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In this paper, we formulate a mathematical model for malaria transmission that
includes incubation periods for both infected human hosts and mosquitoes. We
assume humans gain partial immunity after infection and divide the infected human
population into subgroups based on their infection history. We derive an explicit
formula for the reproductive number of infection, $R_0$, to determine threshold
conditions whether the disease spreads or dies out. We show that there exists an
endemic equilibrium if $R_0>1$. Using an numerical example, we demonstrate that
models having the same reproductive number but different numbers of progression
stages can exhibit different transient transmission dynamics.