📂Dynamics

AIDS Transmission Model

Overview

AIDS, also known as Acquired Immunodeficiency Syndrome, is a contagious disease that has been tormenting humanity for decades, caused by the virus HIV. The transmission routes of AIDS include homosexual and heterosexual interactions, drug use, among others, making the mathematical modeling of its spread inevitably include the structure of the entire population. However, the simplest model introduced by Castillo, Chavez, and others, which is the [ODE](../../categories/Differential Equations) model, will be discussed first.

Model ¹

$$ \begin{align*} T(t) =& S(t) + I(t) + Y(t) \\ {{d S} \over {d t}} =& - \mu S + \Lambda - \lambda {{ C(T) } \over { T }} (I + Y) S \\ {{d I} \over {d t}} =& - \mu I + p \lambda {{ C(T) } \over { T }} (I + Y) S - k_{I} I \\ {{d Y} \over {d t}} =& - \mu Y + (1-p) \lambda {{ C(T) } \over { T }} (I + Y) S - k_{Y} Y \\ {{d A} \over {d t}} =& - (\mu + \delta) A + k_{I} I \\ {{d Z} \over {d t}} =& - \mu Z + k_{Y} Y \end{align*} $$

Variables

• $S(t)$: Represents the number of individuals susceptible to AIDS at time $t$.
• $I(t)$: Represents the number of individuals within the infectious group who are particularly progressing to severe AIDS at time $t$.
• $Y(t)$: Represents the number of individuals within the infectious group who are particularly progressing to mild AIDS at time $t$.
• $A(t)$: Represents the number of individuals who have progressed to severe AIDS at time $t$.
• $Z(t)$: Represents the number of individuals who have progressed to mild AIDS at time $t$.
• $T(t)$: Represents the number of individuals actually involved in transmitting AIDS at time $t$.

Functions

• $C(T)$: The average number of sexual partners an individual has per unit of time.

Parameters

• $\Lambda>0$: The inflow of population, including immigration and birth, which need not be constant.
• $\lambda>0$: The transmission rate of AIDS.
• $p \in [0,1]$: The proportion that progresses to severe condition when infected. $(1-p)$ becomes the naturally progressing to mild condition.
• $k_{I}, k_{Y}>0$: The ratio of infected individuals reaching the symptomatic phase after the asymptomatic phase.
• $\mu>0$: The base mortality rate unrelated to AIDS.
• $\delta>0$: The mortality rate due to AIDS.

Explanation ²

One of the characteristics of AIDS is that it is a chronic disease that is difficult to treat and has a very long asymptomatic period. Initially, one may experience cold-like symptoms, but then there is generally no symptom for 8-10 years, during which the HIV in the body continues to proliferate and develops into AIDS. Naturally, at this point, legal restrictions on daily activities may be applied even before symptoms appear.

In the mathematical model, it is assumed that $A,Z$ no longer engages in sexual activities. Thus, abstractly, it is not distinguishable from the ordinary SIR model’s $R(t)$, except that it is divided into severe and mild cases, differing only in their mortality rates. In the equations, this distinction is reflected by having a higher mortality rate exclusively for $A$.

Improvement

The parameter $k_{I}, k_{Y}$, which inherently represents the progress of AIDS, needs revision as the progression period varies significantly among individuals, reducing the model’s realism. Referring to the asymptomatic period as Sojourn Time, this duration can be considered as ’the survival time before HIV progresses to AIDS’ and can be incorporated by introducing a survival function.

Survival Function: Defined as a non-increasing function $P : [0,\infty) \to [0,\infty)$, where $P(0)=1$.

If $$ \int_{0}^{\infty} P(s) dx = \tau < \infty $$, then $\tau$ represents the average sojourn time. Given a probability density function $f(s)$, the survival function can be defined as $f(s) = - d P(s) / ds$, allowing consideration of various distributions for the sojourn time. Notably, the Gamma distribution and Weibull distribution are commonly used. Accordingly, in the model above, $I$ and $Y$ can be modified by introducing a survival function $P_{I}, P_{Y}$ as follows. $$ \begin{align*} I(t) =& I_{0} + p \int_{0}^{t} \lambda {{ C(T) } \over { T }} (I + Y) S e^{-\mu (t-s)} P_{I} (t-s) ds \\ Y(t) =& Y_{0} + (1-p) \int_{0}^{t} \lambda {{ C(T) } \over { T }} (I + Y) S e^{-\mu (t-s)} P_{Y} (t-s) ds \end{align*} $$