Fatty acids and physical activity are critical for $\beta$-cell mass and insulin sensitivity: Pathways to T2D and prevention

Abstract

Existing mathematical models investigating the progression of type 2 diabetes (T2D) over time primarily focus on glucose, insulin, $\beta$-cell mass, and other related factors, while often omitting fatty acids (FA) as an explicit variable—despite FA being a major energy source for the body. There exists a complex network of dynamical interactions among glucose, insulin, FA, and $\beta$-cell mass. To gain deeper insights into the metabolic dynamics and pathophysiology of T2D, it is essential to incorporate FA into such models.

In this paper, we extend the classic Topp’s $GI\beta$ model by explicitly incorporating FA and exploring its interactions with glucose, insulin, and $\beta$-cell mass. A new formula for insulin sensitivity $\left(S_I\right)$ is proposed to better capture the impaired effect of FA on $S_I$, enabling the exploration of diabetes development pathways and strategies for prevention or delay. Model simulations align well with clinical data and successfully replicate key characteristics of T2D progression, including declining $S_I$, reduced $\beta$-cell mass, a sedentary lifestyle, and excessive dietary intake. Our results demonstrate a strong positive correlation between glucose and FA levels, indicating that elevated FA is associated with increased glucose concentrations. Model analysis shows that FA levels in diabetic subjects rise significantly as a result of T2D development. Numerical analyses indicate that maintaining adequate physical activity or reducing dietary excess effectively preserves $S_I$ and $\beta$-cell mass, thereby reducing the risk of developing T2D.

Most notably, our detailed simulations reveal a striking pattern: in healthy and non-diabetic individuals, FA levels consistently remain below glucose levels across their lifespan. In contrast, in individuals with T2D, FA levels initially remain lower but begin to increase sharply and tangle with glucose levels, and then surpass glucose concentrations near the time of $\beta$-cell failure. These patterns suggest that elevated FA may play a contributory role in increasing glucose levels, reducing insulin sensitivity, and ultimately leading to hyperglycemia. Our findings support the notion that glucotoxicity and lipotoxicity accelerate $\beta$-cell decline, impair insulin secretion, and drive T2D progression. Elevated FA thus emerges not only as a contributing pathway but also as a potential biomarker for monitoring disease development. Furthermore, the observed relationship between glucose and FA levels seems suggesting a quantitative marker that could be used to track progression toward T2D.

1. Introduction

Diabetes is the fastest-increasing disease worldwide. For several decades, diabetes is classified into two major categories based on the underlying causes of hyperglycemia. Type 1 diabetes (T1D) is due to an autoimmune attack on the insulin-secreting $\beta$-cells, leading to a deficiency or lack of insulin. T2D is associated with a deficit in the mass of $\beta$-cells due to the development of a “resistance” to the action of insulin, and the resulting hyperinsulinemia and/or hyperglycemia (ADA, 2013). However, recent research supports a more nuanced classification system that reflects the underlying pathophysiology of diabetes (Ahlqvist et al., 2018). Mezza et al. (2019) highlight the role of islet plasticity and $\beta$-cell fate, underscoring the importance of understanding the physiological mechanisms involved. Misra et al. (2023) reviewed emerging subclassification strategies and advocated for physiologically informed, precision-based approaches. Their goal is to stratify patients using key physiological markers, such as (i) $\beta$-cell function, (ii) insulin sensitivity or resistance, (iii) inflammatory markers, and (iv) genetic and metabolic signatures. Their ultimate aim of enabling precision diabetes care that targets the individual’s specific pathophysiological profile rather than focusing solely on glycemic levels.

The dynamics of $\beta$ cell mass and $S_I$, are central to the pathophysiology of T2D, which are influenced by various metabolic factors including glucose, insulin, and FA. These factors have inspired many researchers to study the endocrine metabolic regulatory system for the understanding of the mechanisms and causes of metabolic dysfunctions, enhance early detection of diabetes, and ultimately develop more reasonable, effective, efficient, and economical treatments for diabetic patients.

Biologists have developed various hypotheses to inspect the hidden factors for the progression of T2D. Some of these hypotheses are actively debated and researched within the scientific community. Esser et al. (2020) propose that dysfunction of $\beta$-cell, characterized by impaired insulin release, typically manifests early in the disease’s progression. In contrast, Johnson (2021) contends that primary hyperinsulinemia is often initially present as T2D progresses. Notably, both perspectives start from the so called environmental factors, including food, lifestyle, temperature, pollutants, population density, sound, light, and parasites. Besides biological hypotheses, mathematical models have been crucial in diabetes research. Mathematical models provide a simplified representation of the interaction of the complex systems’ components using equations and parameters. These models are indispensable for understanding the dynamics of complex systems that are difficult to explore only by experimental and observational methods. The metabolic system is a complex biological system, and several mathematical models have been used to trace the dynamics of glucose, insulin, and $\beta$-cell mass (Bergman, 1989; Visentin et al., 2014; Pompa et al., 2021; Kumnungkit et al., 2022; Sirlanci et al., 2023; Topp et al., 2000; De Gaetano et al., 2008; Gallenberger et al., 2012). The Food and Drug Administration (FDA) accepted T1D glucose tracer simulator of Visentin et al. (2014). Topp et al. (2000) proposed the well-known model, which serves as a fundamental model to explain the dynamics of glucose, insulin, and $\beta$-cell mass in the development and progression of T2D. De Gaetano et al. (2008), proposed a pancreatic islet compensation model to understand the ups and downs of $\beta$-cell mass, conducted simulations by decreasing insulin resistance, and discussed the replication issues to observe its performance over a lifetime. Gallenberger et al. (2012) introduced a mathematical model connecting the dynamics of glucose and insulin concentration with the $\beta$-cell cycle, where they showed glucose is an important regulator of $\beta$-cell cycle. Hardy et al. (2012) used the publicly available Diabetes Prevention Program (DPP) data set to evaluate a newly developed mechanistic mathematical model’s prediction ability for progression of diabetes and the effectiveness of preventive interventions by the model. Boutayeb et al. (2014), developed a model considering the effect of genetic predisposition to T2D on the dynamics of glucose, insulin, and $\beta$-cell to investigate how $S_I$ is affected by physical activity, obesity, and other factors. To explain how compensation success or fail Ha et al. (2016) extended Topp’s model adding regulation of two aspects of $\beta$-cell function on intermediate time scale, and they depict that bariatric surgery and acute caloric restriction rapidly reversed T2D. Ha and Sherman (2020), presents a mathematical model that can track an individual’s metabolic state over a period of years, from normal glucose tolerance to T2D, at any point in time. Li and his colleagues in Yang et al. (2023) added a pathogenic factor $X$ to explain how therapies, like lifestyle modification and surgical options, may improve the curse of T2D. In support of individualized management therapies, the authors emphasize connection between obesity and T2D.

Despite these considerable research efforts, none of the existing models systematically include the dynamics of FA and their longitudinal effect on $S_I$, the insulin-responsive function for FA, the dependence of $\beta$-cell mass on FA levels. These drawbacks underscore the necessity for a more sophisticated approach in modeling T2D that can accommodate the complex interrelations among glucose, insulin, FA, insulin sensitivity, and $\beta$-cell mass. To develop more effective treatment we need to enhance pathophysiology of T2D by addressing these drawbacks.

In this study, we develop a novel model, which incorporates environmental factors such as food (FA), physical activity (PA), and glucose, insulin, FA-influenced $S_I$. This model enables us to explore the dynamical behavior of glucose, insulin, FA, and $\beta$-cell mass, as well as pathophysiology and preventions of T2D with controlled meal intake and increased physical activities. This paper is organized as follows. In Section 2, we formulate a new physiologically-based model. Section 3 presents an analysis of the equilibria, followed by numerical case studies in Section 4. Finally, we conclude with a comprehensive discussion in Section 5.

2. Model formulation

Most studies in diabetes modeling focus primarily on the dynamics of glucose and insulin. Although, the body gets more energy from FA, its contribution is overlooked in glucose-insulin concentric models. We aim to expand the $GI\beta$ model of Topp et al. (2000). Our extension will incorporate the dynamics of FA, longitudinal dynamics of $S_I$, and physical activity. Let $G$, $I$ be fasting plasma glucose and insulin respectively, and $\beta$ stands for $\beta$-cell mass. The system of ODEs for Topp’s $GI\beta$ model is as follows:

$$\left\{\begin{array}{l}{G}^{\prime }(t)=R_0-\left(S_g+S_{I}I\right)G\\ I^{\prime }(t)=\frac{\beta \sigma G^2}{\alpha +G^2}-kI\\ {\beta }^{\prime }(t)=\left(-d_0+r_{1}G-r_2{G}^2\right)\beta \end{array}\right.$$ (2.1)

where $R_0$ stands for net rate of glucose production at zero glucose level, $S_g$ is the glucose effectiveness at zero insulin level, $S_I$ indicates insulin sensitivity, $\beta$ denotes $\beta$-cell mass, $\sigma$ symbolizes maximal insulin secretion rate by the pancreatic $\beta$-cell, $k$ is short for insulin clearance rate, $\alpha$ represents the inflection point of sigmoidal function, $r_1$, $r_2$ determines $\beta$-cell glucose tolerance range, and $d_0$ indicates $\beta$-cell death rate at zero glucose level.

As mentioned above, we are concerned about the dynamics of FA in the metabolism system. Therefore, the primary factors in this system are the dynamics of glucose, insulin, FA, and $\beta$-cell mass. We have developed a model for the endocrine metabolic regulatory system that encompasses the dynamics of glucose, insulin, FA, and $\beta$-cell mass. This model is formulated using a single-compartment mass balance equation, as illustrated in Fig. 1. We continue to denote fasting glucose, insulin, and $\beta$-cell mass by $G(t)$, $I(t)$, $\beta$, and in addition $F(t)$ is the fasting FA concentration in plasma at time $t\ge 0$, and $t$ is the time in day. We have

Fig. 1.

Diagram for the daily glucose and FA metabolism.

$\frac{dG(t)}{dt}$ = glucose production rate - glucose removal rate.

$\frac{dI(t)}{dt}$ = insulin secretion rate - insulin clearance rate.

$\frac{dF(t)}{dt}$ = fatty acids production rate - fatty acids uptake rate.

$\frac{d\beta (t)}{dt}=\beta$-cell mass formation rate - $\beta$-cell mass loss rate.

Glucose production.

Daily glucose is released into our bloodstream from three sources: dietary carbohydrates, the liver, and the kidney. After a meal, dietary carbohydrates are hydrolyzed by brush border enzymes in the small intestine, then release as glucose into the bloodstream. The liver produces glucose through processes called glycogenolysis and gluconeogenesis, which is then released into the blood. Glucose production by the liver depend on plasma insulin concentration and stimulated by glucagon. Therefore, hepatic glucose production can be a function depend on plasma insulin and glucagon, denoted by $HGP(I,A)$ (where A is plasma glucagon concentration). In our model, given that the unit of time is day and recognizing the need for simplification in complex metabolic processes, we assume the daily contributions of glucose from hepatic synthesis to be constant. Thus, for the purposes of this model, we define hepatic glucose production, $HGP(I,A)$, simply as $HGP$, a constant parameter. Additionally, the kidneys produce glucose via gluconeogenesis and release it into the bloodstream. Given that our unit of time is day, the daily contributions of glucose from dietary intake, and kidneys release are nearly constant. These sources are denoted respectively as $G_{meal}$ for meals and $KGP$ for renal glucose production.

Glucose removal.

Glucose utilization consists of two parts, namely, insulin-independent utilization and insulin-dependent utilization. Brain cells, blood cells, and liver cells absorb glucose from the blood without requiring insulin. We denote this type of uptake by $S_{g}G$ indicating its dependency on the glucose concentration level alone. In contrast, glucose uptake by muscle and adipose cells is dependent on both insulin levels and insulin sensitivity, We denote this type of uptake by $S_{I}IG$ (where $S_I$ is insulin sensitivity), indicating its dependency on the insulin level and insulin sensitivity. Shepherd and Kahn (1999) and Holloszy (2005) discovered that muscle and adipose cells can also absorb glucose through muscle tissue contraction, a process independent of insulin but reliant on physical activity. We denote this type of glucose uptake by $S_{T}v$, where $S_T$ is the number of steps per day and $v$ is the amount of glucose removal by per step. Therefore, we get the daily rate of change of fasting plasma glucose as follows:

$$\frac{dG(t)}{dt}=G_{meal}+HGP+KGP-S_{g}G-S_{I}IG-S_{T}v$$ (2.2)

Fatty acids production.

We chose FA as a crucial state variable, encompassing both free FA in the bloodstream and those encapsulated within triglycerides of the plasma. The main sources of these FA in the bloodstream are dietary intake, FA synthesis in the liver, and FA release by lipolysis from adipose tissue. Dietary fat contains 95 % triglyceride (Yen et al., 2015), and 90 % molecular weight of triglyceride is FA (Flatt, 1995). After meals, fats are hydrolyzed into monoglycerides and free FA by digestive enzymes and subsequently absorbed by enterocytes. Inside these cells, they are reassembled into triglycerides. These triglycerides, in conjunction with cholesterol and fat-soluble vitamins, merge with Apolipoproteins to create chylomicrons. The chylomicrons are then secreted first into the lymphatic system and then into the bloodstream via the thoracic duct. and ultimately reach the cells for absorption as FA. The liver synthesizes FA and releases them into the blood as very low-density lipoprotein triglycerides (VLDL-TG). Additionally, adipose tissue releases FA through a process called lipolysis, which binds with albumin to reach body cells for absorption. Since our unit time is day, the total daily output of FA from dietary intake, synthesis by liver, and adipose tissue is almost constant. These sources are denoted respectively by $F_{meal}$ for dietary intake, $HFP$ for hepatic FA production, and $AFP$ for adipose FA production.

Fatty acids uptake.

FA are taken up by adipocytes, muscle tissues, the liver, the heart, and the kidneys. FA and glucose uptake are almost similar (Hamilton and Kamp, 1999). The uptake of FA depends on insulin and is also independent of insulin (Roy and Parker, 2006). There are two primary mechanisms for FA uptake: (i) diffusion and (ii) protein-mediated uptake (Mitchell and Hatch, 2011). Protein-mediate FA uptake depends on insulin (Stahl et al., 2002; Ye, 2007). We denote $\mu F$ to indicate insulin-independent uptake of FA and $\lambda S_{I}IF$ ($\lambda$ is the ratio of insulin sensitivity for FA to insulin sensitivity for glucose), to indicate FA utilization depend on the insulin sensitivity and insulin level. Consequently, the dynamics of plasma FA can be described as follows:

$$\frac{dF(t)}{dt}=F_{meal}+HFP+AFP-\mu F-\lambda S_{I}IF$$ (2.3)

Insulin secretion.

In response to elevated concentrations of glucose and FA, insulin is produced through $\beta$-cell secretion (Li et al., 2006; Jayanthi et al., 2017). FA induce almost 10 % of insulin secretion (Ye, 2007). The secretion of insulin induced by glucose and FA follows a similar functional shape (Murillo et al., 2019). Like Topp et al. (2000), we also consider insulin secretion function $\frac{\sigma G^2}{\alpha +G^2}$ in response to elevated glucose. We assume insulin secretion function $\frac{\rho F^2}{\eta +F^2}$ in response to elevated FA, where $\rho$ is the FA induced all $\beta$-cells maximal insulin secretion rate and $\eta$ is the inflection point of sigmoidal function for FA.

Insulin clearance.

All insulin sensitive tissues clear insulin. Primarily, muscle and adipose cells mainly remove insulin. Insulin degradation is a regulated process involving insulin binding to its receptor. Topp et al. (2000) show that insulin degradation is proportional to insulin concentration. Hence, as with Topp et al. (2000), we assume the insulin clearance rate $d_i>0$. Therefore, resultant insulin dynamics is as follows:

$$\frac{dI(t)}{dt}=\frac{\sigma G^2}{\alpha +G^2}\beta +\frac{\rho F^2}{\eta +F^2}\beta -d_{i}I$$ (2.4)

$\beta$-cell mass dynamics.

Topp et al. (2000) showed that $\beta$-cell replication and apoptosis are affected by fasting plasma glucose levels. When fasting plasma glucose levels rise, $\beta$-cell mass goes up, but when hyperglycemia is chronic (glucotoxicity), $\beta$-cell mass decreases. Topp et al. (2000) defined the dynamics of $\beta$-cell mass by the equation $d\beta (t)/dt=\left(-d_0+r_{1}G-r_2{G}^2\right)\beta$. Similarly, fasting FA levels influence $\beta$-cell proliferation and apoptosis (Oh, 2015). Acute increases in plasma FA levels lead to a temporary increase in $\beta$-cell mass and insulin secretion, whereas chronic elevations (lipotoxicity) result in their decrease (Oh et al., 2018). That’s why we assume a portion of $\beta$-cell mass is governed by the term $\left(-d_0^{\prime }+r_1^{\prime }F-r_2^{\prime }{F}^2\right)\beta$. Where $d_0^{\prime }$ is the beta cell death rate at zero FA levels, $r_1^{\prime }$ and $r_2^{\prime }$ determines $\beta$-cell FA tolerance range. Consequently, the following expression captures the ultimate dynamics of $\beta$-cell mass:

$$\frac{d\beta (t)}{dt}=\left(-d_0+r_{1}G-r_2{G}^2\right)\beta +\left(-d_0^{\prime }+r_1^{\prime }F-r_2^{\prime }{F}^2\right)\beta$$ (2.5)

Insulin sensitivity $S_I$.

Insulin resistance (IR) and insulin sensitivity (IS) are reciprocal. Matthews et al. (1985) developed a formula for estimating insulin resistance based on fasting glucose and insulin levels, known as HOMA-IR, which is calculated as ${\text{HOMA}}_{\text{IR}}=\frac{I\times G}{405}$, where I is the fasting insulin level (in μU/mL) and G is the fasting glucose level (in mg/dL). Katz et al. (2000) developed a formula for estimating insulin sensitivity based on fasting glucose and insulin levels, known as QUICKI, which is calculated as ${\text{QUICKI}}_{\text{IS}}=\frac{1}{\log (I)+\log (G)}$. Yang et al. (2023) use $S_I$ as a function of only insulin in their model. Esser et al. (2020) highlight that the principle of homeostatic control in endocrinology is closely linked to feedback mechanisms, a concept that applies equally to glucose and FA metabolism. Therefore, $S_I$ depends on glucose, insulin, and FA.

The impact of FA on insulin sensitivity dynamics is notably significant. Roden et al. (1996) discovered that elevated plasma FA levels could induce insulin resistance (lowering IS). This finding is complemented by research Lovejoy (1999), and Sears and Perry (2015), who identified FA as having a markedly detrimental effect on insulin action. Given the substantial influence of FA on insulin sensitivity, there emerges a clear need for a refined approach to calculating insulin sensitivity.

To address this gap, we propose a novel formula for insulin sensitivity index $S_I$ calculation that incorporates FA. This approach not only reflects the intricate balance of factors affecting insulin dynamics but also acknowledges the critical role of FA in modulating insulin sensitivity. Our proposed new formula for $S_I$ by incorporating FA is as follows:

$$S_I(G,I,F)=\frac{\tau {\gamma }^n}{{\gamma }^n+{\left(k_{g}G+k_{i}I+k_{f}F\right)}^n},$$

where $\tau$, $\gamma$, $n$, $k_g$, $k_i$, and $k_f$ are positive constants. The parameters $k_g$, $k_i$, and $k_f$ are the proportional contributions to $S_I$ from glucose, insulin, and FA respectively. We consider equal weight of glucose and FA and double weight of insulin in the insulin sensitivity formula to get the consistency of longitudinal dynamics of insulin sensitivity. So $k_g=k_f=1$, $k_i=2$, and also from Section 2.1 we have $n=3$. Thus, throughout this paper, we assume

$$S_I(G,I,F)=\frac{\tau {\gamma }^3}{{\gamma }^3+{(G+2I+F)}^3}.$$ (2.6)

Summarizing (2.2) to (2.6), we have our model as follows:

$$\left\{\begin{array}{l}{G}^{\prime }(t)=G_{meal}+HGP+KGP-S_{g}G-S_{I}IG-S_{T}v\\ I^{\prime }(t)=\frac{\sigma G^2}{\alpha +G^2}\beta +\frac{\rho F^2}{\eta +F^2}\beta -d_{i}I\\ F^{\prime }(t)=F_{meal}+HFP+AFP-\mu F-\lambda S_{I}IF\\ {\beta }^{\prime }(t)=\left(-d_0+r_{1}G-r_2{G}^2\right)\beta +\left(-d_0^{\prime }+r_1^{\prime }F-r_2^{\prime }{F}^2\right)\beta \end{array}\right.$$ (2.7)

Remark

Our insulin sensitivity formula (2.6) incorporated FA with glucose and insulin. An insulin sensitivity formula that includes FA could potentially identify the risk of the T2D development in individuals more accurately. Incorporating FA might allow for earlier detection of insulin resistance, especially in individuals who exhibit normal glucose tolerance but have dysregulated lipid metabolism.

In the next subsection we attempt to determine the ranges of the model parameters values.

2.1. Ranges of parameter values

The volume of extracellular fluid consists of 11 l of interstitial fluid and 3 l of plasma. The glucose and FA space is 14 l (or 140 dl) of extracellular fluid volume (Hall and Guyton, 2016). Zello (2006) notes that the recommended dietary allowance (RDA) for carbohydrates is 130 g/day. However, the actual carbohydrate intake for most individuals, including athletes, often exceeds these RDA. The median carbohydrates intake of men ranges from 200 to 330 g per day and 180 to 230 g per day for women (Zello, 2006). We assume a daily carbohydrate intake of 180–330 g, So, our parameter is $G_{meal}\in [1285,2357]$ mg/dl/day. For a typical, healthy individual, we estimate a $G_{meal}=1310$ mg/dl/day.

On a typical day, the kidneys produce approximately 15–55 g of glucose via gluconeogenesis (Huhtaniemi, 2018). So, we get the amount of kidney’s glucose production, $KGP\in [107,392.86]$ (mg/dl/day), calculated from [15 × 1000/140, 55 × 1000/140] (mg/dl/day). We assume a typical $KGP=107$ mg/dl/day. Normally, the liver produces 50 % of its glucose through glycogenolysis and 30 % through gluconeogenesis, while the kidneys contribute 20 % of glucose via gluconeogenesis (Martini, 2004). Based on this, the liver’s glucose production by gluconeogenesis ranges between 22.5 and 82.5 g per day, calculated from [(15/20)×30, (55/20)×30] (g/day). In addition, the liver produces 37.5 to 137.5 g per day by glycogenolysis, derived from [(15/20) × 50, (55/20) × 50] (g/day). Therefore, the total glucose production by the liver is 60–220 g/day. Thus hepatic glucose production, $HGP\in [428.57,1571]$ (mg/dl/day), copied from [60 × 1000/140, 220 × 1000/140] (mg/dl/day). Considering a typical fasting duration throughout a day, we assume a standard $HGP=535$ mg/dl/day.

The percentage of fat in daily macronutrient intake for adults is approximately 20 % to 35 %, or about 44 to 77 g (Zello, 2006). Hinsberger and Sandhu (2004) state that a normal Western adult diet contains approximately 60–80 g of fat daily, of which about 95 % are long-chain triglycerides. Frayn (2002) notes that a typical Western diet contain almost 100 g of fat per day, while the typical American diet consists of 100–150 g of fat daily (Mansbach and Siddiqi, 2016). Therefore, we consider the daily fat intake range to be 42–150 g. Given that approximately 95 % of dietary fat is in the form of triglycerides (TAG) (Yen et al., 2015), and 90 % molecular weight of TAG is FA (Flatt, 1995). Hence, the daily FA intake from meals is approximately 35.91–128.25 g. Consequently, the FA input from meal, $F_{meal}\in [256.5,916]$ (mg/dl/day), calculated as $F_{meal}\in [35.91\times 1000/140,128.25\times 1000/140]$ (mg/dl/day). We assume a typical $F_{meal}=380$ mg/dl/day. Adiels et al. (2006) report that the liver produces very low-density lipoprotein triglycerides (VLDL-TG) at an average rate of 249 mg/day/kg. For example, a 70 kg man produces 249 × 70 = 17, 430 mg/day. Since triglycerides (TAG) are composed of 90 % FA, the liver’s production of FA is approximately 17, 430 × 0.9 = 15, 687 mg/day. Consequently, the release of FA into the interstitial fluid per deciliter is calculated by dividing 15,687 mg by 140 dl, resulting in 112 mg/dl/day. Thus, the hepatic FA production, $HFP=112$ mg/dl/day.

In a fasting state, adipose tissue releases FA into the bloodstream through the process of lipolysis, and the body utilizes these FA as a major fuel source. We assume per day adipose tissue releases FA for about 6 h. Once a meal has been completely absorbed (typically three to five hours after a meal), the metabolism changes to a fasting state. A person is assumed to be fasting after 8–12 h. We have collected data from the top panel of Fig. 1 of Mittendorfer et al. (2009), with some outliers excluded. This data was then converted to calculate the average release of FA per day per deciliter by adipose tissue. The converted data are presented in Fig. 2. The average release of FA by adipose tissue is approximately 196 mg per deciliter per day. Consequently, a typical adipose FA production, $AFP=196$ mg/dl/day.

Fig. 2.

Average daily release of FA by adipose tissue (adapted from Mittendorfer et al. (2009), the top panel of Fig. 1).

In average, American walks approximately in the range of 3000 to 4000 steps per day according to Mayo Clinic (https://www.mayoclinic.org/healthy-lifestyle/fitness/in-depth/10000-steps/art-20317391, accessed on 2023.11.12). We assume the total number of steps to be $S_T=3500$. Accordingly we assume the blood sugar removal rate to be $v=0.0031$ mg per step. Topp et al. (2000) established that the glucose-induced maximal insulin secretion rate by all $\beta$-cells is $\sigma =43.2$ μU/ml/day. Ye (2007) noted that FA induce almost 10 % insulin secretion, so the FA-induced maximal insulin secretion rate by all $\beta$-cells is $\rho =10\%$ of 43.2 μU/ml/day ≈ 5 μU/ml/day. The ratio $(\lambda )$ insulin sensitivity for FA to insulin sensitivity for glucose can be derived from the sensitivities reported by Reinehr et al. (2005); for normal people, insulin sensitivity for FA ranges from 0.48 to 0.81, and for glucose, it is from 0.60 to 1.17. We get $\lambda \in [0.41,1.35]$. For the lower end of $\lambda$, we calculate the ratio of the lowest insulin sensitivity of FA to the highest insulin sensitivity of glucose, and for the upper end of $\lambda$, we calculate the ratio of glucose. we estimate a typical value of $\lambda =0.45$ for the consistency of the dynamics of simulated glucose, insulin, FA, and $\beta$-cell mass with clinical observation. Weir and Bonner-Weir (2004) state that there is no precise glucose range for $\beta$-cell adaptation, but a fasting glucose range of 89–130 mg/dl is a reasonable approximation. Based on this, we assume a fasting glucose range of 94–147 mg/dl as an effective approximation for $\beta$-cell adaptation. We estimate $d_0=0.05$ day⁻¹, $r_1=0.00087$ dl/mg/day, and $r_2=0.0000036$ dl²/mg²/day to enable $\beta$-cell adaptation within this fasting glucose range. Jayanthi et al. (2017) highlighted that the optimal cut-off point for plasma triglycerides (TAG) in obesity is ≥ 142.5 mg/dl. Metabolic syndromes, which occur in individuals with obesity, include $\beta$-cell mass reduction, a consequence influenced by lipotoxicity. Since cut-off point for FA bound to triglycerides in obesity is 128.25 mg/dl. Additionally, Arabi et al. (2019), note that a non-esterified plasma FA level greater than 0.6 mmol/L is considered high; we have approximated a high level of non-esterified plasma FA at 0.76 mmol/L (equivalent to 21.5 mg/dl). Thus, the total plasma FA level combining these measures is approximately 149.75 mg/dl, which we propose as the upper limit for $\beta$-cell adaptation. We assume a fasting FA range of 93–149.75 mg/dl as a suitable approximation for $\beta$-cell adaptation. Consequently, we estimate $d_0^{\prime }=0.05$ day⁻¹, $r_1^{\prime }=0.00087$ dl/mg/day, and $r_2^{\prime }=0.00000358$ dl²/mg²/day allowing $\beta$-cell to adapt within this fasting FA range.

Glucose and FA play crucial roles in metabolic disorders. Glucose and FA both influence $\beta$-cell proliferation and apoptosis (Topp et al., 2000; Oh et al., 2018). So, glucose and FA impact $\beta$-cell mass and functionality. We assume glucose and FA have almost similar effect on $\beta$-cell. To get a similar rate and pattern of $\beta$-cell mass dynamics response to incremental increases in plasma concentrations of glucose and FA, we estimate the parameters $d_0$, $r_1$, $r_2$ for glucose and $d_0^{\prime }$, $r_1^{\prime }$, $r_2^{\prime }$ for FA almost identically. The $\beta$-cell’s ability to adapt to these ranges is crucial for maintaining glucose homeostasis, which is consistently challenged by both glucose and FA fluctuations. Thus, the parameters and ranges being almost similar for both metabolites underscore the interconnectedness of glucose and lipid metabolism in $\beta$-cell function and adaptation.

For different individuals, the values of the parameters may differ. We assume parameters $\mu =2$ day⁻¹, $\eta =20,000$ mg²/dl², $\tau =1.8$ μU/ml/day, $\gamma =191.7$, and $n=3$ for the consistency of the dynamics of the simulated glucose, insulin, FA, and $\beta$-cell mass with realistic physiological responses.

We summarize the above in Table 1.

Table 1.

Parameters values and sources.

Parameters	Values	Normal ranges	Unit	References
$G_{meal}$	1310	(1285, 2357)a	mg/dl/day	Zello (2006) b
$HGP$	535	(428.57, 1571)	mg/dl/day	Martini (2004) b Huhtaniemi (2018) b
$KGP$	107	(107, 392.86)	mg/dl/day	Martini (2004) b Huhtaniemi (2018) b
$F_{meal}$	380	(256, 916)	mg/dl/day	Zello (2006) b Mansbach and Siddiqi (2016) b
$HFP$	112		mg/dl/day	Adiels et al. (2006) b
$AFP$	196		mg/dl/day	Mittendorfer et al. (2009) b
$S_g$	8		day−1	Jin et al. (2023)
$S_T$	3500	(3000, 4000)	step/day	Mayoclinicc
$\nu$	0.0031		mg/dl/step	Estimatedb
$\sigma$	43.2		μU/ml/day	Topp et al. (2000)
$\rho$	5		μU/ml/day	Ye (2007) b
$d_i$	432		day−1	Topp et al. (2000)
$\alpha$	20,000		mg2/dl2	Topp et al. (2000)
$\eta$	20,000		mg2/dl2	Estimatedb
$\mu$	2		day−1	Estimatedb
$\lambda$	0.45	(0.41, 1.35)	–	Reinehr et al. (2005) b
$d_0$	0.05		day−1	Weir and Bonner-Weir (2004) b
$r_1$	0.00087		dl/mg/day	Weir and Bonner-Weir (2004) b
$r_2$	3.600e–06		dl2/mg2/day	Weir and Bonner-Weir (2004) b
$d_0^{\prime }$	0.05		day−1	Jayanthi et al. (2017) b Arabi et al. (2019) b
$r_1^{\prime }$	0.00087		dl/mg/day	Jayanthi et al. (2017) b Arabi et al. (2019) b
$r_2^{\prime }$	3.5800e–06		dl2/mg2/day	Jayanthi et al. (2017) b Arabi et al. (2019) b
$\gamma$	191.7		–	Estimatedb
$n$	3		–	Estimatedb
$\tau$	1.8		μU/ml/day	Estimatedb

^aFor RDA carb diet $G_{meal}=892$ mg/dl/day.

^bFor details see Section 2.1.

^chttps://www.mayoclinic.org/healthy-lifestyle/fitness/in-depth/10000-steps/art-20317391 (accessed on Nov. 12, 2023).

3. Equilibria

In this section we determine the number of equilibria of the model (2.7), which correspond the homeostasis of glucose, insulin, FA, and $\beta$-cell mass.

Denote

$$G_{in}=G_{meal}+HGP+KGP-v{S}_T\text{and}{F}_{in}=F_{meal}+HFP+AFP.$$

Clearly, when $\beta =0$, the model (2.7) has an equilibrium point at $E_0\left(G_{in}/S_g,0,F_{in}/\mu ,0\right)$, which represents the diabetic state. The following theorem establishes its stability, and we omit the straightforward proof for brevity.

Theorem 1.

If $r_1/r_2<G_{in}/s_g$ and $r_1^{\prime }/r_2^{\prime }<F_{in}/\mu$, then the equilibrium point $E_0\left(G_{in}/S_g,0,F_{in}/\mu ,0\right)$ is asymptotically stable.

Remark

Given the range of the parameter values in Table 1, the conditions of Theorem 1 are not demanding.

When $\beta \ne 0$, additional equilibrium points must satisfy

$$S_I{I}^{*}{G}^{*}=G_{in}-S_g{G}^{*},$$ (3.1)

$$I^{*}=d_i^{-1}\left(\frac{\sigma {\left(G^{*}\right)}^2}{\alpha +{\left(G^{*}\right)}^2}+\frac{\rho {\left(F^{*}\right)}^2}{\eta +{\left(F^{*}\right)}^2}\right){\beta }^{*},$$ (3.2)

$$\lambda S_I{I}^{*}{F}^{*}=F_{in}-\mu F^{*},$$ (3.3)

$$d_0-r_1{G}^{*}+r_2{\left(G^{*}\right)}^2=-d_0^{\prime }+r_1^{\prime }{F}^{*}-r_2^{\prime }{\left(F^{*}\right)}^2.$$ (3.4)

From (3.1) and (3.3), we have that

$$G^{*}=\frac{\lambda G_{in}{F}^{*}}{F_{in}+\left(\lambda S_g-\mu \right)F^{*}}.$$ (3.5)

Replacing $G^{*}$ in (3.4) by (3.5), we have

$$H(F):=a{F}^4+b{F}^3+c{F}^2+dF+e=0$$ (3.6)

where

$$a=r_2^{\prime }{\left(\lambda S_g-\mu \right)}^2>0,b=2r_2^{\prime }{F}_{in}\left(\lambda S_g-\mu \right)-r_1^{\prime }{\left(\lambda S_g-\mu \right)}^2,$$

$$c=\left(d_0+d_0^{\prime }\right){\left(\lambda S_g-\mu \right)}^2-\left(\lambda S_g-\mu \right)\left(\lambda r_1{G}_{in}+2F_{in}{r}_1^{\prime }\right)+r_2{\lambda }^2{G}_{in}^2+r_2^{\prime }{F}_{in}^2,$$

$$d=F_{in}\left[2\left(d_0+d_0^{\prime }\right)\left(\lambda S_g-\mu \right)-\left(\lambda r_1{G}_{in}+r_1^{\prime }{F}_{in}\right)\right],e=F_{in}^2\left(d_0+d_0^{\prime }\right)>0.$$

To determine the number of positive roots of the quartic equation (3.6) and therefore the number of meaningful equilibrium points of the model (2.7), we need the following lemma that is credited to Gerolamo Cardano (Wikipedia, 2024).

Lemma 1.

Consider the general cubic equation

$$a{x}^3+b{x}^2+cx+d=0,$$

which can be reduced to a depressed cubic equation

$$y^3+py+q=0,$$

by the transformation $x=y-\frac{b}{3a}$, where $p=\frac{3ac-b^2}{3a^2}$ and $q=\frac{2b^3-9abc+27a^{2}d}{27a^3}$. If $p$, $q$ are real and discriminant of the depressed cubic equation $\text{Δ}=-\left(4p^3+\right.\left.27q^2\right)<0$, then the cubic equation has one real root and two complex roots. The real root of the depressed cubic equation is

$$y=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}.$$

Hence, $x=y-\frac{b}{3a}$ is the real root of the general cubic equation.

Theorem 2.

If $a$, $c$, $e>0$, $d<0$, and either

1. $3b^2-8ac<0$, or

2. $3b^2-8ac>0$, $b>0$, and $H^{\prime }\left(F_1\right)<0$,

then $H$ has none, one or two distinct positive zeros, where

$$F_1=-\left(3b+\sqrt{3\left(3b^2-8ac\right)}\right)/(12a).$$

Furthermore,

1. if $H\left(F_0\right)>0$, then $H$ has no positive zero and thus the model (2.7) has unique equilibrium point;

2. if $H\left(F_0\right)=0$, then $H$ has one positive zero with multiplicity two, and thus the model (2.7) has two equilibrium points with one is simple and the other is of multiplicity two;

3. if $H\left(F_0\right)<0$, then $H$ has two distinct positive zeros, and thus the model (2.7) has three equilibrium points,

where,

$$F_0=-\frac{3b}{12a}+\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}},$$

with

$$p=\frac{8ac-3b^2}{16a^2}\mathit{\text{and}}q=\frac{b^3-4abc+8a^{2}d}{32a^3}.$$

Proof.

See the appendix. □

In the next section, we numerically investigate bifurcations by respectively varying the parameters $G_{meal}$ and $F_{meal}$ to examine the effects of carbohydrate ingestion and FA intake on the progression of T2D. Our findings on the bifurcations are consistent with the conditions stated in Theorem 2 regarding the number of equilibrium points.

4. Numerical studies

In this section, we numerically analyze the dynamics of glucose, insulin, FA, and $\beta$-cell mass dynamics using our model (2.7) with the carefully determined parameter values and their ranges in Table 1. First, we perform bifurcation analysis to understand the effects of carbohydrate ingestion and FA intake on the evolving dynamics. Second, we conduct data fitting with longitudinal data from Pima Indian tribes (Mason et al., 2007), optimizing and estimating the initial conditions as well as the parameter values of $G_{meal}$ and $F_{meal}$. This analysis reveals several interesting observations in physiology and clinical practice. Finally, we utilize the model to study three feasible lifestyle changes in diet and/or physical activity to prevent or postpone the onset of T2D.

4.1. Bifurcation analysis

In this subsection, we select the environmental factors $G_{meal}$ and $F_{meal}$ as bifurcation parameters to delve into their variations and analyze the dynamics of glucose, insulin, FA levels, and $\beta$-cell mass, as well as the pathways to T2D. Using XPPAUTO (Ermentrout and Mahajan, 2003), and MATLAB^®, we have created bifurcation diagrams with respect to $G_{meal}$ and $F_{meal}$, as illustrated in Fig. 3.

Fig. 3.

Upper four subfigures: Bifurcation diagram of $G_{meal}\in [300,2500]$. Lower four subfigures: Bifurcation diagram of $F_{meal}\in [0,1000]$. The values of the other parameters generating these profiles are in Table 1.

As shown in Fig. 3, the system maintains a unique stable equilibrium $E_0$ for each $G_{meal}$ in the interval (300, 622.9). It exhibits bi-stability for $G_{meal}\in (622.9,2319)$, undergoing saddle-node bifurcations at the endpoints $P$ and $Q$. In this interval, the upper branch (in red) from the point $P$ to $Q$ represents the unstable equilibrium, while the other two branches in blue represent the stable equilibria, including the respective $E_0$ for each $G_{meal}$. When $G_{meal}>2319$, the system transitions back to a single stable equilibrium $E_0$. Similarly, the system shows one stable equilibrium $E_0$ as $F_{meal}$ varies in (0, 94.7942), and $F_{meal}>878.604$, while $F_{meal}$ varies in (94.7942, 878.604), the system undergoing bi-stability bifurcations with the saddle-nodes at $P$ to $Q$. The observed bifurcation could be characterized as imperfect bi-stability.

The bifurcation diagrams in Fig. 3(A, B) suggest that there exist “sweet spots”, respectively, for carbohydrate and fat intake, along with healthy lifestyle, so that individuals can maintain the blood sugar level in the normal range (lower branches between $P$ and $Q$, which represent the stable equilibrium point $G^{*}$). Beyond which, the system undergoes a qualitative change leading to $E_0$ – the onset of diabetes in long term. That is said, overeating carbohydrate and fat intake influences glucose regulation due to the dual role of glucotoxicity and lipotoxicity in damaging $\beta$–cells mass, and impairing insulin secretion and action. On the other hand, when $G_{meal}\in (300,622.9)$ mg/dl/day, an individual is considered undernourished. Research indicates that prolonged undernutrition can have long-term effects on insulin production and sensitivity, potentially elevating the risk of T2D. While insufficient food intake does not directly cause diabetes, it can disrupt metabolism, promote insulin resistance, and gradually increase the risk-particularly if dietary patterns change drastically later in life (Rao and Menon, 1993). Chronic undernutrition may lead to progressive $\beta$-cell dysfunction or heighten vulnerability to other diabetogenic factors, contributing to diabetes development (Rao, 1984). Furthermore, moderate to severe undernutrition during early life may alter glucose-insulin metabolism, raising the likelihood of developing type 2 diabetes in adulthood (van Abeelen et al., 2012). An individual consuming only 50 g of carbohydrates per day, equivalent to 357.14 mg/dl/day, is considered to be on a very low-carbohydrate diet, which is generally not recommended for long-term health (https://hopkinsdiabetesinfo.org/carbohydrate-goals/). Therefore, the case $G_{meal}<300$ is excluded from consideration in Fig. 3(A), as such extreme undernutrition may compromise long-term survival.

Within the optimal range of carbohydrate and fat intake, the body maintains metabolic stability. However, exceeding this range triggers a shift that makes the onset of T2D inevitable. Clinically, this reinforces the critical role of dietary management and early intervention. The presence of an unstable equilibrium underscores how even small changes can significantly impact long-term metabolic health.

4.2. Data fitting for the subjects from pima indian tribes reported by Mason et al. (2007)

According to the Centers for Disease Control and Prevention, the fasting glucose levels for normal, prediabetes, and diabetes stages are 70–99 mg/dl, 100–126 mg/dl, and above 126 mg/dl, respectively (https://www.cdc.gov/diabetes/basics/getting-tested.html, accessed 02/03/2023). Hammel et al. (2023) and Johnson et al. (2010) note that the normal fasting insulin level ranges from 15 to 25 μU/ml. In normal subjects, the total FA level remains below 100 mg/dl (Bermúdez-Cardona and Velásquez-Rodríguez, 2016). Furthermore, the average $\beta$-cell mass in a healthy human is almost 1000 mg (Saisho et al., 2013).

Longitudinal data tracking the progression of blood sugar levels, insulin, $\beta$-cell mass, and FA dynamics over several decades in individuals with T2D is rarely available. Mason et al. (2007) reported eleven datasets, each documenting 2-h glucose levels spanning three to four decades per subject. While recent studies have explored the longitudinal evolution of glucose and insulin, they do not track the same individuals over time but instead compile data from different individuals across various age groups. Consequently, although the dataset may not provide comprehensive information on fatty acid dynamics and other factors, it provides a foundation for in-silico exploration and hypothesis generation with carefully formulated model based on physiology. In this paper, these datasets are labeled Subject 1 to Subject 11 according to their order of appearance in Figs. 1(A–F) and 3(A–C, E,F) of (Mason et al., 2007) (noting that the data in (D) of both figures are identical). We applied the formula $FPG=\exp \{\exp (1.45+0.00055\times 2HPG)\}$ derived by Mohan et al. (2000) and converted the 2-h glucose levels to fasting glucose levels. The eleven datasets are plotted in Fig. 4 and can be classified into three groups: 1) Subject 1–4 and 7–9 progressed to T2D; 2) Subject 5 and 6 remain in the prediabetic state; and 3) Subject 11 has maintained a non-diabetic state.

Fig. 4.

Blood sugar data over several decades from Pima Indian tripe (adopted from Mason et al. (2007)).

Since the initial conditions of glucose, FA, insulin and $\beta$-cell mass, and the amount of daily intake of carbohydrate and fat may differ for different subjects, we estimated these parameters. We define a parameter space $\text{Θ}=\left\{G_{meal},F_{meal},G(0),I(0),\beta (0),F(0)\right\}\in {\mathbf{R}}_{+}^6$, where $G_{meal}$ and $F_{meal}$ represent daily carbohydrate and fat intake, respectively. The parameter estimation was performed using the particle swarm optimization and <monospace>fmincon algorithms in MATLAB^®.

Table 2 lists the optimized and estimated initial conditions for glucose, FA, insulin, $\beta$-cell mass, and daily carbohydrate and fat intake. The model outputs for all subjects 1–11 and a typical subject are presented in Figs. 5, 6, 7 and 8. Each profile is displayed in two subfigures: the upper subfigure illustrates glucose $(G)$, insulin $(I)$ and fatty acids $(FA)$, while the lower subfigure shows $\beta$-cell mass $\left(\beta \right)$ and insulin sensitivity $\left(S_I\right)$ in the lower figure.

Table 2.

Parameters values used for Pima Indian subject fit.

	$G_{meal}$ (mg/dl/day)	$F_{meal}$ (mg/dl/day)	G(0) (mg/dl)	I(0) (μU/ml)	F(0) (mg/dl)	$\beta \left(0\right)$ (mg)
Subject 1	2322	645	94	13	95	1468
Subject 2	2324	646	94	13	95	1472
Subject 3	2325	644	92	15	92	1450
Subject 4	2306.54	681.525	97.63	9.68	88.93	1798.96
Subject 5	2316.05	647.64	97.38	5.76	98.98	1799.48
Subject 6	2420.97	616.81	97.10	10.65	92.49	1798.68
Subject 7	2215.84	701.86	93.19	16.08	99.64	1799.9
Subject 8	2177.9	710.3	99.00	5	100	1585.1
Subject 9	2197.59	704.48	73.70	15.34	82.85	1542.04
Subject 10	2270.26	674.936	98.87	18.29	99.27	1599.9
Subject 11	1700	450	90	13	90	900

Fig. 5.

Left Panel: Profile of a typical normal subject with initial conditions $G(0)=90$, $I(0)=10$, $\mu \text{U}/dl$, $F(0)=90$, $\beta (0)=800$, $G_{meal}=1310$ and $F_{meal}=380$. Right Panel: Profile of the normal subject 11 with initial conditions $G(0)=90$, $I(0)=13$, $F(0)=90$, $\beta (0)=900$, $G_{meal}=1700$ and $F_{meal}=450$. Other parameter values are from Table 1. Upper figure: concentration of glucose (red), insulin (blue), FA (green). Lower figure: $\beta$-cell mass (magenta), and insulin sensitivity $S_I$ (black).

Fig. 6.

Simulations of the model (2.7) for Subject 1–4. Data show T2D had developed. Model simulations are in agreement to data. Model parameters are in Tables 2 and 1. Upper panel of each subfigure: dynamics of glucose (red), insulin (blue), FA (green), and fasting glucose data $\text{(}\circ \text{)}$ adapted from Mason et al. (2007). Lower panel of each subfigure: $\beta$-cell mass (magenta) and Longitudinal $S_I(t)$ (black).

Fig. 7.

Simulations of the model (2.7) for Subject 7–10. Data show T2D had developed. Model simulations are in agreement to data. Model parameters are in Tables 2 and 1. Upper panel of each subfigure: dynamics of glucose (red), insulin (blue), FA (green), and fasting glucose data $\text{(}\circ \text{)}$ adapted from Mason et al. (2007). Lower panel of each subfigure: $\beta$-cell mass (magenta) and longitudinal $S_I(t)$ (black).

Fig. 8.

Simulations of the model (2.7) for Subject 5 and 6. Data show the subjects were at prediabetes stage. Model simulations show ultimately T2D would develop. Model parameters are in Tables 2 and 1. Upper panel of each subfigure: dynamics of glucose (red), insulin (blue), FA (green), and fasting glucose data (∘) adapted from Mason et al. (2007). Lower panel of each subfigure: $\beta$-cell mass (magenta) and longitudinal $S_I(t)$ (black).

4.2.1. Model simulations for non-diabetic subjects

We perform numerical simulations of the model (2.7) for a typical normal subject using parameter values in Table 1, and display the model outcome in Fig. 5 (Left panel). This simulation shows that a normal subject’s fasting glucose and FA level rises but still remains in the normal range over a 100-year lifespan, aligning with clinical data for normal subjects (Bermúdez-Cardona and Velásquez-Rodríguez, 2016), in addition to that $\beta$-cell mass, insulin secretion, and insulin sensitivity decline along the age, which is in agreement to Szoke et al. (2008).

Among the data reported in Mason et al. (2007), Subject 11 is the only non-diabetic individual. The data indicates that this subject remained in a normal state till 60 years of age. Our simulation shown in Fig. 5 (Right panel) suggests that this normal state will be maintained, assuming no changes in lifestyle. The dynamics of all model state variables align with physiological and clinical observations: specifically, aging leads to an increase in glucose and FA levels, a decline in $\beta$-cell mass, and an elevated insulin secretion to compensate for the rise in blood sugar levels.

Moreover, normal adult insulin levels stay around the 10–25 μU/ml range, which is consistent with clinical levels for normal subjects (Cohen and Li, 2021; Hammel et al., 2023; Johnson et al., 2010; Yang et al., 2023). While specific longitudinal studies tracking basal insulin levels from youth to old age are limited, available research provides some insights. a) Szoke et al. (2008) indicates that insulin delivery rates decline by approximately 0.7 % per year, suggesting that aging intrinsically affects basal insulin secretion; b) Erdmann et al. (2008) observed in 10 healthy male subjects that even modest weight gain can lead to significant increases in fasting insulin concentrations, and hyperinsulinemia and insulin resistance.; c) on the other hand, Lieb et al. (2022) found that fasting insulin levels remained relatively stable over a 21-year period in a cohort of 2140 relatively healthy, non-diabetic participants. These studies suggest that, in the absence of metabolic disorders such as obesity or diabetes, insulin levels do not significantly change with age in normal subjects. Our numerical simulations for these two subjects in Fig. 5 also reflect the dynamics of $\beta$-cell mass and insulin sensitivity, falling within the real-life ranges identified by Saisho et al. (2013). This stability underscores the body’s ability to maintain glucose homeostasis throughout the aging process in healthy individuals.

As part of our numerical analysis, we calculate Pearson’s correlation coefficients among glucose, insulin, FA, and $\beta$-cell mass using data of the typical solution the model (2.7). As shown in Table 3, there is a strong positive correlation between glucose and FA levels, possibly suggesting a mutually reinforcing relationship.

Table 3.

Pearson’s correlation among glucose, insulin, FA and $\beta$-cell mass.

	Glucose	Insulin	FA	$\beta$-cell mass
Glucose	1	−0.7513368	0.9989444	−0.9869426
Insulin	−0.7513368	1	−0.7247444	0.8246777
FA	0.9989444	−0.7247744	1	−0.9809394
$\beta$–cell mass	−0.9869426	0.8246777	−0.9809394	1

4.2.2. Model simulations for diabetic subjects

The simulations for Subject 1–4 and 7–10 are demonstrated in Figs. 6 and 7. In each figure, the horizontal line at $G=126$ marks the diagnostic threshold for diabetes-glucose levels above this line indicate the onset of the disease. For all subjects 1–10 in Figs. 6, 7 and 8, the cyan strips across the starts of the transition from prediabetes to diabetes onset in the simulated glucose curves; while the yellow strips in the figures highlight the period from diabetes onset to the beginning of insulin compensation failure. It can be seen from these simulations that T2D development is associated with a decline in $\beta$-cell mass and a reduction in $S_I$ to very low levels, consistent with established medical knowledge. Additionally, FA levels increase as T2D develops, a common symptom of obesity (Boden, 2001; Kovacs and Stumvoll, 2005).

In the profiles for diabetic subjects 1–4 and 7–10, both data and model outcomes show several years in prediabes stage followed by the onset of diabetes. While insulin sensitivity $\left(S_I\right)$ declines, insulin compensation (by $\beta$-cell mass expansion) begins, which, more or less and sooner or later, leads to hyperglycemia. In the late stage of prediabetes, insulin sensitivity abruptly declines and $\beta$-cell mass further expands attempting to secrete more insulin for compensation. Such endeavor lasts 1–3 years followed by the onset of diabetes. Even after the onset, insulin compensation continues for a few more years before the $\beta$-cell mass abruptly decline.

4.2.3. Model simulations for pre-diabetic subjects

The data of the subjects 5 and 6 exhibit indicating a prediabetic state (see Fig. 8). While not all individuals with prediabetes will develop type 2 diabetes (T2D), the risk is substantial without intervention. Research suggests that 5 % to 10 % of individuals with prediabetes progress to T2D annually, and up to 70 % may eventually develop the disease over their lifetime if no lifestyle modifications are made (Nathan et al., 2007; Bardenheier et al., 2021). It is noted that progression rates can differ depending on factors such as age, sex, and the specific criteria used to define prediabetes (van Herpt et al., 2020). Bennasar-Veny et al. (2020) suggests that maintaining a healthy BMI and engaging in regular physical activity may help prevent the progression to T2D. Our simulations in Fig. 8 predict that these subjects will ultimately develop diabetes, likely due to the assumption that their lifestyle remains unchanged and no interventions are introduced.

Summary.

Interestingly, in the profiles of the healthy, non-diabetic subjects shown in Fig. 5, FA levels remain lower than glucose levels in their whole lifespans. In contrast, for the subjects 1–10 who ultimately developed T2D, all profiles in Figs. 6, 7 and 8 consistently display the opposite trend - FA levels initially stay below glucose levels before the onset of diabetes, then begin to rise sharply, tangled with the glucose levels, and eventually surpass glucose levels near the time of $\beta$-cell failure. Since then, FA levels become significantly higher than glucose levels. Notably, in the subjects 5 and 6, this crossing point - where FA levels exceed glucose levels - occurs noticeably later than that in Subjects 1–4 and 7–10. These patterns suggest that (i) elevated FA levels may contribute to rising glucose levels, reduced insulin sensitivity, and ultimately the development of hyperglycemia; (ii) the gradual variation between FA and glucose levels might have a deeper implications on the complex progression of T2D. This highlights elevated FA as a potential pathway and a biomarker for T2D progression.

Another notable observation from Subjects 3–9 is that insulin secretion begins to accelerate in the middle of prediabetes stage, while FA levels rise later, near T2D onset. This pattern aligns with recent meta-analytic findings showing that increases in fasting insulin precede weight gain, suggesting hyperinsulinemia, rather than obesity, as a primary driver of chronic disease outcomes (Wiebe et al., 2021). Our model outcomes are broadly consistent with this view, lending support to the causal perspective of Johnson (2021), while the profiles of the remaining subjects might align with the perspective of Esser et al. (2020, 2021).

4.3. Interventions

As shown, the model (2.7) robustly captures the dynamics of glucose, insulin, FA, $\beta$-cell mass, and insulin sensitivity – key factors in the development of T2D. We utilize this model extensively to explore realistic and feasible early interventions, such as dietary modifications and increased physical activity, aimed at preventing the onset of T2D. We illustrate such interventions using Subject 1 as an example; similar outcomes were observed for Subjects 2–10. The outputs are presented in a picture-in-picture format, with arrows indicating the zoomed-in regions.

4.3.1. Lifestyle changes in diet

Numerical simulations of the model (2.7) demonstrate that reducing food intake (carbohydrates and fats) by approximately 11 % during the prediabetes stage can effectively control the progression of T2D as shown in Fig. 9. This finding aligns with clinical recommendations that emphasize early lifestyle modifications to delay or prevent diabetes onset. Fig. 9 illustrates how glucose and FA dynamics influence $\beta$-cell mass over time. Without intervention, glucotoxicity and lipotoxicity accelerate $\beta$-cell decline, impairing insulin secretion and leading to hyperglycemia. In contrast, dietary intervention helps maintain insulin sensitivity $\left(S_I\right)$ and $\beta$-cell mass at optimal levels, ensuring that glucose and FA concentrations remain within the normal range. The inset highlights the critical transition period, showing that intervention results in a more gradual and controlled transition.

Fig. 9.

Solid line: no intervention. Dash line: approximately 11 % food intervention at prediabetes stage. Left Panel: dynamics of glucose (red), insulin (blue), FA (green), $\beta$-cell mass (magenta) and fasting glucose data $\text{(}\circ \text{)}$ of Subject 1 adapted from Mason et al. (2007). Right Panel: longitudinal $S_I(t)$.

To compare the effects of dietary interventions at different stages, we simulated meal reductions initiated either at the beginning of the prediabetic stage or at the onset of T2D as illustrated in Fig. 10. Blood glucose $(G)$ levels are shown in the left panel, while the corresponding insulin sensitivities $\left(S_I\right)$ are displayed in the right panel, using the same colors and line styles. The solid red curve represents glucose levels without any dietary intervention. The dashed and solid blue curves correspond to an 11 % dietary reduction, initiated at the beginning of the prediabetic stage and at diabetes onset, respectively. The dashed magenta curve represents glucose dynamics under a 25 % dietary reduction, initiated at the onset of diabetes.

Fig. 10.

Red: no intervention. Blue: approximately 11 % intervention on dietary intake of carbohydrates and fats. Magenta: approximately 25 % intervention on dietary intake of carbohydrates and fats. Solid line indicate diabetes and dash line indicate glucose in normal range. Left Panel: glucose concentration. Right panel: longitudinal $S_I(t)$.

These simulations demonstrate that even a modest (11 %) dietary reduction, if initiated early in the prediabetic stage (blue dashed curve), can effectively maintain blood glucose within the normal range and prevent the onset of T2D. In contrast, initiating the same reduction later-at the onset of diabetes (blue solid curve)-fails to reverse hyperglycemia. Achieving a similar glucose-lowering effect in the diabetic stage requires a more substantial intervention, such as a 25 % reduction (magenta dashed curve). This underscores that early intervention is a significantly more effective strategy. Clinically, this supports the role of dietary modification in reversing prediabetes, preventing or delaying the onset of T2D, and managing the disease. The graded response to varying levels of intervention suggests a dose-dependent relationship between dietary intake and metabolic regulation.

4.3.2. Lifestyle change in physical activity

Physical activity contributes to increase $S_I$ (Holloszy, 2005). On average, an American walks about 3000 to 4000 steps per day, roughly equivalent to 1.5 to 2 miles. This is significantly below the commonly recommended 10,000 steps per day for maintaining good health. A “sweet spot” for longevity somewhere between 6000 and 10,000 steps depending on lifestyle, occupation, age, and fitness habits (https://howdyhealth.tamu.edu/walking-how-many-steps-a-day/?utm_source=chatgpt.com). Sedentary individuals may take fewer than 2500 steps, while active people (especially those who exercise regularly) may exceed 10,000 steps daily. Anyhow, walking a less than 5000 steps per day is considered sedentary lifestyle. To compare the effect of PA, we simulate under two conditions: sedentary lifestyle $\left(S_T=3500\right)$ and active lifestyle $\left(S_T=10,000\right)$ from the early age. Throughout these simulations, we set the initial conditions $G(0)=94$, $I(0)=13$, $F(0)=95$ and $\beta (0)=1468$, and keep the daily intake of carbohydrates $G_{meal}=2322$ and fats $F_{meal}=645$ consistently, while other parameter values are specified in Table 1.

Shown in the upper panels in Fig. 11, where the red curves represent a sedentary lifestyle and the blue curves represent an active lifestyle, the simulations reveal a striking contrast. When $S_T=3500$, corresponding to a sedentary lifestyle, we observe a progressive decline in insulin sensitivity $S_I$ and $\beta$-cell mass, ultimately leading to the onset of T2D in Subject 1. In contrast, an active lifestyle $\left(S_T=10,000\right)$ effectively maintains both $S_I$ and $\beta$-cell mass at optimal levels, thereby keeping the blood sugar level under control. The lower panel compares the corresponding $\beta$-cell mass vs insulin sensitivity $\left(S_I\right)$: when T2D is developed, both $\beta$-cell mass and $S_I$ decline to a pathophysiological states; while if the subject remains healthy, they are stabilized at a optimized states. These simulation results support the hypothesis that adopting an active lifestyle from an early age can significantly delay or prevent the progression toward T2D by preserving insulin sensitivity and $\beta$-cell function.

Fig. 11.

Dynamics of Glucose (upper left panel) and $\left(S_I\right)$ (upper right panel) and $\beta$-cell mass and $\left(S_I\right)$ bottom panel (red $S_T=3500$ steps (a little physical activity)), (blue $S_T=10,000$ steps (with more physical activity)). Red curve shows compensation fail (T2D developed) and blue curve shows compensation success (T2D did not developed). More active lifestyle could control development of T2D, preserve $\beta$-cell mass and insulin sensitivity in the subject 1. The lower panel shows the corresponding $\beta$-cell mass vs insulin sensitivity $\left(S_I\right)$: when T2D is developed, both $\beta$-cell mass and $S_I$ decline to a pathophysiological states; while when the subject remains healthy, they are stabilized at a optimized states.

To further investigate the impact of physical activity intensity on blood glucose regulation, we simulated a scenario for Subject 1, who entered the prediabetic stage at age 45. We assumed that the subject maintained a sedentary lifestyle (3,500 steps per day) prior to age 45 and subsequently increased physical activity to 14,000 steps per day. As illustrated in Fig. 12, the solid lines represent the dynamics of glucose ($G$, red), insulin ($I$, blue), FA ($F$, green), and $\beta$-cell mass ($\beta$, magenta) under continued sedentary behavior, while the dashed lines depict the corresponding dynamics following the transition to an active lifestyle. The results show that this substantial increase in physical activity effectively prevented the onset of T2D. However, simulation (not shown) also reveals that if physical activity had only been increased to 10,000 steps per day at the prediabetic stage, it would not have been sufficient to maintain $\beta$-cell mass and insulin sensitivity at levels necessary to avert T2D progression.

Fig. 12.

Left Panel: Dynamics of glucose (red), Insulin (blue), FA (green), $\beta$-cell mass (magenta), fasting glucose data $\text{(}\circ \text{)}$ of Subject 1 adapted from Mason et al. (2007). Right Panel: Corresponding longitudinal $S_I(t)$ (orange). Solid lines are dynamics obtained when assuming PA $S_T=3500$ steps. Dashed lines are the dynamics after physical activity was changed to 14,000 steps at the age 45.

4.3.3. Lifestyle changes in both diet and physical activity

Naturally, a combination of both dietary control and enhancement of PA is an effective intervention strategy on preventing the onset of T2D.

This strategy comprised a nearly 6 % reduction in daily dietary intake (carbohydrates and fats) and an increase in physical activity (with daily steps (ST) escalating from 3500 to 10,000) from the age of 45 (identified as the prediabetes stage in Subject 1) and beyond. The outcomes displayed in Fig. 13 demonstrate that the dual approach of dietary modification and enhanced physical activity effectively maintained insulin sensitivity $\left(S_I\right)$ and $\beta$-cell mass at optimal levels. Consequently, this approach significantly postpones the progression towards T2D in Subject 1, while the intervention failed without reducing dietary intake as discused in 4.3.2.

Fig. 13.

Dynamics of glucose (red), Insulin (blue), FA (green), $\beta$-cell mass (magenta), fasting glucose data $\text{(}\circ \text{)}$ (left panel) and corresponding longitudinal $S_I(t)$ (right panel) with a ($S_T=3500$ steps, $G_{meal}=2322$ mg/dl/day and $F_{meal}=645$ mg/dl/day all time) little physical activity and more food intake (solid line) and more physical activity ($S_T=3500$ steps, $G_{meal}=2322$ mg/dl/day and $F_{meal}=645$ mg/dl/day up to 45 years and after 45 years $S_T=10,000$ steps, $G_{meal}=2250$ mg/dl/day and $F_{meal}=625$ mg/dl/day) (dash line). More Physical activity and 6 % food intervention together in prediabetes state can keep the blood sugar level under control, improve $S_I$, and preserve $\beta$-cell mass at optimum level in the subject 1.

5. Discussions

T2D, as a chronic disease, has become one of the major health issues nowadays. In this paper, we carefully extend a well accepted dynamical system model, Topp’s $GI\beta$ model, by incorporating FA as an explicit factor, and aim to investigate the role of FA in this metabolic regulation. This enhancement provides more insights for the development of T2D through an in silico study in the aspect of the effects of FA in the progression of T2D.

Model analysis reveals that FA dynamics are closely intertwined with glucose dynamics, a key driver in the development of T2D. Early interventions-such as dietary reduction and increased physical activity intensity-are critical for preventing or at least delaying the onset of T2D.

Insulin sensitivity-or its reciprocal, insulin resistance-is a complex physiological concept involving multiple organs and tissues, with varying strength across different regions of the body. While its root causes remain unclear, it is evidently influenced by glucose and insulin levels, which form the basis for widely used indices such as HOMA-IR and QUICKI. Moreover, the impact of FA on the dynamics of insulin sensitivity has been shown to be significant (Roden et al., 1996; Lovejoy, 1999; Sears and Perry, 2015). By incorporating FA as a dependent factor alongside glucose and insulin in the longitudinal formulation of insulin sensitivity, our model more accurately captures human physiology and the progression toward T2D.

A bistability bifurcation for a set of typical parameter values demonstrate that a possible “sweet spot” exists for food intake supplying adequate carbohydrate and fat for sustained and healthy blood sugar level in lifespan (refer to Section 4.1 and Fig. 3). That can be interpreted as – overeating influences glucose regulation due to the dual role of glucotoxicity and lipotoxicity in damaging $\beta$−cells mass, and impairing insulin secretion and action; while chronic undernutrition may lead to progressive $\beta$-cell dysfunction or heighten vulnerability to other diabetogenic factors, contributing to diabetes development. Our findings are in agreement with Rao (1984); Rao and Menon (1993); van Abeelen et al. (2012).

By fitting our model to the longitudinal data reported in Mason et al. (2007), several noteworthy observations emerge:

1. In the profiles of healthy and non-diabetic subjects shown in Fig. 5, FA levels remain consistently lower than glucose levels throughout their lifespan. In contrast, diabetic profiles in Figs. 6 and 7 show that FA levels initially stay below glucose levels but begin to rise and intertwined with glucose levels, and eventually surpass glucose levels around the time of $\beta$-cell failure.

2. In 7 out of 10 diabetic subjects, insulin secretion begins to accelerate during mid-prediabetes, while FA levels rise later-closer to the onset of T2D. Eventually, FA levels exceed glucose levels. (Refer to Sections 4.2.1, 4.2.2 and 4.2.3.)

3. The difference between glucose and FA levels exhibits an accelerating downward trend, shifting from positive to negative. (Refer to Sections 4.2.2 and 4.2.3.)

These patterns not only highlight the roles of glucotoxicity and lipotoxicity in accelerating $\beta$-cell decline-impairing insulin secretion and promoting hyperglycemia-but also suggest that annual monitoring of both fasting glucose and FA concentrations could provide valuable predictive insights into glycemic progression, enabling timely intervention when unfavorable trends emerge.

The trend of simulations for Subject 3–9 shown in Figs. 6 and 7 reveals certain consistency with what Wiebe et al. (2021) recently found: changes in fasting insulin levels precede changes in body weight in a large cohort. This implies that many adverse outcomes currently attributed to obesity may be driven by hyperinsulinemia or another proximate factor, which challenges the widely held belief that obesity is the primary driver of noncommunicable chronic diseases and premature death through elevated fasting insulin levels. Our model simulations revealing that, in 7 out of 10 subjects, insulin levels begin rising before FA levels during the transition from prediabetes to T2D.

We also simulated interventions involving reduced carbohydrate and fat intake, combined with increased physical activity intensity, applied at different stages-across the entire lifespan, from the onset of prediabetes, and from the diagnosis of T2D in Section 4.3. The simulation results indicate that, upon early signs of prediabetes, dietary adjustments and increased physical activity can effectively bring blood sugar levels under control or even reverse hyperglycemia. The optimal integration of dietary reduction and increased physical activity yields greater effectiveness. This underscores that early intervention is a significantly more effective strategy. Clinically, this supports the role of dietary modification in reversing prediabetes, preventing or delaying the onset of T2D, and managing the disease. The graded response to varying levels of intervention suggests a dose-dependent relationship between dietary intake and metabolic regulation.

Although our model simulations align well with findings reported by physiologists and clinicians, certain physiological details remain unaccounted for, as the model incorporates only the major factors. For simplicity, we model daily hepatic production of glucose and FA as constant parameters throughout the lifespan, even though these could realistically vary as functions of insulin, glucose, and/or FA levels. In addition, due to the unavailability of longitudinal data on $\beta$-cell mass-and limited time series data for insulin and FA-fitting the model precisely remains challenging. The dynamics of $\beta$-cell mass are influenced by both glucose (glucotoxicity) and FA (lipotoxicity), and we currently assume their effects follow quadratic polynomial relationships, as adopted from Topp’s model. However, exploring more accurate functional forms informed by additional endogenous signals would be a valuable direction for improving the model. Additionally, explicitly treating age as a spatial variable and formulating a hyperbolic partial differential equation (PDE) model will allow for a more detailed investigation into the effects of age on the progression of insulin resistance and type 2 diabetes. We pursue this study in future.

A more nuanced subclassification of T2D has recently emerged, aiming to reflect the underlying pathophysiology of diabetes. This approach seeks to deepen our understanding of the physiological mechanisms involved and supports precision diabetes care by targeting an individual’s specific pathophysiological profile, rather than focusing solely on glycemic control (Ahlqvist et al., 2018; Mezza et al., 2019; Misra et al., 2023). Thoughtful integration of related factors into our model might help uncover novel pathophysiological insights.

Last, but not the least, regarding to the two perspectives on the causal primacy of hyperglycemia proposed by Esser et al. (2020, 2021) and Johnson (2021) fundamentally differ: does excessive insulin secretion lead to insulin resistance, or does early $\beta$-cell dysfunction initiate the metabolic disturbances that culminate in T2D? This distinction is crucial for informing effective prevention and treatment strategies. While our model is not positioned to provide a definitive answer, fitting with limited available data has produced intriguing profiles and suggestive trends (Figs. 6 and 7). A more refined model incorporating additional physiological detail is needed and a larger cohort of data is required to enable deeper in silico exploration of this important question.

Appendix A. Proof of Theorem 2

Differentiating (3.6) thrice with respect to F, we get

$$H^{\prime }(F):=4a{F}^3+3b{F}^2+2cF+d$$ (A.1)

$$H^{″}(F):=12a{F}^2+6bF+2c$$ (A.2)

$$H^{‴}(F):=24aF+6b$$ (A.3)

Firstly, we will show that if $a$, $c$, $e>0$, $d<0$, and $3b^2<8ac$, then $H$ has zero, one or two distinct positive zeros. Since $\text{Δ}\left(H^{″}\right)={(6b)}^2-4\times 12a\times 2c<0$ i.e., $3b^2<8ac$ and $H^{″}(F)\to \infty$ as $F\to \pm \infty$, we know that $H^{″}(F)>0$ for all $F\in (-\infty ,\infty )$. Thus $H^{\prime }(F)$ increases for all $F\in (-\infty ,\infty )$. As $H^{\prime }(0)=d<0$, $H^{\prime }(F)$ increases for all $F\in (-\infty ,\infty )$. We can conclude that $\exists F_0>0$ such that $H^{\prime }\left(F_0\right)=0$ and $H^{\prime }(F)<0$ for $F<F_0$ and $H^{\prime }(F)>0$ for $F>F_0$. Therefore, $H(F)$ decreases for $F<F_0$, and $H(F)$ increases for $F>F_0$. This implies that $H(F)$ assumes its unique global minimum value at $F_0$. If the zeros of $H$ exist, these must be positive, because $H(0)=e>0$. Therefore, $H$ can have zero, one or two distinct positive zeros.

Secondly, we will show that if $a$, $c$, $e>0$, $d<0$, and $3b^2>8ac$, $b>0$, and $H^{\prime }\left(F_1\right)<0$, where $F_1=\left(-3b-\sqrt{3\left(3b^2-8ac\right)}\right)/(12a)$, then $H$ has zero, one or two distinct positive zeros. Now solving (A.2), we have $F_1=\left(-3b-\sqrt{3\left(3b^2-8ac\right)}\right)/(12a)$ and $F_2=\left(-3b+\sqrt{3\left(3b^2-8ac\right)}\right)/(12a)$. As $b>0$, so $F_1$, $F_2<0$. Plug in $F_1$ and $F_2$ in (A.3), we get $H^{‴}\left(F_1\right)=-2\sqrt{3\left(3b^2-8ac\right)}$ and $H^{‴}\left(F_2\right)=2\sqrt{3\left(3b^2-8ac\right)}$, since $3b^2-8ac>0$, so $H^{\prime }(F)$ has a maximum at $F=F_1$ and a minimum at $F=F_2$.

Since maximum $H^{\prime }\left(F_1\right)<0$ and $H^{\prime }(0)=d<0$. we have $H^{\prime }(F)<0$ for $F<0$. As $a>0$, we get $H^{\prime }(F)\to \infty$ as $F\to \infty$; and $H^{\prime }(0)=d<0$, so we can conclude that $\exists F_0>0$ such that $H^{\prime }\left(F_0\right)=0$, and $H^{\prime }(F)<0$ for $F<F_0$ and $H^{\prime }(F)>0$ for $F>F_0$. Therefore, $H(F)$ decreases for $F<F_0$, and $H(F)$ increases for $F>F_0$. This implies that $H(F)$ assumes its unique global minimum value at $F=F_0$. Moreover, we have $H(0)=e>0$, so if zeros of $H$ exist, these must be positive. Hence $H$ can have zero, one or two distinct positive zeros. Finally, using Lemma 1 we have $F_0=-\frac{3b}{12a}+\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$, which is the real root of $H^{\prime }(F)=0$, with $p=\left(8ac-3b^2\right)/\left(16a^2\right)$ and $q=\left(b^3-4abc+8a^{2}d\right)/\left(32a^3\right)$. Now, if $H\left(F_0\right)>0$, then $H(F)$ does not intersect the $F$–axis. So $H(F)$ has zero positive zero. If $H\left(F_0\right)=0$, then $H(F)$ touches the $F$–axis. So $H(F)$ has one positive zero with multiplicity two. If $H\left(F_0\right)<0$, then $H(F)$ intersects the $F$–axis at two points. So $H(F)$ has two positive zeros.

Data availability

No data was used for the research described in the article.