There is a rich variety of types of stellar evolution models. The modern model of a star is the spherically symmetric quasi-static gas assumed to be in local thermodynamic equilibrium (Chandrasekhar, 1967). There are four equations for how mass, pressure, temperature, and luminosity vary with radius as the star's thermal and radiative pressures support its self-gravity. Assumptions about convective and radiative energy transport, stability of nuclear burning, gas equations of state, gas composition, etc. and other processes must be accounted to model the evolution of a star through its life. A completely comprehensive stellar simulation is beyond computational ability. Numerous types of models exist: e.g., evolving stellar structure magneto-hydrodynamically, numerically evolving the stellar equations, interpolating between tables of known stellar properties derived from detailed simulations, using fitted formulae based on tabulated values, or some combination. Modern models incorporate the effects of rotation and mass loss on the star's
This seminal paper by Hurley et al. (2000) describes formulas that are fitted (and parameterized in terms of the initial stellar mass and metallicity) from numerically computed stellar evolutionary models. The tracks that you see in the HR diagram are made as a visual representation. See this paper for details about those stellar evolutionary numerical computations.
In their Discussion section 8, they describe how to use these parameterized formulas for fast-evaluations of single stellar evolution (which they implemented in the publicly available Single Stellar Evolution (SSE)). This was also implemented in the rapid binary evolution simulation Binary Stellar Evolution (BSE), which has many modified successors now, for example MOBSE.