Table 10. Coupling mechanism of within-host and between-host scales of HIV dynamics.

The within-host and between-host layers of HIV multi-scale model are linked using partial differential equations. The HIV viral immune dynamics model (see Table 2) determines the time-varying within-host viral load, which impacts the transmission rate (β(τ) = r.V(τ); r is a constant coefficient). Another method to determine the HIV transmission rate is based on the viral load equilibrium.

Equations
$\frac{dS}{dt}=b-S{\int }_0^{\infty }\beta \left(\tau \right)I\left(\tau ,t\right)d\tau -\delta S$
$\frac{\partial I}{\partial t}+\frac{\partial I}{\partial \tau }=-m\left(V\left(\tau \right)\right)I\left(\tau ,t\right)$
$I\left(0,t\right)=S{\int }_0^{\infty }\beta \left(\tau \right)I\left(\tau ,t\right)d\tau$
Parameters
S	Number of individuals in the susceptible class
I(τ, t)	Number of infected individuals structured by time since infection (τ)
b	Natural birth rate in the population
β(τ)	HIV transmission rate (r.V(τ))
m	Coefficient on dependence of induced mortality due to disease on the host viral load.