Problem 1 Write model (7.1) in dimensionless form by defining T = Akbt /82 and appropriate vari- ables x, y and v. Explain what you are doing and check that T is dimensionless.Viral infections We first model the dynamics of a viral infection, such as hepatitis B or C, and are inter- ested in describing how the corresponding virus can spread and multiply in a person's body. Denote by X the average number of uninfected cells, which virions will try to in- fect; let Y be the average number of infected cells, and V be the average viral load (or the number of free virions in the body). Consider that uninfected cells are produced at a constant rate > by the body, die at rate S X, and become infected at rate f(X, V) X, where f is some function of X and V. As a consequence, infected cells Y are created at rate f(X, V) X, and we assume they die at rate aY. Finally, free virions are pro- duced at a rate proportional to the number of infected cells KY, and are removed or destroyed at rate K V. If we consider that, as a first approximation, f is a linear func- tion of V, i.e. f(X, V) = bV, we have the following model.(!] dx = 1 - 8X - bVX, dt dy =bVX - aY, (7.1) dt dV = KY - KV, dt This model has six parameters and four variables. We can rescale time and V, but it is not a good idea to rescale X and Y independently since they both count cells and the term in bV X transfers cells from the X compartment to the Y compartment. We can therefore reduce Equations (7.1) to a model with three parameters. More precisely, let kb kb O T = ot, c = 82 A , y = Y, $2 V = V