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Songbo Hou

Publications and source records attributed to Songbo Hou.

At least 19 recordsLinked to original sources
arXiv2026

Stability and Hopf Bifurcation of a Delayed SVIRS Epidemic Model with Media Coverage

This paper formulates and analyzes a delayed SVIRS epidemic model incorporating media coverage effects, vaccination, waning immunity, temporary post-recovery immunity, saturated treatment, and delayed behavioral responses induced by media coverage. The positivity and uniform boundedness of solutions are established, the basic reproduction number is derived, and the local and global asymptotic stability of the disease-free and endemic equilibria is investigated. Taking the media-induced behavioral delay as the Hopf bifurcation parameter, a critical delay threshold is obtained, beyond which the endemic equilibrium loses stability and periodic oscillations emerge. Center manifold and normal form theories are applied to determine the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions, while a global Hopf bifurcation theorem is used to establish the unbounded continuation of the periodic solution branch. Numerical simulations confirm the theoretical results and indicate that stronger media intervention can suppress epidemic oscillations and enhance system stability. These findings reveal the coupled effects of multiple epidemiological mechanisms and delayed media responses, providing theoretical support for the design of effective infectious disease control strategies.

math.DS↗
arXiv2025

Stability and Hopf bifurcation analysis of an age-structured SVIRS epidemic model with temporary immunity

In this paper, we investigate an SVIRS epidemic model that incorporates both temporary immunity and an age-structured recovery process. By reformulating the system as a non-densely defined abstract Cauchy problem, we establish the existence and uniqueness of solutions and derive the basic reproduction number $ \mathcal{R}_0 $. The stability of the equilibria is analyzed through the associated characteristic equations, and the occurrence of Hopf bifurcation near the endemic equilibrium is rigorously demonstrated. Our theoretical results reveal that temporary immunity plays a crucial role in shaping the stability of the endemic state. Finally, numerical simulations are carried out to verify and illustrate the analytical findings.

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arXiv2025

Existence and asymptotic analysis of topological solutions for generalized Chern--Simons equations on discrete lattice graphs

We study a class of generalized Chern-Simons equations on discrete lattice graphs. By an iterative scheme combined with an exhaustion argument, we establish the existence of topological solutions, which is also the maximal topological solution. We further examine the behavior of the maximal topological solution as the parameter tends to either infinity or zero. The present work extends the results of Hua et al., arXiv:2310.13905 (2023) and Hou and Kong, Calc. Var. Partial Differ. Equ. 64(3), 77 (2025).

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arXiv2025

Stability and Hopf bifurcation analysis of an HIV infection model with latent reservoir, immune impairment and delayed CTL immune response

In this paper, we develop a dynamic model of HIV infection that incorporates latent hosts, cytotoxic T lymphocyte (CTL) immunity, saturated incidence rates, and two transmission mechanisms: virus-to-cell and cell-to-cell transmission. The model has three kinds of delays: intracellular delay, replication of viruses delay, immune response delay. Initially, the model's solutions are confirmed to be both nonnegative and bounded for nonnegative initial values. Subsequently, two biologically critical parameters were identified: the virus reproduction number $\mathcal{R}_0$ and the immune reproduction number $\mathcal{R}_1$. Thereafter, by invoking LaSalle's principle of invariance alongside Lyapunov functionals, we establish stability criteria for each equilibrium. The results indicate that the stability of the endemic equilibrium may be altered by a positive immune delay, whereas intracellular and viral replication delays do not affect the equilibria. By considering the delay in the immune response as a bifurcation-inducing threshold, we derive the exact conditions necessary for these stability transitions. Further analysis shows that increasing the immune delay destabilizes the endemic equilibrium, inducing a Hopf bifurcation. Additionally, using the center manifold theorem and normal form theory, we explored the direction and stability of Hopf bifurcations in detail. To corroborate these theoretical results, numerical simulations are systematically conducted.

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arXiv2024

On topological solutions to a generalized Chern-Simons equation on lattice graphs

For $n \geq 2$, consider $\mathbb{Z}^n$ as a lattice graph. We explore a generalized Chern-Simons equation on $\mathbb{Z}^n$. Employing the method of exhaustion, we prove that there exists a global solution that also qualifies as a topological solution. Our results extend those of Hua et al. [arXiv:2310.13905] and complement the findings of Chao and Hou [J. Math. Anal. Appl. $\bf{519}$(1), 126787(2023)], as well as those of Hou and Qiao [J. Math. Phys. $\bf{65}$(8), 081503(2024)].

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arXiv2024

A cytokine-enhanced viral infection model with CTL immune response, distributed delay and saturation incidence

In this paper, we propose a delayed cytokine-enhanced viral infection model incorporating saturation incidence and immune response. We compute the basic reproduction numbers and introduce a convex cone to discuss the impact of non-negative initial data on solutions. By defining appropriate Lyapunov functionals and employing LaSalle's invariance principle, we investigate the stability of three equilibria: the disease-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium. We establish conditions under which these equilibria are globally asymptotically stable. Numerical analyses not only corroborate the theoretical results but also reveal that intervention in virus infection can be achieved by extending the delay period.

math.DS↗
arXiv2024

Solutions to a generalized Chern-Simons Higgs model on finite graphs by topological degree

Consider a finite connected graph denoted as $G=(V, E)$. This study explores a generalized Chern-Simons Higgs model, characterized by the equation: $$ Δu = λe^u (e^u - 1)^{2p+1} + f,$$ where $Δ$ denotes the graph Laplacian, $λ$ is a real number, $p$ is a non-negative integer, and $f$ is a function on $V$. Through the computation of the topological degree, this paper demonstrates the existence of a single solution for the model. Further analysis of the interplay between the topological degree and the critical group of an associated functional reveals the presence of multiple solutions. These findings extend the work of Li, Sun, Yang (arXiv:2309.12024) and Chao, Hou (J. Math. Anal. Appl. (2023) 126787).

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arXiv2024

Existence theorems for a generalized Chern-Simons equation on finite graphs

Denote by $G=(V,E)$ a finite graph. We study a generalized Chern-Simons equation $$ Δu=λ\mathrm{e}^u(\mathrm{e}^{bu}-1)+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ on $G$, where $λ$ and $b$ are positive constants; $N$ is a positive integer; $p_1, p_2, \cdot\cdot\cdot, p_N$ are distinct vertices of $V$ and $δ_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there exists a critical value $λ_c$ such that the equation has a solution if $λ\geq λ_c$ and the equation has no solution if $λ<λ_c$. We also prove that if $λ>λ_c$ the equation has at least two solutions which include a local minimizer for the corresponding functional and a mountain-pass type solution.

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arXiv2024

Existence and asymptotic behaviors of solutions to Chern-Simons systems and equations on finite graphs

In this paper, we consider a system of equations arising from the $\text{U}(1)\times \text{U}(1)$ Abelian Chern-Simons model \begin{eqnarray*}\left\{\begin{aligned} Δu &=λ\left(a(b-a)\mathrm{e}^u-b(b-a)\mathrm{e}^{\upsilon}+a^2\mathrm{e}^{2u}-ab\mathrm{e}^{2\upsilon}+b(b-a)\mathrm{e}^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\\ Δ\upsilon&=λ\left(-b(b-a)\mathrm{e}^u+a(b-a)\mathrm{e}^{\upsilon}-ab\mathrm{e}^{2u}+a^2\mathrm{e}^{2\upsilon}+b(b-a)\mathrm{e}^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. \end{eqnarray*} on finite graphs. Here $λ>0$, $b>a>0$, $m_j>0\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j>0\,(j=1,2,\cdot\cdot\cdot,k_2)$, $δ_{p}$ is the Dirac delta mass at vertex $p$. We establish the iteration scheme and prove existence of solutions. We also develop a new method to get the asymptotic behaviors of solutions as $λ$ goes to infinity. This method is also applicable to the Chern-Simons system $$\left\{\begin{aligned} Δu &=λ\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1) +4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\\ Δ\upsilon&=λ\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. $$ and the classical Chern-Simons equation $$ Δu=λ\mathrm{e}^u(\mathrm{e}^u-1)+4π\sum\limits_{j=1}^{N}δ_{p_j}.$$

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arXiv2022

Existence of solutions to a generalized self-dual Chern-Simons system on finite graphs

We study a system of equations arising in the Chern-Simons model on finite graphs. Using the iteration scheme and the upper and lower solutions method, we get existence of solutions in the non-critical case. The critical case is dealt with by priori estimates. Our results generalize those of Huang et al. (Journal of Functional Analysis 281(10) (2021) Paper No. 109218).

math.AP↗
arXiv2022

Existence of solutions to Chern-Simons-Higgs equations on graphs

Let $G=(V,E)$ be a finite graph. We consider the existence of solutions to a generalized Chern-Simons-Higgs equation $$ Δu=-λe^{g(u)}\left( e^{g(u)}-1\right)^2+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ on $G$, where $λ$ is a positive constant; $g(u)$ is the inverse function of $u=f(\upsilon)=1+\upsilon-e^{\upsilon}$ on $(-\infty, 0]$; $N$ is a positive integer; $p_1, p_2, \cdot\cdot\cdot, p_N$ are distinct vertices of $V$ and $δ_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there is critical value $λ_c$ such that the generalized Chern-Simons-Higgs equation has a solution if and only if $λ\geq λ_c$ . We also prove the existence of solutions to the Chern-Simons-Higgs equation $$ Δu=λe^{u}(e^{u}-1)+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ on $G$ when $λ$ takes the critical value $λ_c$ and this completes the results of An Huang, Yong Lin and Shing-Tung Yau (Commun. Math. Phys. 377, 613-621 (2020)).

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arXiv2021

Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds

Let $λ(t)$ be the first eigenvalue of $-Δ+aR\, (a>0)$ under the backward Ricci flow on locally homogeneous 3-manifolds, where $R$ is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of $λ(t)$. In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, $λ^{+}(t)$ approaches zero, where $λ^{+}(t)=\max\{λ(t),0\}$.

math.DG↗
arXiv2021

Gradient estimates for the nonlinear parabolic equation with two exponents on Riemannian manifolds

In this paper, we study the nonlinear parabolic equation with two exponents on complete noncompact Riemannian maniflods. The special types of such equation include the Fisher-KPP equation, the parabolic Allen-Cahn equation and the Newell-Whitehead equation. We get the Souplet-Zhang's gradeint estimates for the positive solutions to such equation. We also obtain the Liouville theorem for positive ancient solutions. Our results extend those of Souplet-Zhang (Bull. London. Math. Soc. 38:1045-1053, 2006) and Zhu (Acta Mathematica Scientia 36B(2): 514-526, 2016).

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arXiv2019

Extremal functions for a singular Hardy-Moser-Trudinger inequality

In this paper, using blow-up analysis, we prove a singular Hardy-Morser-Trudinger inequality, and find its extremal functions. Our results extend those of Wang-Ye (Adv. Math. 2012), Yang-Zhu ( Ann. Glob. Anal. Geom. 2016), Csató- Roy (Calc. Var. 2015), and Yang-Zhu (J. Funct. Anal. 2017).

math.FA↗
arXiv2019

Eigenvalues under the Ricci flow of model geometries

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for the eigenvalue.

math.DG↗
arXiv2019

Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.

math.DG↗