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This paper extends the next-generation matrix method from epidemiology to mathematical biochemistry by reinterpreting its components as autocatalysis, leak, and transition reactions. A novel ALT-graph decomposition and splitting algorithm are introduced to systematically derive a valid reproduction number, $\rho(FV^{-1})$, while reducing computational complexity by lowering the rank of $FV^{-1}$. The method is validated on both biochemical reaction networks and infectious disease models.
A new ALT-graph framework decomposes biochemical reactions into autocatalytic, leak, and transition components, enabling a mathematically guaranteed and computationally efficient reproduction number calculation for stability analysis.
The next-generation matrix method is a powerful tool for computing the basic reproduction number in compartmental mathematical models of infectious diseases. The method has been recently extended to mathematical biochemistry, where it can be used to establish parameter regions of stability and instability for boundary steady states. Several significant challenges in its application remain, however, particularly around establishing conditions under which the method is mathematically valid, computationally tractable, and biologically meaningful in the biochemical setting. In this paper, we address these challenges by shifting the interpretation from new infections in the epidemiological setting to autocatalysis in the biochemical setting. We introduce a graph, called the ALT-graph (autocatalysis-leak-transition graph), which decomposes the contribution of each reaction to the Jacobian as autocatalytic, leak, or transition edges. We then present a systematic algorithm for splitting the ALT-graph, which is guaranteed to produce a valid reproduction number, $\rho(FV^{-1})$, while also decreasing computational complexity by lowering the rank of $FV^{-1}$. We apply the method to models of both biochemical reaction networks and infectious disease spread.