Communicable diseases are primarily caused by pathogens such as bacteria, viruses, fungi, and parasites, and are transmitted through the air, water, food, and other routes. Their spread affects not only human physical and mental health but also social and economic development. Human T-cell leukemia virus type I (HTLV-I) is a pathogenic retrovirus that is prevalent primarily in the Caribbean, Japan, Central Africa, and South America [1]. Following infection, HTLV-I can persist in the host for a long time, causing HTLV-I-associated myelopathy (HAM), also known as tropical spastic paraplegia (TSP), a chronic inflammatory disease of the central nervous system (CNS) [2][3]. Additionally, HTLV-I infection can lead to malignant hematological diseases such as adult T-cell leukemia/lymphoma (ATL) [4][5].

Individuals infected with HTLV-I typically display no obvious clinical manifestations; only 0.25% - 3.8% develop HAM/TSP, and 2% - 3% develop ATL. These patients have very high levels of CD8⁺ cytotoxic T lymphocytes (CTLs) in their peripheral blood [6]. Experiments have demonstrated that CTLs can reduce viral load, clear infected cells, and protect the body [7][8]; however, when CTL levels are excessively high, they release large amounts of toxins, leading to the symptoms of HAM/TSP [9]. Therefore, incorporating the CTL immune response into the development of a dynamic model of HTLV-I infection is of great significance for gaining a deeper understanding of the pathogenesis of ATL and HAM/TSP and for formulating treatment strategies.

In 1999, Wodarz et al. first constructed an HTLV-I infection model that accounted for interactions between healthy and infected CD4⁺T cells [10]. Subsequently, Gómez-Acevedo and Li demonstrated the existence of the backward branch described in reference [10] through their analysis [11]. In 2002, Wodarz et al. established an HTLV-I infection model incorporating CTL immune responses to study the impact of CTL immunity on viral replication [12], using the bilinear function c y z to describe the proliferation of CTL immune cells. Sun and Wei studied a class of HTLV-I infection models incorporating a CTL immune response and demonstrated that the global dynamical behavior of the system is determined by two threshold parameters, R 0 and R 1 [13].

In 2012, Li and Shu [14] established and studied a model of HTLV-I infection involving a CTL immune response; they introduced a time delay into the model and demonstrated that this delay can destabilize the HAM/TSP equilibrium, leading to a Hopf bifurcation and stable periodic oscillations. Jia and Xu studied a class of HTLV-I models featuring a Beddington-DeAngelis incidence rate of β x y / ( 1 + a 1 y + a 2 y ) and infection delays, investigating the effects of single and double delays on the system dynamics [15]. Chen et al. studied a class of infection models featuring Logistic growth of healthy cells and CTL immune proliferation described by the function f ( y , z ) = c y z / ( 1 + q z ) ( 1 + ε z ) , and investigated the local and global stability of the model’s equilibrium points [16]. Li and Zhou constructed a mathematical model featuring Logistic growth of healthy cells and an infection rate given by β x y / 1 + ω x , and investigated the existence of backward branches in the model [17]. Zhang et al. studied the global dynamical behavior of a class of HTLV-I infection models featuring intracellular time delays and saturated CTL immune responses, proving the existence and local asymptotic stability of viable equilibria [18]. Xu and Yang studied the dynamics of an HTLV-I infection model featuring a saturated immune response and immune impairment. By calculating the unactivated and activated reproduction numbers, they discussed the stability of equilibrium points with (and without) time delays, as well as branching [19].

Building on the research in references [16], [17], [19] and based on the work in reference [14], we account for host resource limitations by incorporating a logistic growth model for the number of healthy cells; furthermore, since the immune response does not increase indefinitely with rising viral load, we adopt a saturated CTL immune response. We aim to study a model of HTLV-I infection characterized by logistic growth and a saturated CTL immune response

In this model, x , y , and z represent, respectively, the densities of healthy CD4⁺T cells, infected CD4⁺T cells, and HTLV-I-specific CD8⁺CTL cells at time t ; r represents the natural proliferation rate of healthy CD4⁺T cells; K represents the carrying capacity of healthy CD4⁺T cells; β represents the viral infection rate; p represents the intensity of the CTL response, i.e., the rate at which the CTL immune response clears infected cells; c represents the proliferation rate of CTL s; q > 0 represents the CTL suppression rate, that is, CTL cells are stimulated by the virus and are generated at a rate of c y z 1 + q z ; μ 1 and μ 2 represent the natural death rates of infected CD4⁺T cells and CTL cells, respectively.

For convenience in calculation, Equation (1.1) is nondimensionalized as follows:

Define the dimensionless parameters:

Continuing to use t to denote τ , Equation (1.1) can be written as

Individual cells often exhibit significant spatial heterogeneity; therefore, spatial heterogeneity plays an indispensable role in infectious disease modeling and helps elucidate the spatial complexity of infectious disease dynamics. References [20][21] have investigated the role of convection in the dynamics of infectious disease models, while references [22]-[24] have discussed the impact of chemotaxis on infectious disease modeling. The introduction of chemotaxis makes the established infectious disease models more realistic and can significantly enhance the effectiveness of disease prevention. Therefore, considering the impact of spatial heterogeneity on the dynamics between immune cells and infected cells, this paper investigates an HTLV-I infection model incorporating immune chemotaxis:

Here, − ξ   ∇ ⋅ ( w ∇ v ) describes the movement of CTL immune cells toward regions of high infected cell concentration. From a biological perspective, ξ > 0 indicates that CTL immune cells move toward regions of high infected cell density, whereas ξ < 0 implies that CTL immune cells move away from the habitats of infected cells to prevent large numbers of infected cells from aggregating and forming a defense.

This paper primarily investigates the existence of Hopf branches in model (1.3), as well as the branch direction and stability of periodic solutions on these branches.

The positive equilibrium of model (1.2) is

When a f b f + a 2 e > 1 , the positive equilibrium of model (1.2) exists.

Let U * = ( u * , v * , w * ) be a positive equilibrium point of model (1.3). At U * , the following conclusions hold:

Let Ω ⊂ ℝ n ( n ≥ 1 ) be a bounded domain with smooth boundary, and let the initial data ( u 0 , v 0 , w 0 ) ∈ [ C ( Ω ¯ ) ] 3 be nonnegative and not identically zero. Under homogeneous Neumann boundary conditions, the theory of parabolic equations ensures that system (1.3) admits a unique local classical solution, and u , v , w > 0 for all x ∈ Ω and t ∈ ( 0 , T max ) .

Theorem 2.1 Suppose that a f b f + a 2 e > 1 holds, then

1) When ξ > 0 , ( u * , v * , w * ) is locally asymptotically stable;

2) When 0 > ξ > ξ 0 = max k ∈ ℕ + { ξ k S , ξ k H } , ( u * , v * , w * ) is locally asymptotically stable;

3) When ξ < ξ 0 = max k ∈ ℕ + { ξ k S , ξ k H } < 0 , then ( u * , v * , w * ) is unstable. Where

We now discuss the existence and stability of periodic solutions for model (1.3). For computational convenience, this paper primarily considers the one-dimensional case of (1.3), namely the model

The symbol ' denotes the derivative with respect to x . Combining Theorem 2.1, we see that the periodic solution of model (2.1) branches off from the positive

equilibrium point at ξ = ξ k H . When ξ passes through ξ 0 = max k ∈ ℕ + { ξ k S , ξ k H } , the equilibrium point becomes unstable via a Hopf bifurcation. To apply bifurcation theory to model (2.1) at ξ = ξ k H , this paper must verify that the real part of the characteristic root at ξ = ξ k H lies on the imaginary axis. When ξ = ξ k H and Q ( ξ , k ) > 0 , the linearization matrix of model (1.3) has the following purely imaginary roots

where

Thus, model (1.3) may generate a Hopf branch at a positive equilibrium point. Below, we explain when Q ( ξ k H , k ) > 0 . Let ξ k M be

the unique root of Q ( ξ k H , k ) = 0 , whose expression is

where μ k = ( k π l ) 2 , k = 0 , 1 , 2 , ⋯ , Calculations show that the following order relations hold among ξ k S , ξ k M , and ξ k H .

Lemma 2.1 For ∀ k ∈ ℕ + , either ξ k S < ξ k M < ξ k H or ξ k H < ξ k M < ξ k S . Furthermore, if the former holds, then Q ( ξ k H ) > 0 > Q ( ξ k S ) . If the latter holds, then Q ( ξ k S ) > 0 > Q ( ξ k H ) .

By Lemma 2.1, the linearization matrix of model (1.3) has a pair of purely imaginary roots if and only if ξ k S < ξ k H . This implies that a Hopf branch of model (2.1) at ( U * , ξ k H ) can occur only if ξ k S < ξ k H . Therefore, in the subsequent analysis of Hopf branches, this paper consistently assumes that ξ k S < ξ k H .

This manuscript investigates the Hopf bifurcation problem in a reaction-diffusion model of HTLV-I infection that incorporates CTL immune responses and chemotactic effects, and systematically analyzes the existence of Hopf bifurcations at positive equilibrium points as well as the stability of bifurcation periodic solutions.

Mathematically, this paper derives the critical conditions for the occurrence of a Hopf bifurcation, proves the existence of periodic solutions near the bifurcation point, and determines the bifurcation direction and stability of the periodic solutions: if ξ k 0 H ″ ( 0 ) > 0 , the bifurcation direction is supercritical, the periodic solutions are stable; if ξ k 0 H ″ ( 0 ) < 0 , the bifurcation direction is subcritical, the periodic solutions are unstable.

From a biological perspective, this result implies that the system undergoes a transition from a stable equilibrium to sustained periodic oscillations as the CTL chemotaxis intensity varies.

First and corresponding author, Hongyan Kang: Conceptualization, mathematical derivation for bifurcation analysis, numerical simulation, original draft writing, interpretation of results, manuscript editing and review. Co‑author, Yinji Huang: Model nondimensionalization, equilibrium point analysis.