Fig 1.
A schematic diagram illustrating the diabetes model with five compartments: S denotes susceptible individuals, D represents individuals with diabetes, corresponds to those receiving non-pharmacological treatment,
indicates individuals under pharmacological treatment and R stands for those who have controlled or restrain.
Fig 2.
2D phase portraits showing the stability of the equilibrium point for the pairs (a)
and (b) (D, R). Each trajectory is initiated from a different initial condition, and all trajectories converge to the same equilibrium point.
This indicates local asymptotic stability of E1 in these two-dimensional projections.
Fig 3.
3D phase portraits illustrating the convergence of trajectories toward the equilibrium point for the combinations (a) (R,
), (b)
, (c)
and (d)
.
Each plot displays ten trajectories starting from different initial conditions, with TN = 0 initially, consistent with . The consistent convergence in all subplots confirms the asymptotic stability of E1.
Table 1.
Model parameters, their descriptions, values, and sources.
Fig 4.
2D phase portraits showing the stability of the endemic equilibrium point for the variable pairs (a)
, and (b) (D, R).
Each trajectory starts from a different initial condition and converges to the same equilibrium point, indicating asymptotic stability in these phase planes.
Fig 5.
3D phase portraits illustrating the convergence of trajectories toward the endemic equilibrium point for various variable combinations: (a)
, (b)
, (c)
, and (d)
.
Ten trajectories are shown in each plot, each starting from a different initial condition. The consistent convergence across all plots supports the asymptotic stability of E2.
Fig 6.
The diabetes model is fitted to the number of new cases, with a 95% confidence interval.
Yearly diabetes data from the United States, covering the years 2000 to 2022, is used.
Fig 7.
Partial Rank Correlation Coefficients (PRCC) for various model parameters with respect to: (a) individuals with diabetes, (b) individuals undergoing non-pharmacological treatment, (c) individuals undergoing pharmacological treatment, and (d) restrained individuals.
Fig 8.
Contour plots showing the effect of parameter combinations on the diabetic population D(t): (a) and (b)
.
Fig 9.
Contour plots showing the combined effect of the transmission rate and relapse parameters on the diabetic population D(t).
(a) , where ω is the relapse rate from Restrain back to Diabetes, (b)
, where
is the failure rate of non-pharmacological treatment and (c)
, where
is the failure rate of pharmacological treatment.
Fig 10.
Time variation plots showing the effect of key parameters on individuals with diabetes: (a) the baseline rate at which individuals develop diabetes , (b) risk factors
and
and (c) treatment rates
and
.
Fig 11.
Effect of treatment initiation rates (,
) and restrain rates (
, γ) on restrained compartment.
Fig 12.
Effect of relapse and treatment failure (
) on restrained individuals and diabetic prevalence.
Table 2.
Spectral radius ρ for different step sizes and numerical schemes.
Table 3.
Convergence behavior (C: Convergent, D: Divergent) of each method for different step sizes.
Fig 13.
Comparison of different numerical techniques for various step sizes (h) (a) , (b)
, (c)
, (d)
, (e)
and (f)
in simulating the dynamics of individuals with diabetes.
Fig 14.
Comparison of different numerical techniques for various step sizes (h) in simulating the dynamics of individuals: (a) under non-pharmacological treatment, (b) under pharmacological treatment, and (c) restrained individuals.
Fig 15.
Comparison of NSFD techniques for various step sizes (h): (a) , (b)
, and (c)
in simulating the dynamics of the entire population.
Table 4.
Comparison of error and convergence rates for Euler, RK4, and NSFD methods.
Fig 16.
Effect of individual control strategies on diabetes dynamics: (a) shows the variation in the diabetic population under different strategies, with Strategy 1 resulting in the lowest diabetes burden; (b) illustrates the restrained population, where Strategy 3 leads to the highest number individuals in the restrained (controlled) state.
Fig 17.
Control profiles corresponding to the three strategies.
The control u2 (non-pharmacological treatment) is maintained over a longer period, highlighting its sustained importance in effective diabetes management.
Fig 18.
IAR and ACER Analysis for Scenario-I.
Table 5.
ICER(Incremental cost-effectiveness ratio) for scenario I.
Table 6.
ICER (Incremental cost-effectiveness ratio) for scenario I.
Fig 19.
Effect of two combined control strategies on diabetes dynamics: (a) depicts the diabetic population, with Strategy 4 yielding the lowest number of cases; (b) shows the restrained population, where Strategy 6 achieves the highest restrain level.
Fig 20.
Control profiles corresponding to the three strategies.
Fig 21.
IAR and ACER Analysis for Scenario-II.
Table 7.
ICER(Incremental cost-effectiveness ratio) for scenario II.
Table 8.
ICER(Incremental cost-effectiveness ratio) for scenario II.
Fig 22.
Effect of Strategy 7 (all controls applied) on diabetes and restrain dynamics compared to the no-control scenario.
Fig 23.
Contour profiles of control measures for Strategy 7 over time.
Table 9.
ICER(Incremental cost-effectiveness ratio) for scenario III.
Table 10.
Incremental Cost-Effectiveness Ratio (ICER) for optimal strategies across scenarios.
Table 11.
Incremental Cost-Effectiveness Ratio (ICER) for optimal strategies across scenarios.