Dynamic Analysis and Optimal Control Strategy for the Impact of Stem Cell Therapy on Type 1 Diabetes

Abstract

Type 1 diabetes (T1D) is a chronic autoimmune disease in which autoreactive effector T cells destroy insulin-producing pancreatic β-cells, leading to impaired insulin production and long-term metabolic complications. This study develops a mathematical model of the interactions among pancreatic β-cells, autoreactive effector T cells, regulatory T cells (Tregs), and stem cell therapy, treating stem cell administration as an immunomodulatory and regenerative strategy that suppresses autoimmune activity and promotes β-cell recovery. The existence, uniqueness, positivity, and boundedness of solutions are established. For the auxiliary subsystem with $S_{E }=0$ and constant treatment input, the disease-free equilibrium and the threshold quantity $R_{0 }$ are derived, and local asymptotic stability of the DFE is established for $R_{0 }<1$. For the full model with $S_{E }>0$, the existence of a unique biologically feasible positive equilibrium is established, together with its local asymptotic stability under constant treatment input. An optimal control problem, formulated using Pontryagin’s Maximum Principle, is used to guide therapeutic dosing, and one-year numerical simulations compare two administration protocols: high-dose pulse injections and continuous infusion. Both reduce autoimmune activity and improve β-cell dynamics, with pulse administration producing stronger transient responses and continuous infusion producing smoother treatment-period dynamics, while both protocols approach similar long-term levels. A local sensitivity analysis identifies immune activation, β-cell destruction, and regulatory T-cell activity as the parameters most strongly shaping disease progression and treatment outcomes. By unifying stem cell therapy, stability analysis, sensitivity analysis, and optimal control, and directly comparing pulse and infusion protocols, this framework offers new insight for designing stem cell-based treatments for autoimmune diabetes.

1. Introduction

Type 1 diabetes mellitus (T1D) is a chronic autoimmune disorder that selectively targets pancreatic β-cells, leaving patients with a near-total loss of insulin production and a lifelong dependence on external insulin therapy [1,2,3]. Unlike type 2 diabetes, the etiology of T1D is driven primarily by immunological dysregulation, in which autoreactive effector T cells target β-cell autoantigens and progressively impair pancreatic function [4,5,6,7]. Although insulin delivery and glucose monitoring technologies have improved significantly, current treatments do not address the underlying autoimmune mechanisms that initiate and sustain the disease [8,9]. The global prevalence of type 1 and type 2 diabetes among individuals under 20 years is projected to peak in 2045 [10], and diabetes continues to affect individuals worldwide regardless of ethnicity, gender, nationality, or socioeconomic background [10].

Stem cell therapy is a promising strategy in regenerative medicine for treating a range of disorders, including T1D. Any viable stem cell-based remedy for T1D must address two goals at once: replacing lost β-cells and regulating the autoimmune response to insulin-producing cells [11]. Clinical intervention therefore aims to prevent or halt the onset and progression of autoimmunity, reverse existing cellular damage, and restore glycometabolic and immunological balance. Despite encouraging results from islet transplantation and progress in immunomodulatory medications, a durable cell-replacement strategy to treat T1D remains elusive, and stem cell treatment offers a promising alternative to the challenges associated with islet transplantation [12]. Advances in stem cell research have since led to substantial progress in stem cell-based therapies for T1D [13].

Over the past two decades, mathematical modeling has become an important tool for studying the complex interactions between the immune system and pancreatic β-cells in T1D [14,15,16,17,18,19]. Several hypotheses have been proposed to explain β-cell function, the mechanisms by which the immune system damages β-cells, and the interplay between effector and regulatory T cells [20], yielding important insights into disease progression, immune tolerance disorders, and the effects of immunomodulatory therapies. However, most existing models focus primarily on immunodynamics and do not incorporate modern regenerative or immunomodulatory treatments. Separately, several studies have incorporated time-dependent control functions into mathematical models of biological and pharmacological systems to represent dosing regimens [21,22,23,24]. For instance, pulse-modulated feedback dosing has been used to study the nonlinear effects of discrete drug administration on system dynamics [25], and impulsive controls representing short drug-release windows have been applied to pharmacokinetic models of periodic dosing. Related work has used ODE models to represent impulsive insulin injections as bolus doses [26], and clinical pharmacokinetic studies have examined mixed continuous and pulsed dosing regimens to assess their effects on drug concentration and clinical outcomes [27]. More recent modeling studies have investigated immune-response dynamics in viral infections and the effects of time delays in epidemic models [28,29], highlighting the importance of incorporating temporal and immunological mechanisms into mathematical models of disease progression.

The core immune subsystem describing pancreatic β-cells, autoreactive effector T cells, and regulatory T cells is adapted from the established T1D modeling frameworks in [30,31], and no novelty is claimed for these basic immune interactions. The present study extends this framework by introducing a dynamic stem-cell compartment that contributes to β-cell recovery and enhances regulatory T-cell activity, together with a time-dependent stem-cell administration mechanism. In this extended model, pulse-based and continuous-infusion schedules are compared using the same baseline parameter set, and an optimal-control problem is formulated to investigate treatment administration. Thus, the principal contribution of this work is integrating stem-cell-mediated regenerative and immunomodulatory mechanisms with treatment scheduling and optimal control, rather than proposing a new core T1D immune model.

The main contributions of this work are summarized as follows:

• Starting from the established immune-dynamics framework in [30,31], a dynamic stem-cell compartment is introduced to represent both β-cell regenerative support and regulatory T-cell enhancement within a single extended system.
• For the auxiliary autonomous subsystem with no basal recruitment of pathogenic effector T cells and constant therapeutic input, a threshold quantity $R_0$ is derived using the next-generation matrix approach, and the local stability of the DFE is established. For the full model with $S_{E }>0$, a unique biologically feasible positive equilibrium is established and its local asymptotic stability is proved independently of $R_0$.
• An optimal control framework compares two representative stem-cell administration strategies, pulse injections and continuous infusion, providing quantitative guidance for the design of stem cell-based treatment protocols.
• A sensitivity analysis identifies the key biological parameters that govern treatment effectiveness, offering additional insights into optimizing therapeutic outcomes.

Overall, the proposed framework offers new insights into the therapeutic potential of stem cell therapy for restoring immune homeostasis and preserving pancreatic β-cell mass in T1D, helping bridge the gap between mathematical modeling and clinical application.

The rest of this paper is structured as follows. Section 2 develops the mathematical model and discusses its biological assumptions. Section 3 establishes the model’s reliability by proving the uniqueness of the solution and the positivity and boundedness of the state variables. Section 4 analyzes the model, including the calculation of the reproduction number and the local stability analysis. Section 5 summarizes the results of the sensitivity analysis. Section 6 formulates an optimal control problem for the proposed model. Section 7 presents the numerical simulations, compares the pulse and infusion treatment strategies, and discusses their biological and clinical implications. Section 8 presents the conclusions.

2. Formulation of an Optimal Mathematical Model

T1D results from a disruption of immunological tolerance, leading to the proliferation of autoreactive effector T cells that specifically destroy pancreatic β-cells. The reduction in β-cell mass impairs insulin production, leading to persistent hyperglycemia. At the same time, regulatory T cells (Tregs), which usually control pathogenic immunological activation, are known to be both quantitatively and functionally inadequate in people at risk for T1D. Understanding the dynamics of β-cells, autoreactive T cells, and regulatory T-cell populations is crucial for defining disease development and evaluating prospective immunomodulatory therapy. This study presents a mathematical model that explains the connections among β-cells $B\left(t\right)$, autoreactive effector T cells $E\left(t\right)$, regulatory T cells $R\left(t\right)$, and stem cell transplants $S\left(t\right)$. To account for therapeutic interventions based on stem cell infusion or stem cell-derived factors, variables indicating circulating stem cells $S\left(t\right)$ are included. These variables represent the potential contribution to β-cell regeneration and support the functions of regulatory T cells.

In the β-cell equation for $B\left(t\right)$, the first term denotes the natural supply or replacement rate of β cells. The second term represents the decline in the β-cell population due to interactions between β-cells and autoreactive effector T cells. The third term reflects a decline in β-cell numbers due to natural mortality. The fourth term denotes the regenerative and functional support provided by stem cell transplantation, which promotes the differentiation of stem cells into progenitor β-cells.

$${\displaystyle \frac{dB\left(t\right)}{dt}}={\mathrm{S}}_B-K_{E}EB-{\mu }_B B+{\rho }_B S.$$

(1)

The dynamics of autoreactive effector T cells $E\left(t\right)$ are governed by four physiologically motivated components: basal production, antigen-driven expansion, regulatory repression, and natural turnover.

$${\displaystyle \frac{dE\left(t\right)}{dt}}=S_{E }+{\alpha }_B{\displaystyle \frac{B}{B+h_B}}E-{\phi }_R RE-{\mu }_{E}E.$$

(2)

The term $S_E$ refers to the thymus’s background generation of autoreactive T cells as well as peripheral activation. This baseline influx is independent of β-cell antigen levels because, even in healthy individuals, a small percentage of self-reactive T cells evade thymic negative selection. The second term ${\alpha }_B{\displaystyle \frac{B}{B+h_B}}$ refers to clonal growth of autoreactive effector T cells in response to β-cell antigen presentation. The factor is a saturating Hill-type function that increases with β-cell mass $B\left(t\right)$, indicating that larger β-cell populations present more autoantigen and thus activate pathogenic T cells more efficiently. The parameter ${\alpha }_B$ controls the intensity of this antigen-dependent proliferation, while multiplication by $E$ guarantees that expansion is proportional to the current effector pool. Regulatory T cells $R\left(t\right)$ inhibit autoreactive effector T cells via contact-dependent and cytokine-mediated pathways. The bilinear equation ${\phi }_{R}RE$, describes the reduction in effector T-cell activity that is proportional to the amounts of Tregs and effector cells present. The coefficient ${\phi }_R$, indicates the extent to which Tregs reduce pathogenic effector function. The last term accounts for the natural deterioration, death, and turnover of effector T cells at a rate ${\mu }_E$.

The dynamics of regulatory T cells, which serve as a preventative tactic, are described by the third equation.

$${\displaystyle \frac{dR\left(t\right)}{dt}}=S_R+{\rho }_R S-{\mu }_R R.$$

(3)

The rate at which regulatory T cells are supplied to the islet from the thymus is represented by the first term $S_R$. The second term represents the increase in regulatory T cells induced directly by stem cells, where ${\rho }_R$ is the rate at which stem cells stimulate regulatory T-cell expansion. The final term ${\mu }_R$ denotes the rate of regulatory T cell inactivation or natural death.

The fourth equation, $S\left(t\right)$, describes the behavior of the therapeutic stem cells.

$${\displaystyle \frac{dS\left(t\right)}{dt}}={-(K_S+\mathsf{\rho }}_S)S+u(t).$$

(4)

The first term, ${ K}_S$, represents the natural clearance, degradation, or elimination of injected stem cells from the system. A lower $K_S$ value indicates longer stem cell survival after injection. The second parameter, ${\rho }_S$, represents the effective transition or depletion rate associated with the loss of stem cells from the therapeutic stem-cell compartment through differentiation-related processes. Thus, both $K_S$ and ${\rho }_S$ reduce the available stem-cell population after administration. The last term $u\left(t\right)$ denotes the time-dependent stem-cell administration rate, with units of ${\mathrm{m}\mathrm{m}}^{-3}{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$. It functions as a control input that specifies the dosage regimen, allowing the model to simulate various clinical or experimental treatment schedules. These assumptions are consistent with experimental observations suggesting that transplanted stem cells contribute to tissue repair primarily through differentiation and immune modulation rather than long-term accumulation.

The model consists of four interacting populations representing pancreatic β-cells, autoreactive effector T cells, regulatory T cells, and stem cells. The interactions among these populations describe the autoimmune destruction of β-cells, immune regulation, and the therapeutic effects of stem cell administration.

After describing each component separately, Equations (1)–(4) are combined into the following system representing the proposed optimal $T1DS$ model:

$$\begin{array}{l}{\displaystyle \frac{dB}{dt}}={\mathrm{S}}_B-K_{E}EB-{\mu }_B B+{\rho }_B S,\\ {\displaystyle \frac{dE}{dt}}= S_{E }+{\alpha }_B{\displaystyle \frac{B}{B+h_B}}E-{\phi }_R RE-{\mu }_{E}E,\\ {\displaystyle \frac{dR}{dt}}=S_R+{\rho }_R S-{\mu }_R R, \end{array}$$

$${\displaystyle \frac{dS}{dt}}=-(K_S+{\rho }_S)S+u,$$

(5)

Furthermore, the initial conditions are

$$B\left(t_0\right)=B\left(0\right), E\left(t_0\right)=E\left(0\right), R\left(t_0\right)=R\left(0\right), S\left(t_0\right)=S\left(0\right).$$

The state variables and their biological interpretations are summarized in

Table 1. State variables and their biological interpretation.

Variables	Description
$B\left(t\right)$	Pancreatic β-cells
$E\left(t\right)$	Autoreactive effector T cells
$R\left(t\right)$	Regulatory T cells
$S\left(t\right)$	Stem cell population

.

Dimensional consistency: The state variables $B, E, R$, and $S$ are expressed as cell densities $\left({\mathrm{m}\mathrm{m}}^{-3}\right)$, while time is measured in days. Accordingly, all terms in Equations (1)–(4) have units of ${\mathrm{m}\mathrm{m}}^{-3}{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$. The corresponding parameter units are reported consistently in

Table 2. Values and explanation of the parameters for the $T1DS$ model.

Parameter	Description	Value	Units	References
${\mathrm{S}}_B$	Supply of β-cells	20	${\mathrm{m}\mathrm{m}}^{-3}{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	[30,31]
$S_{E }$	Supply of autoreactive T cells	20	${\mathrm{m}\mathrm{m}}^{-3}{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	[30,31]
$S_R$	Supply rate of Regulatory T cells	20	${\mathrm{m}\mathrm{m}}^{-3}{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	[30]
$K_E$	Damage of autoreactive cells on β-cells	$2\times {10}^{-6}$	${\mathrm{m}\mathrm{m}}^3{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	[30,31]
$K_S$	Stem cells clearance	0.75	${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	Assumed
${\rho }_B$	Effective stem-cell-mediated β-cell regeneration	0.30	${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	Assumed
${\rho }_R$	Effective stem-cell-mediated Treg enhancement	0.15	${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	Assumed
${\rho }_S$	Effective stem-cell transition/depletion rate	0.90	${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	Assumed
${\alpha }_B$	Activation rate of autoreactive T cells by $\beta$-cell antigen	0.1	${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	Assumed
$h_B$	Antigen threshold	1	${\mathrm{m}\mathrm{m}}^{-3}$	[32]
${\phi }_R$	Autoreactive cell death due to R	0.0001	${\mathrm{m}\mathrm{m}}^3{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	[30]
${\mu }_B$	Death rate of β-cells	0.02	${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	[30,31]
${\mu }_E$	Death rate of E cells	0.02	${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	[30,31]
${\mu }_R$	Death rate of R cells	0.02	${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$	[30]

Note: For clarity, $S_E=0$ is used only in the auxiliary DFE and $R_0$ analyses and in the corresponding threshold simulations of Section 7.1, whereas $S_E=20$ is used in the positive-equilibrium calculations, sensitivity analysis, and full-model treatment simulations.

.

Parameter Selection and Justification

The parameters adopted directly from previous T1D models are identified by their corresponding references [30,31,32] in Table 2. In contrast, several coefficients introduced to represent the stem-cell therapeutic mechanisms in the present model lack direct quantitative estimates. These parameters were therefore treated as phenomenological coefficients and assigned a nominal baseline set. Their selection was guided by three considerations: the experimentally supported direction of the corresponding biological mechanism, the characteristic time scales implied by the model equations, and the requirement that the resulting trajectories remain biologically interpretable over the one-year simulation horizon. Consequently, these values should not be interpreted as patient-specific clinical measurements.

In particular, $K_S=0.75 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1},$ represents effective loss of administered stem cells from the modeled therapeutic compartment. The transition/depletion coefficient ${\rho }_S=0.90 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$ was selected jointly with $K_S$. Their combined value, $K_S+{\rho }_S=1.65 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$, corresponds to a characteristic residence time of approximately $0.61$ days and an effective compartmental half-life of approximately 0.42 days in the absence of additional administration. This time scale produces transient stem-cell exposure following pulse administration and prevents unrealistic accumulation of the therapeutic compartment.

The coefficient ${\rho }_B=0.30 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$ is interpreted as an effective regenerative coefficient rather than a direct differentiation rate. It aggregates the modeled contribution of stem-cell therapy to β-cell support and regeneration. Similarly, ${\rho }_R=0.15 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$ represents an effective MSC-mediated enhancement of regulatory T-cell activity. Experimental studies support both the regenerative potential of stem-cell-derived β-cells and the immunomodulatory capacity of MSCs, including enhancement of regulatory T-cell responses; however, directly transferable rate constants for the present model are unavailable. Therefore, the selected coefficients provide nominal strengths for these two therapeutic pathways without being interpreted as experimentally measured rates.

Finally, ${\alpha }_B=0.10 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$ was selected as the baseline coefficient for antigen-dependent autoreactive effector T-cell activation. Within the effector-cell equation, this value allows antigen-driven expansion to compete with regulatory suppression and natural turnover, thereby permitting persistent autoimmune activity in the untreated system while preserving responsiveness to therapeutic intervention.

The same nominal parameter set was used in both the pulse and continuous-infusion simulations. Thus, differences between the two treatment protocols arise from the administration function $u\left(t\right)$ rather than from changes in the underlying biological parameter values.

3. Mathematical Analysis of the T1DS Model

(Existence and uniqueness, positivity, and boundedness for solutions)

This section contains the fundamental analytical properties of the T1DS model, including theorems on the existence and uniqueness of solutions, their positivity, and their uniform boundedness. These criteria ensure that the model is mathematically sound and biologically consistent [14,23,24].

Rewrite the T1DS model (5) in the following format:

$${\displaystyle \frac{d\varpi }{dt}}=F\left(t,\varpi \left(t\right)\right),\varpi \left(0\right)={\varpi }_0$$

(6)

where $\varpi \left(t\right)\in C^1[0,T]$ and $\varpi \left(t\right):{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}^4$ is a real-valued function that is defined by

$$\begin{array}{l}\varpi \left(t\right)={\left(B\left(t\right),E\left(t\right),R\left(t\right), S\left(t\right)\right)}^T, \mathrm{with}\\ {\varpi }_0{=(B\left(0\right),E\left(0\right),R\left(0\right), S\left(0\right))}^T, \mathrm{and}\\ F\left(\varpi \left(t\right)\right)={\left(F_1\left(\varpi \left(t\right)\right),F_2\left(\varpi \left(t\right)\right),F_3\left(\varpi \left(t\right)\right), F_4\left(\varpi \left(t\right)\right)\right)}^T.\end{array}$$

Theorem 1.

Assume that all parameters of system (5) are non-negative and the initial condition satisfies ${\varpi }_0=\varpi (0){=(B\left(0\right),E\left(0\right),R\left(0\right), S\left(0\right))}^T\in {\mathbb{R}}_{+}^4$. Then the system (5) has a unique solution on $\left[0,T\right]$.

Proof. 

Let

$${\varpi }_n\left(t\right)=\left(\begin{array}{c}\begin{array}{c}B\left(t\right)\\ E\left(t\right)\\ R\left(t\right)\end{array}\\ S\left(t\right)\end{array}\right),\phi \left(\varpi \right)=\left(\begin{array}{c}\begin{array}{c}\dot{B}\\ \dot{E}\\ \dot{R}\end{array}\\ \dot{S}\end{array}\right).$$

System (5) can be written as

$$\phi \left(\varpi \right)=A\varpi +G\left(\varpi \right)=F\left(\varpi \right),$$

where $A= \left(\begin{array}{c}\begin{array}{c}-{\mu }_B\\ 0\\ 0\end{array}\\ 0\end{array} \begin{array}{c}\begin{array}{c}0\\ -{\mu }_E\\ 0\end{array}\\ 0\end{array} \begin{array}{c}\begin{array}{c}0\\ 0\\ -{\mu }_R\end{array}\\ 0\end{array} \begin{array}{c}\begin{array}{c}{\rho }_B\\ 0\\ {\rho }_R\end{array}\\ {-(K_S+\mathsf{\rho }}_S)\end{array}\right),$ and $G\left(\varpi \right)=\left(\begin{array}{c}\begin{array}{c}{\mathrm{S}}_B-K_{E}EB\\ {\mathrm{S}}_E+{\alpha }_B{\displaystyle \frac{B}{B+h_B}}E-{\phi }_R RE\\ {\mathrm{S}}_R\end{array}\\ u\end{array} \right)$.

Thus

$$F\left(\varpi \right)=A\varpi +G\left(\varpi \right).$$

Now let

$${\varpi }_1={\left(B_1,E_1,R_1,{ S}_1\right)}^T, {\varpi }_2={\left(B_2,E_2,R_2,{ S}_2\right)}^T.$$

Then

$$F\left({\varpi }_1\right)-F\left({\varpi }_2\right)=A \left({\varpi }_1-{\varpi }_2\right)+G\left({\varpi }_1)-G({\varpi }_2\right)$$

Therefore,

$$‖F\left({\varpi }_1\right)-F\left({\varpi }_2\right)‖\le ‖A \left({\varpi }_1-{\varpi }_2\right)‖+‖G \left({\varpi }_1)-G({\varpi }_2\right)‖.$$

Since

$$‖A \left({\varpi }_n-{\varpi }_{n-1}\right)‖\le \left|A\right|‖{\varpi }_1-{\varpi }_2‖,$$

(7)

it remains to estimate $‖G \left({\varpi }_1)-G({\varpi }_2\right)‖.$

$$G \left({\varpi }_1)-G({\varpi }_2\right)=\left(\begin{array}{c}{\alpha }_B\begin{array}{c}-K_E\left(E_1{B}_1-E_2{B}_2\right)\\ \left({\displaystyle \frac{B_1}{B_1+h_B}}{E}_1-{\displaystyle \frac{B_2}{B_2+h_B}}{E}_2\right)-{\phi }_R \left(R_1{E}_1-{R_{2}E}_2\right)\\ 0\end{array}\\ 0\end{array}\right).$$

(8)

Since the solution is considered in the feasible bounded region $\Omega$, there exist positive constants

$$B_{max}, E_{max}, R_{max}, S_{max,}$$

Such that $0\le B_i\le B_{max}$, $0\le E_i\le E_{max},$ $0\le R_i\le R_{max},$ $0\le S_i\le S_{max}$, $i=1,2$.

For the first nonlinear term,

$$E_1{B}_1-E_2{B}_2=B_1\left(E_1-E_2\right)+E_2\left(B_1-B_2\right).$$

Thus

$$\left|E_1{B}_1-E_2{B}_2\right|\le B_{max}\left|{ E}_1-E_2\right|+E_{max}\left|{ B}_1-B_2\right|.$$

Hence

$$\left|-K_E\left(E_1{B}_1-E_2{B}_2\right)\right|\le K_E B_{max}\left|{ E}_1-E_2\right|+K_E E_{max}\left|{ B}_1-B_2\right|.$$

For the saturation term, define

$$q\left(B\right)={\displaystyle \frac{B}{B+h_B}}, q^{\prime }\left(B\right)={\displaystyle \frac{h_B}{{\left(B+h_B\right)}^2}},$$

and $h_B>0,$ the following is obtained

$$\left|q\left(B_1\right)-q\left(B_2\right)\right|\le {\displaystyle \frac{1}{h_B}}\left|{ B}_1-B_2\right|.$$

Now

$$q\left(B_1\right)E_1-q\left(B_2\right)E_2=q\left(B_1\right)\left(E_1-E_2\right)+E_2\left(q\left(B_1\right)-q\left(B_2\right)\right).$$

Since $0\le q\left(B\right)\le 1,$

Then

$$\left|q\left(B_1\right)E_1-q\left(B_2\right)E_2\right|\le \left|E_1-E_2\right|+{\displaystyle \frac{E_{max}}{h_B}}\left|{ B}_1-B_2\right|.$$

Therefore

$${\alpha }_B\left|{\displaystyle \frac{B_1}{B_1+h_B}}{E}_1-{\displaystyle \frac{B_2}{B_2+h_B}}{E}_2\right|\le {\alpha }_B\left|E_1-E_2\right|+{\displaystyle \frac{{\alpha }_B E_{max}}{h_B}}\left|{ B}_1-B_2\right|.$$

(9)

For the regulatory interaction term

$$R_1{E}_1-{R_{2}E}_2=R_1\left(E_1-E_2\right)+E_2\left(R_1-R_2\right).$$

Thus,

$$\left|R_1{E}_1-{R_{2}E}_2\right|\le R_{max}\left|E_1-E_2\right|+E_{max}\left|R_1-R_2\right|.$$

Hence,

$${\phi }_R \left|R_1{E}_1-{R_{2}E}_2\right|\le {{\phi }_R R}_{max}\left|E_1-E_2\right|+{\phi }_R E_{max}\left|R_1-R_2\right|.$$

Combining the above estimates, there exists a positive constant $K$ such that

$$‖G\left({\varpi }_1\right)-G\left({\varpi }_2\right)‖\le K‖{\varpi }_1-{\varpi }_2‖.$$

(10)

For example, $K$ may be chosen large enough to dominate all coefficients appearing in the above inequalities.

$$‖F\left({\varpi }_1\right)-F\left({\varpi }_2\right)‖\le \left(\left|A\right|+K\right)‖{\varpi }_1-{\varpi }_2‖.$$

(11)

From (11), let $L=\left|A\right|+K$.

Then

$$‖F\left({\varpi }_1\right)-F\left({\varpi }_2\right)‖\le L ‖{\varpi }_1-{\varpi }_2‖.$$

Thus, F is Lipschitz continuous on the feasible bounded region $\Omega$. Therefore, by the Picard–Lindelöf theorem, system (5) admits a unique solution corresponding to the initial condition ${\varpi }_0=\varpi \left(0\right)$ [22,33,34]. □

Theorem 2.

Assume that the control function $u\left(t\right)$ is measurable, nonnegative, and bounded on $\left[\left.0,\infty \right),\right.$ satisfying $0\le u\left(t\right)\le u_{max}, t\ge 0.$ Let $\left(B\left(t\right),E\left(t\right),R\left(t\right), S\left(t\right)\right)$ be a nonnegative solution of the system (5). Define ${\delta }_S=K_S{+\rho }_S$, $S_{max}=max\left\{S\left(0\right),{\displaystyle \frac{u_{max}}{{\delta }_S}}\right\}$, $R_{max}=max\left\{R\left(0\right),{\displaystyle \frac{S_R+{\rho }_R S_{max}}{{\mu }_R}}\right\},$ and let $c={\displaystyle \frac{K_E h_B}{{\alpha }_B}}>0.$ Furthermore, define $\mu =min\left\{{\mu }_B, {\mu }_E\right\},$

$$C_W={\mathrm{S}}_B+{cS}_{E }+{\rho }_B S_{max}, \mathrm{and} W_{max}=max\left\{B\left(0\right)+cE(0),{\displaystyle \frac{C_W}{\mu }}\right\}.$$

Then, for every $t\ge 0,$ $0\le R\left(t\right)\le R_{max},$ $0\le S\left(t\right)\le S_{max},$ and $0\le B\left(t\right)+cE\left(t\right)\le W_{max.}$ Consequently $0\le B\left(t\right)\le W_{max},$ and $0\le E\left(t\right)\le {\displaystyle \frac{W_{max}}{c}}$.

Hence, all state variables of system (5) remain uniformly bounded for $t\ge 0.$

In particular, every nonnegative solution of system (5) remains bounded within the region

$$\mathsf{\Omega }=\left\{\left(B,E,R,S\right)\in {\mathbb{R}}_{+}^4:S\le S_{max},R\le R_{max},B+cE\le W_{max}\right\}.$$

Proof. 

From the fourth equation of system (5).

$${\displaystyle \frac{dS}{dt}}=-\left(K_S{+\rho }_S\right)S+u(t).$$

Since $0\le u\left(t\right)\le u_{max},$ it follows that

$${\displaystyle \frac{dS}{dt}}\le u_{max}-\left(K_S{+\rho }_S\right)S.$$

Using ${\delta }_S=K_S{+\rho }_S>0,$ it follows that

$${\displaystyle \frac{dS}{dt}}\le u_{max}-{\delta }_{S}S.$$

Consider the scalar comparison equation

$${\displaystyle \frac{dy}{dt}}=u_{max}-{\delta }_{S}y, y\left(0\right)=S\left(0\right).$$

Its solution is

$$y\left(t\right)={\displaystyle \frac{u_{max}}{{\delta }_S}}+\left(S\left(0\right)-{\displaystyle \frac{u_{max}}{{\delta }_S}}\right)e^{-{\delta }_S t}.$$

Therefore, by the comparison principle,

$$S\left(t\right)\le {\displaystyle \frac{u_{max}}{{\delta }_S}}+\left(S\left(0\right)-{\displaystyle \frac{u_{max}}{{\delta }_S}}\right)e^{-{\delta }_S t}.$$

Hence,

$$S\left(t\right)\le \mathrm{m}\mathrm{a}\mathrm{x}\left\{S\left(0\right),{\displaystyle \frac{u_{max}}{{\delta }_S}}\right\}=S_{max}, t\ge 0.$$

Thus, $S\left(t\right)$ is uniformly bounded on $\left[\left.0,\infty \right).\right.$

Next, from the third equation of system (5), it follows that

$${\displaystyle \frac{dR}{dt}}=S_R+{\rho }_R S-{\mu }_R R.$$

Since $S\left(t\right)\le S_{max},$ it follows that

$${\displaystyle \frac{dR}{dt}}\le S_R+{\rho }_R S_{max}-{\mu }_R R.$$

Consider the scalar comparison equation

$${\displaystyle \frac{dz}{dt}}=S_R+{\rho }_R S_{max}-{\mu }_R z.$$

Its solution is

$$z\left(t\right)={\displaystyle \frac{S_R+{\rho }_R S_{max}}{{\mu }_R}}+\left(R\left(0\right)-{\displaystyle \frac{S_R+{\rho }_R S_{max}}{{\mu }_R}}\right)e^{-{\mu }_R t}.$$

Therefore, by the comparison principle,

$$R\left(t\right)\le {\displaystyle \frac{S_R+{\rho }_R S_{max}}{{\mu }_R}}+\left(R\left(0\right)-{\displaystyle \frac{S_R+{\rho }_R S_{max}}{{\mu }_R}}\right)e^{-{\mu }_R t}.$$

$$R\left(t\right)\le \mathrm{m}\mathrm{a}\mathrm{x}\left\{R\left(0\right),{\displaystyle \frac{S_R+{\rho }_R S_{max}}{{\mu }_R}}\right\}=R_{max}, t\ge 0.$$

Thus, $R\left(t\right)$ is uniformly bounded on $\left[\left.0,\infty \right).\right.$

To establish the boundedness of $B\left(t\right)$ and $E\left(t\right)$, define the weighted auxiliary function

$$W\left(t\right)=B\left(t\right)+cE\left(t\right), \mathrm{where} c={\displaystyle \frac{K_E h_B }{{\alpha }_B}}>0.$$

Differentiating $W\left(t\right)$ along the solutions of system (5), it follows that

$${\displaystyle \frac{dW}{dt}}={\displaystyle \frac{dB}{dt}}+c {\displaystyle \frac{dE}{dt}}.$$

Using the first and second equations of system (5), it follows that

$${\displaystyle \frac{dW}{dt}}={\mathrm{S}}_B-K_{E}EB-{\mu }_B B+{\rho }_B S+c\left[ S_{E }+{\alpha }_B{\displaystyle \frac{B}{B+h_B}}E-{\phi }_R RE-{\mu }_{E}E\right].$$

The two interaction terms involving $B$ and $E$ can be combined as follows:

$$\begin{array}{c}-K_{E}EB+c{\alpha }_B{\displaystyle \frac{B}{B+h_B}}E=-K_{E}EB+K_E{h}_B{\displaystyle \frac{B}{B+h_B}}E\\ {=-K}_{E}BE\left(1-{\displaystyle \frac{h_B}{B+h_B}}\right)=-{\displaystyle \frac{K_E{B}^{2}E}{B+h_B}}.\end{array}$$

Since $B\left(t\right)\ge 0$ and $E\left(t\right)\ge 0,$ it follows that

$$-{\displaystyle \frac{K_E{B}^{2}E}{B+h_B}}\le 0.$$

Moreover, since $R\left(0\right)\ge 0$ and $E\left(t\right)\ge 0,$ it follows that

$$-c {\phi }_{R}R E\le 0.$$

Consequently,

$${\displaystyle \frac{dW}{dt}}\le {\mathrm{S}}_B+c{S}_{E }+{\rho }_B S-{\mu }_B B-c {\mu }_{E}E.$$

Using the previously established bound $S\left(t\right)\le S_{max},$

$${\displaystyle \frac{dW}{dt}}\le {\mathrm{S}}_B+c{S}_{E }+{\rho }_B S_{max}-{\mu }_B B-c {\mu }_{E}E.$$

Define $C_W={\mathrm{S}}_B+c{S}_{E }+{\rho }_B S_{max},$ and $\mu =min\left\{{\mu }_B,{\mu }_E\right\}>0.$

Since $B\left(t\right)\ge 0$ and $E\left(t\right)\ge 0,$ it follows that

$$\begin{array}{c}{\mu }_B B+c {\mu }_{E}E\ge \mu B+\mu cE\\ =\mu \left(B+cE\right)=\mu W.\end{array}$$

Applying the comparison principle to

$${\displaystyle \frac{dW}{dt}}\le C_W-\mu W.$$

Therefore,

$$W\left(t\right)\le {\displaystyle \frac{C_W}{\mu }}+\left(W\left(0\right)-{\displaystyle \frac{C_W}{\mu }}\right)e^{-mt}, t\ge 0.$$

Thus,

$$W\left(t\right)\le max\left\{W\left(0\right),{\displaystyle \frac{C_W}{\mu }}\right\}.$$

Since

$$W\left(0\right)=B\left(0\right)+cE\left(0\right),$$

The definition

$$W_{max}=max\left\{B\left(0\right)+cE\left(0\right),{\displaystyle \frac{C_W}{\mu }}\right\},$$

implies that

$$0\le W\left(t\right)\le W_{max}, t\ge 0.$$

Because

$$W\left(t\right)=B\left(t\right)+cE\left(t\right),$$

and $B\left(t\right), E\left(t\right)\ge 0,$ it follows that

$$B\left(t\right)\le W\left(t\right)\le W_{max},$$

and

$$c E\left(t\right)\le W\left(t\right)\le W_{max}$$

Consequently,

$$0\le B\left(t\right)\le W_{max},$$

and

$$0\le E\left(t\right)\le {\displaystyle \frac{W_{max}}{c}}, t\ge 0.$$

Hence, $B\left(t\right)$ and $E\left(t\right)$ are uniformly bounded on $[0,\infty ).$

It remains to verify that the upper bounds defining $\mathsf{\Omega }$ cannot be crossed outward. On the boundary $S=S_{max}$, it follows that

$${\displaystyle \frac{dS}{dt}}\le u_{max}-{\delta }_S{S}_{max}\le 0.$$

On the boundary $R=R_{max}$, using $S\left(t\right)\le S_{max}$, it follows that

$${\displaystyle \frac{dR}{dt}}\le S_R+{\rho }_R S_{max}-{\mu }_R R_{max}\le 0.$$

Similarly, on the boundary $B+cE=W_{max}$, it follows that

$${\displaystyle \frac{dW}{dt}}\le C_W-\mu W_{max} \le 0.$$

Consequently, the trajectories cannot cross the boundary of $\mathsf{\Omega }$ outward, and $\mathsf{\Omega }$ is positively invariant. It follows that every nonnegative solution of system (5) remains uniformly bounded for all $t\ge 0$. Since the vector field is locally Lipschitz and the solution remains bounded on any finite time interval, the standard extension theorem rules out finite-time blow-up; hence the solution exists and is uniformly bounded for all $t\in [0,\infty )$. □

Theorem 3.

Let the initial data satisfy $B\left(0\right)\ge 0, E\left(0\right)\ge 0, R\left(0\right)\ge 0,$ and $S\left(0\right)\ge 0.$

Then all solutions of system (5) remain non-negative for all $t\ge 0$.

Proof. 

To prove positivity, it is sufficient to show that the vector field of system (5) is directed inward or tangent to the boundary of the non-negative orthant.

When $B=0,$ the first equation gives

$${\left.{\displaystyle \frac{dB}{dt}}\right|}_{B=0}={\mathrm{S}}_B+{\rho }_B S.$$

Since ${\mathrm{S}}_B\ge 0,{\rho }_B\ge 0,$ and $S\ge 0,$ the following is obtained:

$${\left.{\displaystyle \frac{dB}{dt}}\right|}_{B=0}\ge 0.$$

Thus, $B\left(t\right)\ge 0,$ for all $t\ge 0.$

The second equation gives

$${\left.{\displaystyle \frac{dE}{dt}}\right|}_{E=0}={\mathrm{S}}_E.$$

Since ${\mathrm{S}}_E\ge 0,$ the following is obtained:

$${\left.{\displaystyle \frac{dE}{dt}}\right|}_{E=0}\ge 0.$$

Thus, $E\left(t\right)$ cannot become negative.

When $R\left(t\right)=0,$ the third equation gives

$${\left.{\displaystyle \frac{dR}{dt}}\right|}_{R=0}=S_R+{\rho }_R S.$$

Since ${\mathrm{S}}_R\ge 0, {\rho }_R\ge 0,$ and $S\ge 0,$ the following is obtained:

$${\left.{\displaystyle \frac{dR}{dt}}\right|}_{R=0}\ge 0.$$

Thus, $R\left(t\right)$ cannot become negative. When $S\left(t\right)=0,$ the fourth equation gives

$${\left.{\displaystyle \frac{dS}{dt}}\right|}_{S=0}=u(t).$$

Since $u\left(t\right)\ge 0,$ the following is obtained:

$${\left.{\displaystyle \frac{dS}{dt}}\right|}_{S=0}\ge 0.$$

Thus, $S\left(t\right)$ cannot become negative.

Therefore, on each boundary face of the non-negative orthant, the vector field is directed inward or tangent to the boundary of the non-negative orthant. Hence, ${\mathbb{R}}_{+}^4$ is positively invariant for system (5). Consequently

$$B\left(t\right),E\left(t\right),R\left(t\right), S\left(t\right)\ge 0, \forall t\ge 0.$$

Thus, the feasible region for the proposed model (5) is defined as

$\mathsf{\Omega }=\left\{\left(B\left(t\right),E\left(t\right),R\left(t\right), S\left(t\right)\right)\in {\mathbb{R}}_{+}^4\right\}$ is positively invariant. □

Corollary 1.

By Theorems 2 and 3, every solution of system (5) with nonnegative initial conditions remains nonnegative and uniformly bounded for all $t\ge 0$. Consequently, the region $\mathsf{\Omega }=\left\{\left(B,E,R,S\right)\in {\mathbb{R}}_{+}^4:S\le S_{max},R\le R_{max},B+cE\le W_{max}\right\}$ is positively invariant under the flow of system (5).

4. Stability Analysis and Equilibrium Points

The equilibrium analysis is carried out under two related but mathematically distinct regimes of system (5). First, the disease-free-equilibrium DFE analysis is performed for the auxiliary autonomous subsystem obtained by setting $S_{E }=0$ and assuming a constant treatment input $u\left(t\right)\equiv u^{*}$. This restriction is necessary because, if $S_{E }>0$, then

$${\left.{\displaystyle \frac{dE}{dt}}\right|}_{E=0}={\mathrm{S}}_E>0.$$

and hence $E=0$ cannot be an equilibrium of the full system. Accordingly, the threshold quantity $R_0$ derived from the DFE is used only to characterize the local behavior of this auxiliary $S_{E }=0$ subsystem. Second, the full system with $S_{E }>0$, which is the regime used in the endemic-equilibrium calculations, sensitivity analysis, and numerical simulations, is analyzed separately through its unique positive equilibrium. Thus, $R_0$ is not used as an existence or bifurcation criterion for the positive equilibrium of the full system.

4.1. Disease-Free Equilibrium Point (DFE)

The full model includes a basal recruitment term $S_{E }$ for autoreactive effector T cells. Consequently, $E=0$ cannot be an equilibrium when $S_{E }>0$. Therefore, the DFE and its local stability are examined only for the auxiliary autonomous subsystem obtained by setting $S_{E }=0$ and $u\left(t\right)=u^{\mathrm{*}},$ where $u^{\mathrm{*}}$ is a nonnegative constant. The endemic equilibrium and numerical simulations are considered separately for the full model with $S_{E }>0$.

In the present model, the variable $E\left(t\right)$ denotes pathogenic autoreactive effector T cells that are actively engaged in β-cell destruction. Therefore, the disease-free equilibrium is defined by the absence of these pathogenic effector cells, i.e., $E_0^{*}=0$.

To define the DFE, set all derivatives of the T1DS model (5) to zero, with $S_{E }=0$, $u\left(t\right)=u^{\mathrm{*}}$ and $E_0^{*}=0$ as:

$$\begin{array}{c}{\mathrm{S}}_B-{\mu }_B B_0^{*}+{\rho }_B S_0^{*}=0 ⟹{ B}_0^{*}={\displaystyle \frac{{\mathrm{S}}_B+{\rho }_B S_0^{*}}{{\mu }_B}},\\ S_R+{\rho }_R S_0^{*}-{\mu }_R R_0^{*}=0 ⟹ R_0^{*}= {\displaystyle \frac{{\mathrm{S}}_R+{\rho }_R{S}_0^{*}}{{\mu }_R}}, \\ {-(K_S+\mathsf{\rho }}_S) S_0^{*}+ u^{*}=0, ⟹ S_0^{*}={\displaystyle \frac{ u^{*}}{K_S+{\rho }_S}}.\end{array}$$

The DFE of the T1DS model (5) is given by

$${\chi }_0=\left(B_0^{*},E_0^{*},R_0^{*},S_0^{*}\right)=\left({\displaystyle \frac{{\mathrm{S}}_B+{\rho }_B S_0^{*}}{{\mu }_B}}, 0, {\displaystyle \frac{{\mathrm{S}}_R+{\rho }_R{S}_0^{*}}{{\mu }_R}}, {\displaystyle \frac{u^{*}}{K_S+{\rho }_S}}\right).$$

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Remark 1.

The condition $S_{E }=0$ is imposed only for the auxiliary DFE analysis, including the derivation of $R_0$ and the local stability analysis. In contrast, the full model with $S_{E }>0$ is used for the endemic equilibrium analysis and full-model treatment simulations.

4.2. Endemic Equilibrium Point (EE)

The endemic equilibrium represents a steady state associated with the sustained presence of autoimmune activity within the host. For the full model with $S_{E }>0$ and constant treatment input $u\left(t\right)=u^{\mathrm{*}},$

The EE is denoted as ${\chi }_1=\left(B_1^{*},E_1^{*},R_1^{*},S_1^{*}\right)$ where

$$\begin{gathered} B_1^{*}={\displaystyle \frac{C}{K_E{E}^{* }+{\mu }_B}}, E_1^{*}={\displaystyle \frac{-{ϵ}_1+\sqrt{{ϵ}_1^2-4{ϵ}_2{ϵ}_0}}{2{ϵ}_2}}, R_1^{*}={\displaystyle \frac{{\mathrm{S}}_R+{\rho }_R S^{*}}{{\mu }_R}}, S_1^{*}={\displaystyle \frac{ u^{*}}{K_{S }+{\rho }_S}}, \\ C=S_B+{\rho }_B{S}_1^{*}, A={\phi }_R R_1^{*}+{\mu }_E, { ϵ}_2=A h_B{K}_E, \\ {ϵ}_1=A\left(C+h_B{\mu }_B\right)-\left({\alpha }_{B}C+S_E{h}_B{K}_E\right), {ϵ}_0=-S_E\left(C+h_B{\mu }_B\right). \end{gathered}$$

For $S_{E }>0,$ and positive model parameters, ${ϵ}_2>0$ and ${ϵ}_0<0.$ Hence, ${\displaystyle \frac{{ϵ}_0}{{ϵ}_2}}<0,$ which implies that the two roots of the quadratic equation for $E_1^{*}$ have opposite signs. Therefore, exactly one of the two roots is positive. Moreover, ${ϵ}_1^2-4{ϵ}_2{ϵ}_0>{ϵ}_1^2\ge 0.$

Thus, the roots are real, and the biologically feasible root is

$$E_1^{*}={\displaystyle \frac{-{ϵ}_1+\sqrt{{ϵ}_1^2-4{ϵ}_2{ϵ}_0}}{2{ϵ}_2}}>0.$$

Consequently, $B_1^{*}>0, R_1^{*}>0, S_1^{*}\ge 0.$ Therefore, for $S_{E }>0,$ the full system admits a unique biologically feasible endemic equilibrium with $E_1^{*}>0.$

4.3. Threshold Quantity $R_0$ for the Auxiliary DFE Subsystem

The threshold quantity $R_0$ introduced in this section is defined exclusively for the auxiliary autonomous subsystem with $S_{E }=0$ and constant stem-cell administration $u\left(t\right)\equiv u^{*}.$ It is not a threshold quantity for the full model with $S_{E }>0$ and is not used to characterize the numerical treatment simulations performed with $S_{E }=20.$ The local stability of the auxiliary disease-free subsystem was assessed by examining the model reproduction number $R_0$. In the context of T1D, the basic reproduction number $R_0$ is defined as a threshold quantity associated with the β-cell-driven expansion of autoreactive effector T cells $E\left(t\right)$ near the disease-free equilibrium. Thus, $R_0$ quantifies the balance between antigen-driven proliferation of pathogenic effector T cells and their removal through natural turnover and regulatory T-cell-mediated suppression.

If $R_0<1,$ small perturbations in the autoreactive effector T-cell population decay near the DFE, whereas if $R_0>1,$ the DFE becomes locally unstable. This interpretation applies exclusively to the auxiliary autonomous subsystem with $S_{E }=0.$ For the full model with $S_{E }>0,$ an exact disease-free equilibrium does not exist; therefore, $R_0$ is not interpreted as an existence or stability threshold for the endemic equilibrium.

The basic reproduction number $R_0$ is derived for the auxiliary disease-free subsystem using the next-generation matrix approach:

$$F={\left[{\displaystyle \frac{\partial \mathcal{F}}{\partial E}}\right]}_{DFE}, V={\left[{\displaystyle \frac{\partial \nu }{\partial E}}\right]}_{DFE}.$$

The Jacobian of $F$ and $V$ is expressed as

$$\begin{gathered} F=\left[{\displaystyle \frac{{\alpha }_B B_0^{*}}{B_0^{*}+h_B}}\right], V=\left[{\phi }_R{R}_0^{*}+{\mu }_E\right], \\ F{V}^{-1}=\left[{\displaystyle \frac{{\alpha }_B{B}_0^{*}}{\left(B_0^{*}+h_B\right)\left({\mu }_E+{\phi }_R{R}_0^{*}\right)}}\right] \end{gathered}$$

Therefore, the reproduction number is evaluated to be

$$R_0=\rho \left(F{V}^{-1}\right)={\displaystyle \frac{{\alpha }_B{B}_0^{*}}{\left(B_0^{*}+h_B\right)\left({\mu }_E+{\phi }_R{R}_0^{*}\right)}}={\displaystyle \frac{{\alpha }_B({\mathrm{S}}_B+{\rho }_B\frac{u^{*}}{K_S+{\rho }_S})}{\left({\mathrm{S}}_B+{\rho }_B\frac{u^{*}}{K_S+{\rho }_S}+{\mu }_B{h}_B\right)\left({\mu }_E+\frac{{\phi }_R}{{\mu }_R}(S_R+{\rho }_{R }\frac{u^{*}}{K_S+{\rho }_S})\right)}} .$$

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4.4. Local Stability of Disease-Free Equilibrium

Theorem 4.

Assume that $S_E=0$ and the stem cell administration input is constant, $u\left(t\right)\equiv u^{\mathrm{*}}\ge 0.$ Let ${\chi }_0=\left({\displaystyle \frac{{\mathrm{S}}_B + {\rho }_B S_0^{*}}{{\mu }_B}},0,{\displaystyle \frac{{\mathrm{S}}_R +{ \rho }_R{S}_0^{*}}{{\mu }_R}},{\displaystyle \frac{ u^{*}}{K_S +{ \rho }_S}}\right)$ be the DFE point of the auxiliary $T1DS$ subsystem. Then ${\chi }_0$ is locally asymptotically stable if $R_0 < 1$, and unstable if $R_0 > 1$.

Proof. 

The Jacobian matrix of the $T1DS$ model (5) is computed at the DFE ${\chi }_0$:

$$J\left({\chi }_0\right)=\left(\begin{array}{cccc}-{\mu }_B& -K_E{B}_0^{*}& 0& {\rho }_B\\ 0& {\alpha }_B{\displaystyle \frac{B_0^{*}}{B_0^{*}+h_B}}-{\phi }_R{R}_0^{*}-{\mu }_E& 0& 0\\ 0& 0& -{\mu }_R& {\rho }_R\\ 0& 0& 0& {-(K_S+\mathsf{\rho }}_S)\end{array}\right)$$

The eigenvalues of the matrix $J\left({\chi }_0\right)$ are computed as

$${\lambda }_1=-{\mu }_B, { \lambda }_2=-{\mu }_R, {\lambda }_3={-(K_S+\mathsf{\rho }}_S), {\lambda }_4={\alpha }_B{\displaystyle \frac{B_0^{*}}{B_0^{*}+h_B}}-{\phi }_R{R}_0^{*}-{\mu }_E.$$

Since ${\mu }_B>0, {\mu }_R>0, {K_S+\mathsf{\rho }}_S>0.$ The first three eigenvalues are strictly negative. Using the definition of $R_0,$ the fourth eigenvalue can be written as

$$\begin{array}{l}{\lambda }_4={\alpha }_B{\displaystyle \frac{B_0^{*}}{B_0^{*}+h_B}}-\left({\phi }_R{R}_0^{*}+{\mu }_E\right),\\ =\left({\phi }_R{R}_0^{*}+{\mu }_E\right)\left(R_0-1\right).\end{array}$$

Therefore, if $R_0<1,$ then ${\lambda }_4<0.$

Hence, all eigenvalues of $J\left({\chi }_0\right)$ have strictly negative real parts, and the DFE ${\chi }_0$ is locally asymptotically stable.

Conversely, if $R_0>1,$ then ${\lambda }_4>0.$

Therefore, $J\left({\chi }_0\right)$ has a positive eigenvalue, and the DFE ${\chi }_0$ is unstable.

When $R_0=1,$ the fourth eigenvalue is zero, and the linearization method is inconclusive. □

The threshold quantity $R_0$ compares the antigen-driven proliferation of autoreactive effector T cells with their natural removal and regulatory T-cell-mediated suppression. When $R_0<1$, regulatory suppression and effector-cell turnover dominate antigen-driven activation, and small perturbations of the DFE decay over time. When $R_0>1$, effector-cell activation dominates, causing the DFE to become unstable. In the present formulation, the immunoregulatory effect of stem cell therapy acts through ${\rho }_R$, which increases the disease-free regulatory T-cell level $R_0^{\mathrm{*}}$ and consequently reduces $R_0$. In contrast, the regenerative effect represented by ${\rho }_B$ increases the disease-free β-cell level $B_0^{\mathrm{*}}$, which may increase the antigen-dependent activation term ${\displaystyle \frac{B_0^{\mathrm{*}}}{B_0^{\mathrm{*}}+h_B}}$. Therefore, the model represents potentially competing regenerative and immunoregulatory effects of stem cell therapy. This interpretation and the associated local stability result apply only to the auxiliary autonomous DFE subsystem with $S_E=0$ and constant treatment input $u\left(t\right)\equiv u^{\mathrm{*}}$. They should not be extended to the full model with $S_E>0$, the time-varying controlled system, or the numerical treatment simulations performed with $S_E=20$.

4.5. Local Stability of the Positive (Endemic) Equilibrium Point

Theorem 5.

Assume that $S_{E }>0, u\left(t\right)\equiv u^{*}\ge 0$, and all model parameters are positive. Let ${\chi }_1=\left(B_1^{*},E_1^{*},R_1^{*},S_1^{*}\right)$ be the unique biologically feasible positive (endemic) equilibrium of system (5), with $E_1^{*}>0$. Then the endemic equilibrium ${\chi }_1$ is locally asymptotically stable.

Proof. 

The Jacobian matrix for $T1DS$ system (5) at ${\chi }_1$ is derived as follows:

$$J\left({\chi }_1\right)=\left(\begin{array}{cccc}-{\mathsf{\mu }}_B-{\mathrm{K}}_E{E}_1^{*}& -{\mathrm{K}}_E{B}_1^{*}& 0& {\mathsf{\rho }}_B\\ {\displaystyle \frac{h_B{\mathsf{\alpha }}_B{E}_1^{*}}{{(h_B+{\mathrm{B}}^{*})}^2}}& -{\mathsf{\mu }}_E+{\displaystyle \frac{{\mathsf{\alpha }}_B{B}_1^{*}}{h_B+B_1^{*}}}-{\mathsf{\varphi }}_R{R}_1^{*}& -{\mathsf{\varphi }}_R{E}_1^{*}& 0\\ 0& 0& -{\mathsf{\mu }}_R& {\mathsf{\rho }}_R\\ 0& 0& 0& -(K_S+{\rho }_S)\end{array}\right)$$

The matrix $J\left({\chi }_1\right)$ has the block upper-triangular form

$$J\left({\chi }_1\right)=\left(\begin{array}{cc}M& P\\ 0& N\end{array}\right),$$

where

$$\begin{gathered} M=\left(\begin{array}{cc}-{\mathsf{\mu }}_B-{\mathrm{K}}_E{E}_1^{*}& -{\mathrm{K}}_E{B}_1^{*}\\ {\displaystyle \frac{h_B{\mathsf{\alpha }}_B{E}_1^{*}}{{(h_B+B_1^{*})}^2}}& -{\mathsf{\mu }}_E+{\displaystyle \frac{{\mathsf{\alpha }}_B{B}_1^{*}}{h_B+B_1^{*}}}-{\mathsf{\varphi }}_R{R}_1^{*}\end{array}\right), \\ P=\left(\begin{array}{cc}0& {\mathsf{\rho }}_B\\ -{\mathsf{\varphi }}_R{E}_1^{*}& 0\end{array}\right), \mathrm{and}\hspace{1em}\hspace{1em}N=\left(\begin{array}{cc}-{\mathsf{\mu }}_R& {\mathsf{\rho }}_R\\ 0& -(K_S+{\rho }_S)\end{array}\right). \end{gathered}$$

Therefore, the eigenvalues of $J\left({\chi }_1\right)$ consist of the two eigenvalues of $M$ together with the two eigenvalues of $N$. Since $N$ is upper triangular, its eigenvalues are

$${\lambda }_1=-{\mathsf{\mu }}_R<0,ߓߓ{\lambda }_2=-\left(K_S+{\rho }_S\right)<0.$$

It remains to examine the two eigenvalues associated with $M$.

At the endemic equilibrium, the second equation of system (5) satisfies

$$S_{E }+\left({\displaystyle \frac{{\alpha }_B B_1^{*}}{B_1^{*}+h_B}}-{\phi }_R R_1^{*}-{\mu }_E\right)E_1^{*}=0.$$

Since $S_{E }>0$ and ${\mathrm{E}}^{*}>0,$ it follows that:

$$\left({\displaystyle \frac{{\alpha }_B B_1^{*}}{B_1^{*}+h_B}}-{\phi }_R R_1^{*}-{\mu }_E\right)=-{\displaystyle \frac{S_{E }}{E_1^{*}}}<0.$$

Hence, the matrix $M$ can equivalently be written as

$$M=\left(\begin{array}{cc}-{\mathsf{\mu }}_B-{\mathrm{K}}_E{E}_1^{*}& -{\mathrm{K}}_E{B}_1^{*}\\ {\displaystyle \frac{h_B{\mathsf{\alpha }}_B{E}_1^{*}}{{(h_B+B_1^{*})}^2}}& -{\displaystyle \frac{S_{E }}{E_1^{*}}}\end{array}\right),$$

The trace of $M$ is therefore

$$Tr(M)=- \left({\mathsf{\mu }}_B+{\mathrm{K}}_E{E}_1^{*}+{\displaystyle \frac{S_{E }}{E_1^{*}}}\right)<0.$$

Next, the determinant of $M$ is

$$\begin{array}{c}\mathit{det}\left(M\right)=\left({\mathrm{a}}_{11}{\mathrm{a}}_{22}-{\mathrm{a}}_{12}{\mathrm{a}}_{21}\right), \hfill \\ \hfill =\left({\mathsf{\mu }}_B+{\mathrm{K}}_E{E}_1^{*}\right)\left({\displaystyle \frac{S_{E }}{E_1^{*}}}\right)+\left({\mathrm{K}}_E{B}_1^{*}\right)\left({\displaystyle \frac{h_B{\mathsf{\alpha }}_B{E}_1^{*}}{{\left(h_B+B_1^{*}\right)}^2}}\right)>0.\end{array}$$

Since all parameters are positive and $B_1^{*}>0, E_1^{*}>0, S_{E }>0.$

Thus, the determinant $\mathit{det}\left(M\right)>0.$

The characteristic polynomial associated with $M$

$${\lambda }^2-Tr\left(M\right)\lambda +\mathit{det}\left(M\right)=0.$$

Since $\mathrm{T}\mathrm{r}\left(\mathrm{M}\right)<0$ and $\mathit{det}\left(M\right)>0,$ the Routh–Hurwitz criterion for a second-order polynomial implies that both eigenvalues of $M$ have strictly negative real parts [35].

Therefore, all eigenvalues of $J\left({\chi }_1\right)$ have negative real parts, and the endemic equilibrium ${\chi }_1$ is locally asymptotically stable. □

This stability result is obtained directly for the full system with $S_E>0$ and is independent of the threshold quantity $R_0$, which is defined only for the auxiliary DFE subsystem with $S_E=0$.

5. Sensitivity Analysis

All sensitivity calculations in this section are performed for the full model with $S_E$ = 20. Thus, the sensitivity results correspond to the persistent-effector regime analyzed through the positive equilibrium in Section 4.2 and Section 4.5 and are not interpreted using the auxiliary DFE threshold $R_0$. A local sensitivity analysis of the model parameters was performed using a $\pm 1\%$ perturbation scheme. For each parameter $p_i$, two simulations were computed using $p_i+0.01p_i$ and $p_i-0.01p_i,$ and the corresponding sensitivity index was evaluated as [28,36,37]:

$$S_i={\displaystyle \frac{F\left(p_i+∆p\right)-F\left(p_i-∆p\right)}{2∆p}}·{\displaystyle \frac{p_i}{F_0}},$$

where $∆p_i=0.01p_i, F=B\left(t_{end}\right)$ is the β-cell population at the final simulation time $\left(t_{end}=365 \mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}\right),$ and $F_0$ is the baseline output.

Two control protocols were considered:

• Pulse protocol (high dose): three discrete injections at days 7, 14, and 21.
• Infusion protocol (high dose): continuous infusion over the entire simulation horizon.

The sensitivity indices obtained for both protocols are summarized in

Table 3. Sensitivity indices for the model parameters under the pulse and infusion protocols.

Parameters	Sensitivity Index
	Pulse Protocol	Infusion Protocol
${\mathrm{S}}_B$	$0.99904$	$0.84582$
${\mathrm{S}}_E$	$-0.09342$	$-0.06640$
${\mathrm{S}}_R$	$0.54467$	$0.27967$
${\alpha }_B$	$-0.56834$	$-0.29534$
$K_E$	$-0.09540$	$-0.06794$
${\phi }_R$	$0.55431$	$0.30645$
$h_B$	$0.00064$	$0.00029$
$K_S$	$-0.00354$	$-0.08142$
${\rho }_S$	$-0.00425$	$-0.09771$
${\rho }_B$	$0.00017$	$0.15379$
${\rho }_R$	$0.00758$	$0.02536$
${\mu }_B$	$-0.89540$	$-0.91774$
${\mu }_E$	$0.11344$	$0.05899$
${\mu }_R$	$-0.52084$	$-0.28342$

Note: Positive sensitivity indices indicate that an increase in the corresponding parameter increases the final β-cell population $B\left(t_{end}\right)$, whereas negative indices indicate a decrease in the final β-cell population.

and illustrated in Figure 1. The analysis indicates that a small subset of parameters dominates the system dynamics. For the pulse protocol, the most influential parameters are the β-cell source rate and β-cell death rate, activation coefficient, regulatory suppression rate, and the regulatory T-cell source rate, reflecting their large sensitivity magnitudes. Specifically, ${\mathrm{S}}_B$ reveals a strong positive effect, i.e., increases in β-cell production greatly increase the ultimate β-cell population. On the other hand, ${\mu }_B$ has a strong negative effect, identifying β-cell loss as the most important harmful driver. Under the infusion protocol, a similar trend is observed, where the major parameters are still ${\mathrm{S}}_B, {\mu }_B,{\alpha }_B, {\phi }_R$, and ${\mathrm{S}}_R$. However, the relative contributions differ slightly due to the treatment’s persistent nature. The infusion approach tends to smooth out short-term immunological changes, resulting in a more gradual but sustained effect across the influential parameters.

The sensitivity analysis further shows that the two treatment options are driven by the same core biological mechanisms, i.e., β-cell generation and destruction, and immune modulation. This suggests that the drug distribution technique influences the system’s time course but does not substantially alter the parameters’ relative importance.

The relative sensitivity indices are also shown in Figure 1. Parameters with positive sensitivity indices improve β-cell preservation, whereas negative sensitivity indices increase β-cell loss. The strong dominance of $S_B$ and ${\mu }_B$ suggests the importance of tuning the balance between β-cell renewal and destruction when designing effective therapeutic strategies.

Taken together, these results imply that treatment strategies should focus not only on reducing β-cell death and promoting regeneration but also on modulating immune-mediated activation pathways. The present study employs a local sensitivity analysis to identify the most influential parameters around the baseline parameter set. Although this approach provides useful information regarding parameter importance, it does not fully capture nonlinear interactions over the entire feasible parameter space. Future work may consider global sensitivity techniques, such as Latin Hypercube Sampling and Partial Rank Correlation Coefficients, to further investigate parameter uncertainty and nonlinear effects.

6. Optimal Control Problem Formulation

To investigate a time-dependent stem cell administration strategy, the control function $u\left(t\right)$ is introduced into the stem cell equation of system (5). The control represents the rate of therapeutic stem cell administration over the fixed treatment interval $\left[0,T\right]$. The admissible control set is defined by

$$U=\left\{u\in L^{\infty }\left(0,T\right):0\le u\left(t\right)\le u_{max} \mathrm{for} \mathrm{almost} \mathrm{every} t\in \left[0,T\right]\right\},$$

where $u_{max}>0$ denotes the prescribed maximum admissible stem cell administration rate. The objective is to determine an admissible control that suppresses autoreactive effector T cells, preserves pancreatic β-cells, enhances regulatory T-cell activity, and avoids unnecessarily large treatment intensities.

6.1. Well-Posedness of the Controlled System and Existence of an Optimal Control

For every $u\in U$, the right-hand side of the controlled state system (5) is measurable with respect to time and locally Lipschitz continuous with respect to the state variables on bounded subsets of the nonnegative region. Moreover, the Lipschitz estimate used in Theorem 1 remains valid for every admissible control because $u\left(t\right)$ is uniformly bounded by $u_{max}$.

Consequently, the argument established in Theorem 1, together with the positivity and boundedness results obtained in Section 3, guarantees that the controlled system has a unique nonnegative absolutely continuous solution

$$(B, E, R, S) \in AC\left(\right[0,T], {\mathbb{R}}_{+}^4)$$

corresponding to every $u\in U$. Therefore, each admissible control determines a unique state trajectory, and the objective functional can be evaluated without ambiguity.

Theorem 6.

Let $T>0$ be fixed, let all model parameters and weighting constants be positive, and assume that the initial conditions are nonnegative. Define the admissible control set by

$$U=\left\{u\in L^{\infty }\left(0,T\right):0\le u\left(t\right)\le u_{max} \mathrm{for} \mathrm{almost} \mathrm{every} t\in \left[0,T\right]\right\},$$

where $u_{max}>0.$ Then there exists at least one optimal control $u^{*}$ $\in U$ such that:

$$J\left(u^{*}\right)=\underset{u\in U}{\mathit{min}}J\left(u\right),$$

where $J$ is the objective functional defined in (14).

Proof. 

The admissible control set $U$ is nonempty, closed, and convex. Since the treatment horizon $T$ is finite and every $u\in U$ satisfies

$0\le u\left(t\right)\le u_{max} \mathrm{for} \mathrm{almost} \mathrm{every} t\in \left[0,T\right],$ it follows that

$${‖u‖}_{L^2\left(0,T\right)}\le u_{max}\sqrt{T}.$$

Hence, $U,$ regarded as a subset of $L^2\left(0,T\right),$ is bounded. Moreover, the pointwise constraints

$0\le u\left(t\right)\le u_{max}$ defined a closed and convex subset of $L^2\left(0,T\right).$ Since $L^2\left(0,T\right)$ is a reflexive Banach space, $U$ is weakly sequentially compact.

Let $\left\{u_n\right\}\subset U,$ be a minimizing sequence such that

$$J\left(u_n\right)\to \underset{u\in U}{\mathit{inf}}J\left(u\right).$$

By weak sequential compactness, there exist a subsequence, still denoted by $\left\{u_n\right\},$ and a control $u^{*}\in U$ such that $u_n\to u^{*}$ weakly in $L^2\left(0,T\right).$ For each $u_n,$ let ${\chi }_n=\left(B_n\left(t\right), E_n\left(t\right), R_n\left(t\right), S_n\left(t\right)\right)$ denote the corresponding solution of the controlled state system (5). By the existence and uniqueness result established in Theorem 1, together with the positivity and boundedness results of Theorems 2 and 3, each admissible control determines a unique nonnegative state trajectory on $\left[0,T\right].$ In particular, the bounds established in Theorem 2 depend only on the model parameters, the initial data, and $u_{max}$, and are therefore uniform with respect to $n$. Consequently, there exists a constant $M>0$, independent of $n,$ such that

$$0\le B_n\left(t\right), E_n\left(t\right), R_n\left(t\right), S_n\left(t\right)\le M, t\in \left[0,T\right].$$

Since the state variables and the controls are uniformly bounded, all right-hand sides of system (5) are uniformly bounded on $[0,T].$ Therefore, the derivatives of the state trajectories are uniformly bounded, and the sequence $\left\{{\chi }_n\right\}$ is uniformly bounded and equicontinuous on $\left[0,T\right]$.

By the Arzelà–Ascoli theorem, there exists a further subsequence and a continuous function.

${\chi }^{*}=\left(B^{*}\left(t\right), E^{*}\left(t\right), R^{*}\left(t\right), S^{*}\left(t\right)\right)$, ${\chi }_n\to {\chi }^{*}$ uniformly on $[0,T].$

To verify that ${\chi }^{*}$ corresponds to the control $u^{*}$, write the state system in integral form. The nonlinear terms involving only the state variables converge to their corresponding limits because of the uniform convergence of ${\chi }_n$ and the continuity of the model functions on the bounded feasible region.

For the stem-cell equation, the control appears linearly. Since $u_n\to u^{*}$ weakly in $L^2\left(0,T\right),$ for each fixed $t\in [0,T],$ the characteristic function $1_{[0,t]}$ belongs to $L^2\left(0,T\right)$, and therefore

$${\int }_0^t{u}_n\left(s\right) ds\to {\int }_0^t{u}^{*}\left(s\right) ds.$$

It follows that the limit ${\chi }^{*}$ satisfies the integral form of system (5) with control $u^{*}$. By uniqueness of the state solution, ${\chi }^{*}$ is precisely the state trajectory associated with $u^{*}$. It remains to show that $u^{*}$ minimizes the objective functional. Since the state trajectories converge uniformly,

$${\int }_0^T\left[w_E{E}_n\left(t\right)-w_B{B}_n\left(t\right)-w_R R_n\left(t\right)\right]dt\to {\int }_0^T\left[w_E{E}^{*}\left(t\right)-w_B{B}^{*}\left(t\right)-w_{R }{R}^{*}\left(t\right)\right]dt.$$

Moreover, because the mapping $u\to {\displaystyle \frac{{\upsilon }_w}{2}}{\int }_0^T{u\left(t\right)}^2 dt$ is convex and weakly lower semicontinuous in $L^2\left(0,T\right)$

$${\displaystyle \frac{{\upsilon }_w}{2}}{\int }_0^T{u^{*}\left(t\right)}^2 dt\le \underset{n\to \infty }{\mathit{lim}}\mathit{inf}{\displaystyle \frac{{\upsilon }_w}{2}}{\int }_0^T{u_n\left(t\right)}^2 dt.$$

Therefore, $u^{*}$ is an optimal control,

$$J\left(u^{*}\right)\le \underset{n\to \infty }{\mathit{lim}}\mathit{inf} J\left(u_n\right)=\underset{u\in U}{\mathit{inf}} J\left(u\right).$$

Since $u^{*}\in U,$ it follows that

$$J\left(u^{*} \right)=\underset{u\in U}{\mathit{min}} J\left(u\right).$$

Hence, at least one optimal control exists.

The result follows from the standard existence theory for optimal control problems [38]. □

6.2. Objective Functional

The optimal control problem is formulated by introducing a time-dependent control variable $u\left(t\right)$, which represents the rate of stem cell administration. This control aims to modulate autoimmune dynamics by promoting β-cell regeneration and immune regulation while suppressing harmful autoreactive effector T cells. The control $u\left(t\right)$ is assumed to be measurable and bounded, reflecting the biological and clinical limitations of stem cell delivery.

The primary therapeutic objectives are to reduce the number of autoreactive effector T-cells $E\left(t\right)$, expedite the regeneration of β-cells $B\left(t\right)$, and enhance the function of regulatory T-cells $R\left(t\right),$ all while minimizing the costs associated with stem cell administration. The objective functional is defined as follows:

$$J\left(u\right)={\int }_0^T\left(w_E E\left(t\right)-w_B B\left(t\right)-w_R R(t)+{\displaystyle \frac{1}{2}}{(\upsilon }_w{u}^2)\right)dt,$$

(14)

where the constants $w_E$, $w_B$, $w_R$ and ${\upsilon }_w$ are positive weighting constants that equilibrate the relative significance of immune suppression, β-cell preservation, immunological regulation, and treatment expenditure, respectively.

Thus, the optimal control problem is to find $u^{*}\in U$ such that

$$J\left(u^{*}\right)=\underset{u\in U}{\mathit{min}}J\left(u\right).$$

6.3. Biological Interpretation of the Cost Functional

The objective functional (14) consists of four terms, each representing a distinct biological or therapeutic cost:

• The term ${\int }_0^T{w}_E E\left(t\right) dt$ represents the cost associated with the presence of autoreactive effector T cells, which are the primary drivers of β-cell destruction in T1DS. Minimizing this term corresponds to suppressing the pathogenic immune response.
• The term $-{\int }_0^T{w}_B B\left(t\right) dt$ represents the benefit of preserving and restoring pancreatic β-cell mass. The negative sign indicates that larger β-cell populations are desirable, and maximizing this term promotes β-cell regeneration and survival.
• The term $-{\int }_0^T{w}_R R\left(t\right) dt$ represents the benefit of enhancing regulatory T cell populations, which are crucial for maintaining immune tolerance and suppressing autoreactive responses. The negative sign indicates that higher Treg levels are beneficial for long-term immune regulation.
• The term ${\int }_0^T{\displaystyle \frac{1}{2}}{\upsilon }_w{u}^{2}dt$ represents the cost of stem cell administration, which includes direct financial costs, potential side effects, and the logistical burden of treatment. The quadratic form is chosen to penalize high-dose therapies more severely, reflecting the principle of diminishing returns and increased risks at higher doses.

The positive weighting constants $w_E, w_B, w_R$ and ${\upsilon }_w$ balance the relative importance of these competing objectives.

6.4. Hamiltonian Formulation

To formulate the necessary optimality conditions based on Pontryagin’s Maximum Principle, the Hamiltonian (H) must be created in the following manner:

$$\begin{gathered} H=w_E E\left(t\right)-w_B B\left(t\right)-w_R R(t)+{\displaystyle \frac{1}{2}}{(\upsilon }_w{u}^2)+{\lambda }_1\left({\mathrm{S}}_B-K_{E}EB-{\mu }_B B+{\rho }_B S\right)+{\lambda }_2(S_{E }+{\alpha }_B{\displaystyle \frac{B}{B+h_B}}E- \\ {\phi }_R RE-{\mu }_{E}E)+{\lambda }_3\left(S_R+{\rho }_R S-{\mu }_R R\right)+{\lambda }_4\left({-(K_S+\mathsf{\rho }}_S)S+u\right), \end{gathered}$$

(15)

where ${\lambda }_k; k=1, 2, 3, 4$ are adjoint variables. By differentiating the Hamiltonian (15) with respect to the state variables and employing the subsequent relation:

$${\displaystyle \frac{d{\lambda }_k}{dt}}=-{\displaystyle \frac{\partial H}{\partial x_k}}; x_1=B, x_2=E, x_3=R, x_4=S.$$

The subsequent system of adjoint variables is obtained:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_1}{dt}}=-{\displaystyle \frac{\partial H}{\partial \mathrm{B}}}=& w_B+{\lambda }_1\left({\mu }_B+K_{E}E\right)-{\lambda }_2 \left({\alpha }_{B}E{\displaystyle \frac{h_B}{{\left(B+h_B\right)}^2}}\right),\hfill \\ \hfill {\displaystyle \frac{d{\lambda }_2}{dt}}=-{\displaystyle \frac{\partial H}{\partial E}}=& -w_E+{\lambda }_1\left(K_E B\right)-{\lambda }_2\left({\alpha }_B{\displaystyle \frac{B}{B+h_B}}-{\mathsf{\varphi }}_{R}R-{\mu }_E\right),\hfill \\ & {\displaystyle \frac{d{\lambda }_3}{dt}}=-{\displaystyle \frac{\partial H}{\partial \mathrm{R}}}=w_R+{\lambda }_2\left({\mathsf{\varphi }}_{R}E\right)+{\mu }_R{\lambda }_3,\hfill \end{array}$$

$${\displaystyle \frac{d{\lambda }_4}{dt}}=-{\displaystyle \frac{\partial H}{\partial \mathrm{S}}}={-{\rho }_B{\lambda }_1-{\rho }_R{\lambda }_3+(K_S+\rho }_S){\lambda }_4,$$

(16)

subject to the terminal conditions ${\lambda }_k\left(T\right)=0; k=1,2,3,4$.

The unique state trajectory established in Section 6.1 provides the forward reference trajectory along which the adjoint equations are solved backward in time. This establishes the explicit link between the existence and uniqueness of the controlled state system and the derivation of the adjoint system.

6.5. Characterization of the Optimal Control

According to Pontryagin’s Maximum Principle, the optimal control satisfies

$$H\left\{B^{*}, E^{*},R^{*},S^{*},u^{*},{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\right\}=\underset{0\le u\le u_{\mathit{max}}}{\mathit{min}}H\left\{B^{*}, E^{*},R^{*},S^{*},u,{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\right\} \mathrm{for} \mathrm{almost} \mathrm{every} t\in \left[0,T\right].$$

The stationarity condition is ${\displaystyle \frac{\partial H}{\partial u}}=0:$

$${\displaystyle \frac{\partial H}{\partial u}}={\upsilon }_{w}u+{\lambda }_4=0,$$

which gives

$$u\left(t\right)=-{\displaystyle \frac{{\lambda }_4}{{\upsilon }_w}}.$$

With the bound constraint $0\le u\left(t\right)\le u_{max},$ the optimal control is obtained by projection:

$$u^{*}\left(t\right)=min\left\{u_{max},max\left\{0,-{\displaystyle \frac{{\lambda }_4\left(t\right)}{{\upsilon }_w}}\right\}\right\}, \mathrm{for} \mathrm{almost} \mathrm{every}\in \left[0,T\right].$$

(17)

The state system (5), the adjoint system (16), the initial and terminal conditions, and the control characterization (17) together constitute the optimality system. This system is solved numerically in Section 7 using the forward–backward sweep method.

7. Numerical Simulation

This section presents numerical simulations of the stem cell–modulated T1D model to examine its dynamical behavior and to evaluate the effectiveness of various treatment strategies. A separate numerical illustration of the auxiliary DFE threshold is first presented for $S_{E }=0$ and $u\left(t\right)\equiv 0$. The subsequent disease-progression and treatment simulations are performed for the full model with $S_{E }=20$. Accordingly, the $R_0$-based threshold analysis is interpreted only for the auxiliary $S_{E }=0$ subsystem and is not used to characterize the full-model treatment trajectories. Disease progression in the absence of therapeutic intervention is then analyzed, followed by fixed stem-cell dosing protocols and the optimal control framework developed in the previous section. The results illustrate how a time-dependent stem cell administration strategy can suppress autoreactive effector T cells, enhance regulatory T-cell responses, and restore β-cell mass. Unless otherwise stated, the full-model simulations use the parameter values listed in Table 2, together with the following initial conditions:

$$B \left(0\right)=700, E\left(0\right)=300, R\left(0\right)=80, \mathrm{and} S \left(0\right)=0.$$

All numerical simulations were performed using Wolfram Mathematica, version 11.2.0 (Wolfram Research, Inc., Champaign, IL, USA), with the Stiffness Switching solver, chosen to capture both the rapid immune-cell transitions and the slower β-cell recovery dynamics inherent to the model.

Several parameters in the model are treated as nominal modeling assumptions because direct quantitative estimates for the corresponding processes in human T1D stem-cell therapy are unavailable. The numerical simulations use $K_S =0.75 {\mathrm{day}}^{-1},$ ${\rho }_B=0.30 {\mathrm{day}}^{-1}$, ${\rho }_R=0.15 {\mathrm{day}}^{-1},$ ${\rho }_S=0.90 {\mathrm{day}}^{-1}$, and ${\alpha }_B=0.10 {\mathrm{day}}^{-1}$. These values are listed in Table 2 and justified in Parameter Selection and Justification Section; they should be interpreted as effective model parameters rather than direct clinical measurements.

7.1. Numerical Verification of the Auxiliary DFE Threshold

To provide a numerical illustration of the local stability result established in Theorem 4, separate simulations were performed for the auxiliary autonomous subsystem under the assumptions

$${\mathrm{S}}_E=0, u\left(t\right)\equiv u^{*}=0.$$

These assumptions are used only for the disease-free threshold analysis and are distinct from the full-model simulations with persistent autoreactive effector-cell recruitment ${\mathrm{S}}_E>0$. Theorem 4 in the current manuscript already establishes local asymptotic stability of the auxiliary DFE for $R_0<1$ and instability for $R_0>1$.

For $u^{*}=0$, the disease-free equilibrium is

$${\chi }_0=\left(B_0^{*},E_0^{*},R_0^{*},S_0^{*}\right)=\left(1000,0,1000,0\right),$$

where

$$B_0^{*}={\displaystyle \frac{{\mathrm{S}}_B}{{\mu }_B}}={\displaystyle \frac{20}{0.02}}=1000, { R}_0^{*}={\displaystyle \frac{{\mathrm{S}}_R}{{\mu }_R}}={\displaystyle \frac{20}{0.02}}=1000, \mathrm{and}{E}_0^{*}=S_0^{*}=0.$$

To examine the local behavior near the disease-free equilibrium, a small perturbation was introduced in the autoreactive effector T-cell population, and the initial condition was selected as

$$\left(B\left(0\right),E\left(0\right),R\left(0\right),S\left(0\right)\right)=\left(1000,10,1000,0\right).$$

Here, $S_{E }=0$ represents the absence of continuous recruitment of autoreactive effector T cells and does not require the initial effector-cell population to be zero. The small value $E\left(0\right)=10$ was therefore introduced as a perturbation around the equilibrium value $E_0^{*}=0$. Starting exactly from $E\left(0\right)=0$ would result in $E\left(t\right)\equiv 0$ and would not provide a numerical illustration of the local stability or instability of the DFE.

For the auxiliary subsystem, the threshold quantity is given by

$$R_0={\displaystyle \frac{{\alpha }_B{B}_0^{*}}{\left(B_0^{*}+h_B\right)\left({\mu }_E+{\phi }_R{R}_0^{*}\right)}}.$$

Using the baseline value ${\alpha }_B=0.10 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$ gives $R_0=0.8325<1.$

The corresponding eigenvalues of the Jacobian matrix evaluated at the DFE are

$${\lambda }_1=-0.02, {\lambda }_2=-0.02, {\lambda }_3=-1.65, {\lambda }_4=-0.02010.$$

Since all eigenvalues have negative real parts, the DFE is locally asymptotically stable. Consistently, the numerical simulation shown in Figure 2a demonstrates that the small initial perturbation in $E\left(t\right)$ progressively decays toward zero, indicating convergence toward the disease-free equilibrium.

To illustrate the behavior on the opposite side of the threshold, the value of ${\alpha }_B$ corresponding to $R_0=1$ was first determined from the threshold expression. Using the parameter values of the auxiliary DFE subsystem gives

$${\alpha }_B={\displaystyle \frac{\left(B_0^{*}+h_B\right)\left({\mu }_E+{\phi }_R{R}_0^{*}\right) }{B_0^{*}}}=0.12012 {\mathrm{day}}^{-1}.$$

Therefore, a slightly larger value, ${\alpha }_B=0.13 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$, was used only in this auxiliary numerical experiment to obtain $R_0=1.0823>1$, while all other parameter values were kept unchanged.

This modified value is used solely to illustrate the $R_0>1$ case and does not replace the baseline value ${\alpha }_B=0.10 {\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$ used in the full-model simulations.

The corresponding eigenvalues are

$${\lambda }_1=-0.02, {\lambda }_2=-0.02, {\lambda }_3=-1.65, {\lambda }_4=0.00987.$$

The positive fourth eigenvalue confirms the local instability of the DFE. As shown in Figure 2b, the small initial effector-cell perturbation grows with time rather than returning to zero, and the trajectory therefore moves away from the disease-free equilibrium. Overall, these simulations numerically support the local threshold behavior predicted by Theorem 4: a small perturbation in the autoreactive effector T-cell population decays when $R_0<1$, whereas it grows when $R_0>1$. This result applies only to the auxiliary subsystem with $S_{E }=0$ and is not used to interpret the treatment simulations of the full model with $S_{E }=20.$

7.2. Model Behavior Without Treatment (u = 0)

In the first scenario, the system is simulated in the absence of any therapeutic intervention, without stem cell administration or immune-modulating control, representing the natural progression of T1D driven by autoimmune processes. Without intervention, the population of autoreactive effector T cells persists at excessive levels, resulting in prolonged immune-mediated damage to pancreatic β-cells. Consequently, the β-cell population progressively declines and fails to recover. At the same time, regulatory T cells remain insufficient to suppress the autoimmune response, resulting in sustained immunological dysregulation. This uncontrolled scenario underscores the immune system’s inability to restore internal balance without external intervention. The results establish a foundation for future simulations of stem cell-based therapy, highlighting the need for therapeutic approaches that both inhibit autoreactive immune activity and preserve β-cell mass.

7.3. Fixed Stem Cell Administration Protocols (Control Measure u ≠ 0)

In this case, the model is simulated under stem cell–based therapeutic strategies. The control variable $u\left(t\right)$ denotes the time-dependent stem cell administration into the stem cell compartment, given by

$${\displaystyle \frac{dS\left(t\right)}{dt}}=-(K_S+{\mathsf{\rho }}_S)S+u\left(t\right).$$

The therapeutic objective is to suppress autoreactive effector T cells, enhance regulatory T-cell function, and promote pancreatic β-cell preservation, while maintaining a clinically appropriate treatment intensity. Two stem cell delivery protocols are investigated over a one-year simulation horizon ($T=365$ days): pulse-based administration and continuous infusion. Each protocol is evaluated at low, medium, and high dose levels.

7.3.1. Pulse-Based Stem Cell Administration (Pulses Protocol)

In this protocol, stem cells are injected in short, separate doses at specific time intervals. In our simulations, injections were provided on specific days.

$$t_1=7, t_2=14, t_3=21,$$

each over a small interval of width $\tau$. The pulse control function takes the form:

$$u_{pluse}\left(t\right)={\displaystyle \sum _{k=1}^3}{\displaystyle \frac{D_k}{\tau }}{1}_{\left[t_k,t_{k+\tau }\right]}\left(t\right),$$

where $D_k$ is the total dose delivered in pulse $k.$ This protocol produces acute transient peaks in the stem-cell population $S\left(t\right),$ which decline rapidly because of the combined clearance and transition terms represented by $K_S+{\rho }_S$. These short-lived peaks produce rapid but temporary increases in regulatory T cells $R\left(t\right)$, temporary suppression of effector T cells $E\left(t\right),$ and a transient enhancement in β-cell mass $B\left(t\right)$.

Figure 3 illustrates the temporal dynamics of the model compartments in the absence of treatment and the effects of low, medium, and high-dose pulse treatments on the state variables. The simulations show that increasing pulse intensity enhances β-cell preservation and immune regulation during the treatment period. However, the therapeutic effect declines rapidly between and after the injections because the stem-cell population decreases quickly once each pulse ends. Thus, the pulse protocol produces a pronounced but predominantly transient therapeutic response.

7.3.2. Continuous Infusion-Based Stem Cell Administration (Infusion Protocol)

In the infusion protocol, stem cells are administered continuously over an extended period, resulting in a more gradual and sustained therapeutic effect. The control function is delineated as follows:

$$u_{inf}\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{D}{T_n-T_0}}, T_0\le t\le T_n,\\ 0 \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}\right.$$

where $D$ is the total dose distributed uniformly across the infusion interval $\left[T_0,T_n\right].$

In the simulations, the continuous-infusion protocol was applied continuously from day 7 to day 30 within the one-year simulation horizon. This protocol produces a smoother stem-cell profile in $S\left(t\right)$ than pulse administration, together with more gradual changes in $R\left(t\right),$ $E\left(t\right)$, and $B\left(t\right)$. Figure 4 shows the system dynamics under low-, medium-, and high-dose infusion therapy. The results indicate that continuous infusion produces a more evenly distributed therapeutic response during the treatment period, whereas pulse administration generates sharper transient responses. After treatment cessation, the trajectories under the two fixed-dose protocols gradually approach similar long-term levels. Therefore, the main difference between the two delivery strategies lies in the temporal pattern and smoothness of the response rather than in a large difference in the final one-year outcome.

7.4. Optimal Control Simulations

The optimal stem cell delivery strategy is investigated based on the optimal control formulation established in the preceding section. The permissible control set is delineated by

$$0 \le u\left(t\right)\le u_{max}, t\in \left[0,T\right],$$

where $u_{max}$ denotes the maximum feasible stem cell administration rate.

The optimal control problem is solved numerically using the forward–backward sweep method. In this procedure, the state system is integrated forward in time using the prescribed initial conditions, whereas the adjoint system is integrated backward from terminal conditions; the control is then updated iteratively using the derived characterization of $u\left(t\right)$ until convergence.

The simulations indicate that the optimal strategy initially applies a moderately elevated stem cell delivery rate to rapidly mitigate autoimmune activation. As the system approaches a more controlled immunological state, the control effort progressively declines. This strategy yields a substantial reduction in autoreactive effector T cells, a sustained increase in regulatory T-cell levels, and marked recovery and stabilization of pancreatic β-cell mass. Compared with fixed-dose protocols, the optimal control technique achieves a more balanced therapeutic response while avoiding unnecessary treatment intensity in the later stages of therapy.

These findings suggest that adaptive stem cell immunomodulation may offer a viable strategy for the long-term management of autoimmune dynamics in T1D. The choice between pulse and continuous infusion protocols carries important implications for treatment timing and complexity. Recent studies have highlighted the importance of multi-stage intervention strategies in autoimmune disease management [39], and the present simulations indicate that sustained delivery produces smoother treatment-period dynamics, whereas intermittent high-dose therapy produces stronger transient responses. However, the optimal protocol may vary depending on patient-specific factors, such as disease stage, age, and immune status.

8. Conclusions

This study developed an extended mathematical framework to investigate the immune processes involved in T1D and the effects of stem-cell-based therapy under pulse-based administration and continuous infusion. The analytical and numerical results demonstrate the framework’s effectiveness. For the auxiliary subsystem with $S_{E }=0$ and constant treatment input, the threshold quantity $R_0$ characterizes the local stability of the disease-free equilibrium. Specifically, the DFE is locally asymptotically stable when $R_0<1$ and unstable when $R_0>1$. This threshold result applies only to the auxiliary disease-free subsystem and is not used to characterize the full model with $S_{E }>0$. The study highlights the importance of regulatory T cells in suppressing the activity of autoreactive effector T cells and restoring the immune system balance. To evaluate the effects of therapy, an optimal control problem was formulated. Numerical simulations showed that both fixed-dose strategies increased β-cell mass but produced different temporal response patterns. Pulse treatment produced stronger transient responses around the administration periods, whereas continuous infusion produced smoother, more evenly sustained dynamics during the infusion interval. After treatment cessation, the trajectories gradually approached similar long-term levels, indicating that the primary difference between the two protocols lies in the timing and smoothness of the therapeutic response rather than in a large difference in the final one-year outcome. Sensitivity analysis indicated that pulse dynamics are significantly affected by factors regulating β-cell proliferation and immune-cell activation, specifically ${\mathrm{S}}_B$, ${\alpha }_B$, and ${\mu }_R$, demonstrating that the system is extremely reactive to sudden therapeutic interventions. In contrast, the infusion strategy produced a smoother and more homogeneous β-cell trajectory during the treatment period, consistent with its continuous delivery profile. The local sensitivity analysis showed that the same core biological mechanisms remained influential under both protocols, although their relative sensitivity magnitudes differed by dosing pattern. Because the analysis is local to the baseline parameter set, interpret these differences as local parameter effects rather than evidence of global robustness or long-term superiority of one protocol over the other. The theoretical analysis distinguishes between two mathematically related regimes. For the auxiliary subsystem with $S_{E }=0$, the threshold quantity $R_0$ characterizes the local stability of the disease-free equilibrium. For the full model with $S_{E }>0$, a unique biologically feasible positive equilibrium exists and is locally asymptotically stable under constant therapeutic input. The full-model disease-progression and treatment simulations were performed with $S_{E }=20$ and are therefore interpreted in terms of suppression of autoreactive effector T-cell activity, enhancement of regulatory T-cell responses, and β-cell recovery, rather than through the auxiliary DFE threshold $R_0$. This contributes to sustained suppression of autoimmune activity and promotes favorable immune regulation. The proposed framework enables quantification and mechanistic understanding of how stem cell-based immunomodulation tunes the underlying mechanisms of autoimmune disease in T1D. The model provides quantitative insights into the design and optimization of stem cell-based therapeutic strategies. Future research can extend the current framework in several directions. Incorporating glucose-insulin dynamics would improve understanding of disease progression and treatment evaluation. The model could be made more realistic by accounting for patient diversity and parameter uncertainty through data-driven parameter estimation and sensitivity analysis. The optimal control framework could also be extended to evaluate other treatment options, such as combination therapies or short-term treatment regimens. The complexity of treatment timing and protocol design further underscores the need for multi-stage intervention strategies [39]. Future work should explore adaptive protocols that adjust stem cell delivery based on real-time immune markers, potentially leading to more personalized and effective therapies. Additionally, extending the current ODE framework to PDE models that capture spatial heterogeneity and the effects of localized stem cell delivery represents a promising direction [29,40]. Such models could provide insights into the spatial distribution of stem cells, their homing to specific tissues, and the local dynamics of immune cell populations in the pancreatic islets. Incorporating patient-specific heterogeneity via data-driven parameter estimation would further enhance the model’s clinical applicability.