This paper mainly studies the limit behavior of Riemann solutions for the macroscopic production model with anti-Chaplygin gas. Firstly, we investigate the Riemann problem associated with this model. Three types of Riemann solutions are obtained: a combination of contact discontinuity and rarefaction wave (J₁ + R₂), a combination of contact discontinuity and shock wave (J₁ + S₂), and Dirac shock wave (δS). Secondly, the pressure vanishing limit of the macroscopic production model of the anti-Chaplygin gas is studied. As the perturbation parameter ε decreases to the parameter ε₀, which is dependent only on the initial data, it is proved that the Riemann solution (J₁ + S₂) converges to the δS of the anti-Chaplygin gas state equation. Moreover, when ε eventually approaches 0, the δS converges to the δS of the transport equation. Additionally, it is proved that the Riemann solution (J₁ + R₂) converges to the vacuum solution of the transport equation. Finally, we present some representative numerical experimental results.

The formation of singularities in the classical solutions to the Aw-Rascle traffic flow system with generalized Chaplygin gas is studied. By using the method of characteristic decomposition, we prove that the density itself of the classical solutions of the Cauchy problem for the traffic flow model will blow up in finite time when the initial data meets certain conditions. Additionally, this physical phenomenon is verified by numerical simulation.

Assuming total population remains constant, this paper establishes a class of MSIR epidemic models with both natural age and infection-age and studies the stability of the equilibrium solutions of this model. Firstly, by normalizing the model and linearizing it at the disease-free equilibrium, demonstrating that when R₁<1, the disease-free equilibrium is locally asymptotically stable. Subsequently, employing the method of characteristics for hyperbolic systems and Fatou's lemma, it is proven that when R₁<1, the disease-free equilibrium is globally asymptotically stable. Finally, the existence of a unique endemic equilibrium solution is established using the intermediate value theorem when R₁>1.