Dynamical behavior and bifurcation analysis for a theoretical model of dengue fever transmission with incubation period and delayed recovery

TL;DR

This work develops a delay-differential equation model for dengue transmission that incorporates intrinsic and extrinsic incubation delays and a delayed recovery term. Using a non-dimensional host–vector framework and next-generation matrix analysis, it identifies a basic reproduction number $R_0$ that governs the existence of a disease-free or endemic state, with $D_0$ globally stable when $R_0\le 1$. The study further analyzes time-delay effects on stability, revealing Hopf bifurcations and potential complex dynamics, including period-doubling, driven by delays in incubation and recovery. The findings highlight the critical role of time delays in shaping dengue dynamics and offer guidance for considering delays in control strategies and qualitative epidemic analyses.

Abstract

As offered by the World Health Organisation (WHO), close to half of the population in the world's resides in dengue-risk zones. Dengue viruses are transmitted to individuals by Aedes mosquito species infected bite (Ae. Albopictus of Ae. aegypti). These mosquitoes can transmit other viruses, including Zika and Chikungunya. In this research, a mathematical model is formulated to reflect different time delays considered in both extrinsic and intrinsic incubation periods, as well as in the recovery periods of infectious individuals. Preliminary results for the non-delayed model including positivity and boundedness of solutions, non-dimensionalization and equalibria analysis are presented. The threshold parameter (reproduction number) of the model is obtained via next generation matrix schemes. The stability analysis of the model revealed that various dynamical behaviour can be observed depending on delay parameters, where in particular the effect of delay in the recovery time of infectious individuals may lead to substantial changes in the dynamics. The ideas presented in this paper can be applied to other infectious diseases, providing qualitative evaluations for understanding time delays influencing the transmission dynamics.

Figures (5)

• Figure 1: Time evolution of each variable in the model \ref{['equ21']}-\ref{['equ27']} in the absence of delay terms, e.g. $\tau_r=\tau_h=\tau_v=0$ (a) in the presence of average extrinsic and intrinsic incubation time with $\tau_r=0$ and $\tau_h=\tau_v=5$ (b), in the presence of delay terms with $\tau_r=1.1$, $\tau_h=\tau_v=0.55$ (c) and $\tau_r=1.2$, $\tau_h=\tau_v=0.6$ (d). In the presence of all three delay terms it is assumed that $\tau_r=2 \tau_h=2 \tau_v$.
• Figure 2: Time evolution of the system given by \ref{['equ21']}-\ref{['equ27']} in the presence of the delay in recovery of the individuals with $\tau_h=\tau_v=0$ and $\tau_r=1.4$ (a), $\tau_r=1.7$ (b), $\tau_r=2.2$ (c) and $\tau_r=2.4$ (d). The initial conditions are set to $(1,0.1,0.1,0.1,1,0.1,0.1)$ and other parameters are given in the text.
• Figure 3: Bifurcation diagram of the variable for susceptible human population with respect to two parameters: $\alpha_1$ (a) and $\beta_{1h}$ (b) in the absence of delay constants, e.g. $\tau_h=\tau_v=\tau_r=0$. The green square represents a transcritical bifurcation where endemic equilibrium and disease free equilibrium intersect. The initial conditions are set to $(1,0.1,0.1,0.1,1,0.1,0.1)$ and other parameters are given in the text.
• Figure 4: Single parameter numerical continuation of susceptiple human population with respect to parameters $\alpha_1$ (a), $b_{1h}$ (b), $b_{1v}$ (c) and $c_{1v}$ (d) with $\tau_h=\tau_v=\tau=0.62$ and $\tau_r=1.24$. The colored lines represent the number of eigenvalues with positive real parts. The dashed lines stand for the branches emanating from Hopf bifurcation and represent the maximum of the periodic orbits. Purple and green squares respectively represent Hopf and transcritical bifurcations.
• Figure 5: Single parameter numerical continuation of susceptible human population with respect to parameters $\alpha_1$ (a), $b_{1h}$ (b), $b_{1v}$ (c) and $c_{1v}$ (d) with $\tau_h=\tau_v=\tau=0$ and $\tau_r=2.4$. The colored lines represent the number of eigenvalues with positive real parts. The dashed lines stand for the branches emanating from Hopf bifurcation and represent the maximum of the periodic orbits. Purple, cyan and green squares respectively represent Hopf, Period doubling and transcritical bifurcations.

Theorems & Definitions (7)

• lemma 1
• proof
• lemma 2
• proof
• theorem 1
• lemma 3
• lemma 4