Global dynamics of a time-delayed malaria transmission model incorporating hospitalized patients and exposed mosquitoes

Abstract Based on the transmission dynamics of malaria, we develop a six-dimensional time-delayed differential equations model that incorporates hospitalized patients and exposed mosquitoes. Subsequently, this paper establishes the local asymptotic stability of both the disease-free equilibrium and the endemic equilibrium. We employ limiting system and Lyapunov functional method to demonstrate the global asymptotic stability of these equilibria. Our findings indicate that the disease-free equilibrium is globally asymptotically stable and globally attractive when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> , respectively; conversely, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> , the endemic equilibrium exhibits global asymptotic stability. To substantiate the global asymptotic stability of the endemic equilibrium, we also establish weak persistence within our model. Finally, sensitivity analysis and numerical simulations are conducted to illustrate how variations in incubation period τ and hospitalization treatment rate α affect infected individuals.