In this paper, we consider a continuous fragmentation–coagulation model in which the reacting particles can be transported in physical space through either advection or diffusion. We prove new results on the generation of C 0 -semigroups with parameter and use them to show that the Abstract Cauchy Problem associated with a more general version of the advection/diffusion–fragmentation problem generates a positive C 0 -semigroup in spaces L 1 ( R + , X x , ( 1 + m r ) d m ) , where m is the particle mass, X x is either the space of integrable or continuous functions with respect to the spatial variable, and the weight exponent r is sufficiently large. These results enable us to prove the classical solvability of a wide range of advection/diffusion–fragmentation–coagulation equations with unbounded coagulation kernels.

In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with Transmission Blocking Drugs (TBDs). The paper’s main aim is to demonstrate the impact the chosen model for demographic growth has on the disease’s transmission and the effect of its treatment with TBDs. We calculate the model’s control reproduction number and equilibria, and perform a global stability analysis of the disease-free equilibrium point. The mathematical analysis reveals that, depending on the model’s demography, the model can exhibit forward, backward and even some unconventional types of bifurcation, where disease elimination can occur for both small and large values of the reproduction number. We also conduct a numerical analysis to explore the short-time behavior of the model. A key finding is that for one type of demographic growth, the population experienced a significantly higher disease burden than the others, and when exposed to high levels of treatment with TBDs, only this population succeeded in effectively eliminating the disease within a reasonable timeframe.

The existence and occurrence, especially by a backward bifurcation, of endemic equilibria is of utmost importance in determining the spread and persistence of a disease. In many epidemiological models, the equation for the endemic equilibria is quadratic, with the coefficients determined by the parameters of the model. Despite its apparent simplicity, such an equation can describe an amazing number of dynamical behaviours. In this paper, we shall provide a comprehensive survey of possible bifurcation patterns, deriving explicit conditions on the equation's parameters for the occurrence of each of them, and discuss illustrative examples.