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Using the differential equations of the SIR model and converting them to numerical discrete forms, one can set up the recursive equations and calculate the S, I, and R populations with any given initial conditions but accumulate errors over a long calculation time from the reference point.
In a deterministic model, individuals in the population are assigned to different subgroups or compartments, each representing a specific stage of the epidemic. [17] The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations.
In its initial form, Kermack–McKendrick theory is a partial differential-equation model that structures the infected population in terms of age-of-infection, while using simple compartments for people who are susceptible (S), infected (I), and recovered/removed (R). Specified initial conditions would change over time according to
The mathematical modelling of epidemics was originally implemented in terms of differential equations, which effectively assumed that the various states of individuals were uniformly distributed throughout space. To take into account correlations and clustering, lattice-based models have been introduced.
SIR model; SIS model; Hagen–Poiseuille equation in blood flow; Hodgkin–Huxley model in neural action potentials; Kardar–Parisi–Zhang equation for bacteria surface growth models; Kermack-McKendrick theory in infectious disease epidemiology; Kuramoto model in biological and chemical oscillations; Mackey-Glass equations
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The Lotka–Volterra system of equations is an example of a Kolmogorov population model (not to be confused with the better known Kolmogorov equations), [2] [3] [4] which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.
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