Asymmetric Dissipativity and Supply Rates for Compartmental Systems with Logarithmic Storage Functions

Many compartmental models in biology, chemistry, ecology, sociology, and related sciences have positive variables, for which equilibrium points of interest are in the interior of the positive orthant. Reactions are often modeled as a bilinearity, which renders a boundary of the positive orthant a separatrix. In such situations, the capability of Lyapunov functions defined on symmetric domains is very limited, and logarithmic functions are sometimes used in mathematical biology. The domain of a logarithm function is a ray, which is asymmetric with respect to the equilibrium of interest. This paper aims to add a useful tool to dissipativity theory for analyzing networks via storage functions defined on asymmetric spaces. This paper focuses on an asymmetric supply rate and discusses its properties and utilities which standard symmetric supply rates do not offer. The power of the asymmetric supply rate is illustrated briefly by establishing asymptotic stability of the endemic equilibrium of the SIQRS model of infectious diseases in an arbitrarily large domain.