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 = b VX - 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, letProblem 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.Summary This chapter illustrates how the modeling concepts discussed in the previous chapters may be applied to the description, in terms of ordinary differential equations, of the dy- namics of infectious diseases and the spread of epidemics. Moreover, it provides the reader with a basic introduction to the terminology and applications of compartmental models. It should be clear by now that models of arbitrary complexity may be built from the simple tools discussed in this text. The modeling process is always the same, no matter how involved the model is. The methods of analysis, in terms of maps or differential equations, are also similar, but become more complicated as the dimension of the model is increased. In particular, three-dimensional continuously differentiable dynami- cal systems may exhibit chaos, the understanding of which requires more advanced techniques than those discussed here. The section on further reading includes texts on dynamical systems and chaos that the reader may want to consult. kb kb T = ot, a = $2 X , y = Y, U = V. Then, the scaled version of Equations (7.1) is dx dT =(-x - vx, dy =vx - ny, (7.2) dT dT =y - uv, Akb where ( = a K 83 ,7 = 8, and u = are dimensionless parameters. We scaled time according to the death rate of normal cells. Alternatively, we could have scaled time according to the rate at which normal cells are produced by the body, i.e. we could have defined T = At/ Xo, with Xo = $2 /(kb). We could also have used a combination of these two time scales. In general, there is more than one possible way of defining dimensionless variables. The most convenient choice is often that which gives dimensionless parameters of order one, if at all possible. We refer the reader to the 1998 article by A.U. Neumann et al., entitled Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy, to see how esti- mating parameters in model (7.1) may be used to understand the role of interferons in hepatitis C therapy