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Study on Dynamical Behavior of Coinfection Infectious Disease Model

Yang Liu

TL;DR

This work presents a coinfection extension of SIS dynamics with two single infections and a co-infected class, identifying equilibrium regimes (disease-free, boundary, interior) and deriving a composite basic reproduction number $R_0=\\max\{\frac{a_1}{r_1}, a_2, k\}$. Using next-generation analysis and bifurcation theory, it characterizes stability conditions for these equilibria, uncovers cusp and saddle-node phenomena, and establishes a codimension-3 Bogdanov–Takens bifurcation with parameter set $(a_1,a_2,k)$ as the bifurcation drivers. The study provides detailed normal-form computations and center-manifold reductions to reveal how small parameter perturbations can trigger complex transitions between disease-free and endemic coinfection states. Numerical simulations validate the theoretical predictions, illustrating stable boundary states and bifurcation-induced regime changes, with practical implications for targeted control of co-infection spread. The results advance understanding of multi-disease interactions and illuminate critical thresholds for intervention in coinfection dynamics.

Abstract

This paper conducts research on the established model and presents the main conclusions . Firstly, by separately considering the infectivity of each of the two infectious diseases and the infectivity of the population simultaneously infected with the two infectious diseases, the existence of three types of boundary equilibrium points is determined, as well as the existence of the interior equilibrium point when the parameters are under specific conditions. Then, the stability of the equilibrium points is analyzed. It is concluded that under different parameter conditions, the stability of the disease free equilibrium point can exhibit various scenarios, such as a stable node or a saddle-node, etc. For the boundary equilibrium points, the situation is more intricate,and a cusp may occur. The stability of the interior equilibrium point under specific conditions is also presented. Finally,the degeneracy of the equilibrium points is studied through the bifurcation theory.Mainly, the saddle-node bifurcation occurring at the interior equilibrium point is obtained, and when the infection rate of the first infectious disease, the infection rate of the second infectious disease, and the infection rate of the co-infected population to other populations are selected as bifurcation parameters, a codimension 3 B-T bifurcation is obtained.

Study on Dynamical Behavior of Coinfection Infectious Disease Model

TL;DR

This work presents a coinfection extension of SIS dynamics with two single infections and a co-infected class, identifying equilibrium regimes (disease-free, boundary, interior) and deriving a composite basic reproduction number . Using next-generation analysis and bifurcation theory, it characterizes stability conditions for these equilibria, uncovers cusp and saddle-node phenomena, and establishes a codimension-3 Bogdanov–Takens bifurcation with parameter set as the bifurcation drivers. The study provides detailed normal-form computations and center-manifold reductions to reveal how small parameter perturbations can trigger complex transitions between disease-free and endemic coinfection states. Numerical simulations validate the theoretical predictions, illustrating stable boundary states and bifurcation-induced regime changes, with practical implications for targeted control of co-infection spread. The results advance understanding of multi-disease interactions and illuminate critical thresholds for intervention in coinfection dynamics.

Abstract

This paper conducts research on the established model and presents the main conclusions . Firstly, by separately considering the infectivity of each of the two infectious diseases and the infectivity of the population simultaneously infected with the two infectious diseases, the existence of three types of boundary equilibrium points is determined, as well as the existence of the interior equilibrium point when the parameters are under specific conditions. Then, the stability of the equilibrium points is analyzed. It is concluded that under different parameter conditions, the stability of the disease free equilibrium point can exhibit various scenarios, such as a stable node or a saddle-node, etc. For the boundary equilibrium points, the situation is more intricate,and a cusp may occur. The stability of the interior equilibrium point under specific conditions is also presented. Finally,the degeneracy of the equilibrium points is studied through the bifurcation theory.Mainly, the saddle-node bifurcation occurring at the interior equilibrium point is obtained, and when the infection rate of the first infectious disease, the infection rate of the second infectious disease, and the infection rate of the co-infected population to other populations are selected as bifurcation parameters, a codimension 3 B-T bifurcation is obtained.

Paper Structure

This paper contains 5 sections, 9 theorems, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Existence and stability of equilibrium points
  3. Bifurcation Analysis
  4. Numerical simulation
  5. Conclusions

Key Result

Theorem 1

(1) Regardless of the values of the parameters, the system always has a disease-free equilibrium point $E_0=(0,0,0)$; (2) When $R_{0,1}>1$, the system has a boundary equilibrium point $E_1 = (\frac{a_1 - r_1}{a_1},0,0)$; (3) When $R_{0,2}>1$, the system has a boundary equilibrium point $E_2=(0,\frac

Figures (3)

  • Figure 1: The graph of parameter $a_1$ with respect to $a_2$
  • Figure 2: The density change curves of three types of populations when $R_2 < 1$
  • Figure 3: The phase diagram of the system near $E_4$ when $k = 1$ and $v_2=0$

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • Remark 2
  • ...and 12 more