Minimal wave speed for a reaction–diffusion tuberculosis model with the fast and slow progression

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This work is concerned with full information about the traveling wave solutions for a diffusive tuberculosis model with fast and slow progression. After relaxing the rigid ordered restriction on the diffusion rates typically employed in the previous classical study, and overcoming some significant technical obstacles to verify the boundedness of the infected components, we first obtain the existence of super-critical traveling waves connecting disease-free equilibrium to endemic equilibrium. Then, on the basis of the uniform boundedness for sequences of super-critical traveling waves and further delicate analysis, the existence of critical traveling waves is thoroughly explored. Finally, the nonexistence of non-negative bounded traveling waves is discussed in two cases.