Analysis of a partially diffusive vector-borne disease model with human-to-human infection in a spatially heterogeneous environment

Abstract In this paper, we deal with the sharp threshold results of a mathematical epidemic model for vector-borne diseases. We construct a degenerate reaction–diffusion equation system with spatially heterogeneous parameters to consider the situation where vectors move around randomly in a spatially heterogeneous environment. We then define the basic reproduction number $$\Re _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℜ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> as a threshold parameter to predict whether the malaria will spread or not. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if $$\Re _0 &lt; 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ℜ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , whereas the system is uniformly persistent if $$\Re _0&gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ℜ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We also confirm that in the case of $$\Re _0=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ℜ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the disease will go extinct. In addition, the global asymptotic stability of the unique constant positive equilibrium is investigated by using a Lyapunov function method in a special case where all parameters are spatially homogeneous.