We formulate an immuno-epidemiological model of coupled “within-host” model of ODEs and
“between-host” model of ODE and PDE, using the Human Immunodeficiency Virus (HIV) for
illustration. Existence and uniqueness of solution to the “between-host” model is
established, and an explicit expression for the basic reproduction number of the
“between-host” model derived. Stability of disease-free and endemic equilibria is
investigated. An optimal control problem with drug-treatment control on the within-host
system is formulated and analyzed; these results are novel for optimal control of ODEs
linked with such first order PDEs. Numerical simulations based on the forward-backward
sweep method are obtained.