Modeling and analysis of a novel two-strain dengue epidemics model considering secondary infections with increased mortality

TL;DR

This paper addresses the complex transmission dynamics of dengue by formulating a deterministic two-strain host–vector model that explicitly includes temporary cross-immunity, ADE through distinct primary and secondary transmission rates, disease-induced mortality in secondary infections, and vector co-infection. It derives the basic reproduction number $R_0 = \left( \frac{α β}{κ (γ + μ)} \right)^{1/2}$ and analyzes disease-free, one-strain, and two-strain equilibria, showing that ADE can enable invasion by a heterologous strain and that backward bifurcation is possible. Using center-manifold theory and COCO numerical continuation, the study uncovers bistability between disease-free and endemic states, a forward bifurcation near $α ≈ 0.33355$, and Hopf-induced oscillations near $α ≈ 0.50012$, with oscillation amplitudes increasing as $α$ grows. The results highlight the interplay of ADE, waning cross-immunity, and vector co-infection in producing rich dengue dynamics, informing control and vaccination strategies in endemic regions.

Abstract

In this study, we develop and analyze a deterministic two-strain host-vector model for dengue transmission that incorporates key immuno-epidemiological mechanisms, including temporary cross-immunity, antibody-dependent enhancement (ADE), disease-induced mortality during secondary infections, and explicit vector co-infection. The human population is divided into compartments for primary and secondary infections, while the mosquito population includes single- and co-infected classes. ADE is modeled through distinct primary ($α$) and secondary ($σ$) transmission rates. Using the next-generation matrix method, we derive the basic reproduction number $R_0$ and establish the local stability of the disease-free equilibrium for $R_0 < 1$. Analytical results show that one-strain endemic equilibria lose stability under ADE conditions ($σ> α$), allowing invasion by a heterologous strain. Employing center-manifold theory and numerical continuation (COCO), we demonstrate the occurrence of backward bifurcation, bistability between disease-free and endemic states, and Hopf-induced oscillations. Numerical simulations confirm transitions among disease-free, endemic, and periodic regimes as key parameters vary. The model highlights how ADE, waning cross-immunity, and vector co-infection interact to generate complex dengue dynamics and provides insights useful for designing effective control and vaccination strategies in dengue-endemic regions.

Figures (4)

• Figure 1: Schematic representation of the host-vector transmission dynamics of a two-strain pathogen. The human population $N$ is divided into compartments based on infection status: susceptible ($S$), infected with strain $i$ ($I_i$), recovered from strain $i$ ($R_i$), partially susceptible after infection with strain $i$ ($S_i$), reinfected with strain $j$ after recovering from strain $i$ ($I_{ij}$), and fully recovered ($R$) where $i,j \in 1,2$. The vector population includes susceptible vectors ($U$) and vectors infected with strain 1, strain 2, or both ($V_1, V_2, V_{12}$). Arrows indicate transitions due to infection, recovery, waning immunity, and disease-induced mortality.
• Figure 2: One- and two-strain transient responses of the dengue epidemics model \ref{['Eq:Model']}, computed for the parameter values given in Table \ref{['tab:params_values']}, with $\sigma=0.45$. In this picture, the time histories for $I_{1}(t)$ and $I_{2}(t)$ are plotted in black and green, respectively.
• Figure 3: (a) One-parameter continuation of the two-strain (blue curve) and one-strain (red curve) equilibria calculated from Fig. \ref{['fig-sol-ini']}. In these diagrams, the vertical axes are given by $I_{\text{\tiny Tot}}=I_{1}+I_{2}+I_{12}+I_{21}$ and $I_{\text{\tiny Sec}}=I_{12}+I_{21}$. The BP and H points stand for branching ($\alpha\approx0.33355$) and Hopf ($\alpha\approx0.50012$) bifurcations, respectively. Solution branches of stable equilibria are plotted with solid lines, while dashed lines denote instability. (b) System responses calculated at the test points P1 ($\alpha=0.15$) and P2 ($\alpha=0.39$). Here, the color code is as follows: $I_{1}(t)$ (black), $I_{2}(t)$ (green), $V_{1}(t)$ (purple) and $V_{2}(t)$ (yellow).
• Figure 4: (a) Family of periodic solutions produced by the Hopf bifurcation detected in Fig. \ref{['fig-bif-diags']}(a), calculated from $\alpha=0.529$ to $\alpha=0.75$. The arrow indicates the direction of increasing $\alpha$. (b) Time histories corresponding to the solution highlighted in red ($\alpha=0.627$) in panel (a). In these plots, solid and dashed lines correspond to state variables with leading subscript 1 and 2, respectively.

Theorems & Definitions (1)

• Theorem 1