The replacement number dynamics in SIR-type epidemic models I: From SSISS to RND picture

Abstract

In SIR-type epidemic models time derivative of prevalence $I$ can always be cast into the form $\dot{I}=(X-1)I$, where $X$ is the replacement number and recovery rate is normalized to one. Assuming $\dot{X}=f(X,I)$ for some smooth function $f$ defines a "replacement number dynamics" (RND). Choosing transmission coefficients $\beta_1>\beta_2$, any such system uniquely maps to an isomorphic "SSISS model", i.e. an abstract SIR-type 3-compartment model. Extending to negative values $\beta_i<0$ takes care of demographic dynamics with compartment dependent birth and death rates. Fixing $f$ and varying $\beta_i$ generates a family of isomorphic SSISS systems, the "SSISS fiber" $\cal{F}(f)$}. A symmetry group $G_\sigma$ acts freely and transitively on fibers $\cal{F}(f)$, so SSISS systems become a principal $G_\sigma$-fiber bundle over the space of RND systems. Choosing two specific 6-parameter polynomials $f$ at most quadratic in $(X,I)$ covers a large class of models in the literature, with in total up to 17 (largely redundant!) parameters, including also reactive social behavior models. Epidemiological admissibility conditions guarantee forward boundedness and absence of periodic solutions in all these models. Part II of this work will prove existence and stability properties of endemic equilibria, which in RND picture simply boils down to analyzing zeros of the parabolas $f|_{I=0}$ and $f|_{X=1}$. This will cover and extend well known results for a wide range of models by a unifying approach while also closing some open issues in the literature.