# American Institute of Mathematical Sciences

## Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays

 Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave, Room 3309, Statesboro, Georgia, 30460, USA

* Corresponding author

Received  July 2020 Revised  October 2020 Published  January 2021

A multipopulation HIV/AIDS deterministic epidemic model is studied. The population structure is a multihuman behavioral structure composed of humans practicing varieties of distinct HIV/AIDS preventive measures learnt from information and education campaigns (IEC) in the community. Antiretroviral therapy (ART) treatment is considered, and the delay from HIV exposure until the onset of ART is considered. The effects of national and multilateral support providing official developmental assistance (ODAs) to combat HIV are represented. A separate dynamics for the IEC information density in the community is derived. The epidemic model is a system of differential equations with random delays. The basic reproduction number (BRN) for the dynamics is obtained, and stability analysis of the system is conducted, whereby other disease control conditions are obtained in a multi- and a finite dimensional phase space. Numerical simulation results are given.

Citation: Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021005
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Shows the behaviors of the modified standard incidence and the ordinary standard incidence rates as the number of infectives continually increase over time. Clearly, the modified standard incidence is more suitable for many real life scenarios where the incidence rate of the disease saturates over time as the number of infections increase in the population
shows the different states of the population in the HIV/AIDS epidemic, and the transition rates between the states. Note that the operators $\mathbb{E}_{\tau_{1}}[\cdots]$ and $\mathbb{E}_{\tau_{2}}[\cdots]$ represent expectations with respect to the random variables $\tau_{1}$ and $\tau_{2}$, respectively
(ⅰ) depicts a continuously rising relationship between the BRN and the delay $\tau_{2}$. (ⅱ) shows a declining relationship between the BRN and the proportion $1-\varepsilon_{0}$ of individuals who are receiving ART treatment
(a-1)-(d-1) shows the behavior of the path of the total susceptible population $S(t) = S_{0}(t)+S_{1}(t)+S_{2}(t)$, over time, whenever the conditions of Theorem 5.3 and Theorem 5.4 are satisfied. Clearly, the path of $S(t)$ is persistent as proven in Theorem 5.4 and approaches the DFE state $S^{*}_{0} = \frac{B}{\mu_{S_{0}}} = 26.13417$. The dotted redline in (d-1) is the value of $S^{*}_{0} = 26.13417$. The figures (e-1), (f-1) and (g-1) also show the paths of the HIV related states $I, T$ and $A$. Clearly, the paths of $I, T$ and $A$ approach the corresponding coordinate $0$ of the DFE $E_{0}$. In other words, the disease is getting extinct over time
Shows the list of model parameters, estimates and their definitions. Note that the parameters are expressed in years and converted to days for all simulations in Section 7
 Parameter Symbol(s) Estimate(s) in years Effective response rate of $S_0(t), S_1(t), S_2(t)$ $\gamma_0, \gamma_1, \gamma_2$ 0.1, 0.1, 0.8 Infection transmission rates $\beta_0, \beta_1, \beta_2$ 0.0211, 0.001055, 0.00844 Natural death rates of $S_0(t), S_1(t), S_2(t)$ $\mu_{S_0}, \mu_{S_1}, \mu_{S_2}$ 0.01568 Natural death rates of $I(t), T(t), A(t), R(t)$ $\mu_I, \mu_T, \mu_A, \mu_R$ 0.01568 Infection related death rates of $I(t)$ $d_I$ 0.1474 Infection related death rates of $T(t)$ $d_T$ 0.03685 Infection related death rates of $A(t)$ $d_A$ 0.2948 Recruitment rate B 0.55 Return rate from $T(t)$ to $I(t)$ $\alpha_{TI}$ 0.01 Failure of treatment rate from $T(t)$ to $A(t)$ $\alpha_{TA}$ 0.01 Proportion of newly infected individuals from the class $S_j, j=0, 1, 2$ who do not receive ART and joins full blown AIDS state $A(t)$ $\epsilon_0, \epsilon_1, \epsilon_2$ 0 - 1 Time delay to progress to full blown AIDS $\tau_1$ 2 - 15 Time delay to begin treatment $\tau_2$ 0.38-15
 Parameter Symbol(s) Estimate(s) in years Effective response rate of $S_0(t), S_1(t), S_2(t)$ $\gamma_0, \gamma_1, \gamma_2$ 0.1, 0.1, 0.8 Infection transmission rates $\beta_0, \beta_1, \beta_2$ 0.0211, 0.001055, 0.00844 Natural death rates of $S_0(t), S_1(t), S_2(t)$ $\mu_{S_0}, \mu_{S_1}, \mu_{S_2}$ 0.01568 Natural death rates of $I(t), T(t), A(t), R(t)$ $\mu_I, \mu_T, \mu_A, \mu_R$ 0.01568 Infection related death rates of $I(t)$ $d_I$ 0.1474 Infection related death rates of $T(t)$ $d_T$ 0.03685 Infection related death rates of $A(t)$ $d_A$ 0.2948 Recruitment rate B 0.55 Return rate from $T(t)$ to $I(t)$ $\alpha_{TI}$ 0.01 Failure of treatment rate from $T(t)$ to $A(t)$ $\alpha_{TA}$ 0.01 Proportion of newly infected individuals from the class $S_j, j=0, 1, 2$ who do not receive ART and joins full blown AIDS state $A(t)$ $\epsilon_0, \epsilon_1, \epsilon_2$ 0 - 1 Time delay to progress to full blown AIDS $\tau_1$ 2 - 15 Time delay to begin treatment $\tau_2$ 0.38-15
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