysis of the basic reproduction number indicates that the transmission rate of unaware infective humans to aware infective, the rate of vertical transmission and horizontal transmission rate are the most sensitive parameters that can be used to control the spread of the disease.

Recommendation
From the above results and discussion we would like to recommend the following to control the spread of HIV/AIDS: the most sensitive parameters like transmission rate of unaware infective humans to aware infective, vertical transmission rate, and horizontal transmission rate of unaware infective and Pre-AIDS individuals are those that should be targeted most by policymakers in the fight against the disease.