A Comparative Analysis of Harmonic Mean, Holling Type II, Beddington–DeAngelis, and Crowley–Martin Incidence Rates of a Piecewise Dengue Fever Dynamics Model
Abstract
1. Introduction
2. Description of the Model
- The natural death rate of humans;
- The natural death rate of mosquitoes;
- The mosquito-to-human transmission probability per bite;
- The human-to-mosquito transmission probability per bite;
- Recovery rate of humans;
- Recruitment rate of mosquitoes;
- The human incubation rate (inverse of latent period);
- The mosquito extrinsic incubation rate (inverse of latent period);
- The mosquito biting rate.
- Harmonic Mean type: This formulation assumes transmission is limited by the less abundant interacting population.
- Holling Type II (Saturated): Proposed by Holling (1959) [7], this rate represents saturation in the infection process, e.g., due to limited mosquito biting capacity or handling time, as the density of infectious individuals ( or ) increases. The form for mosquito-to-human transmission is
- Beddington–DeAngelis (B–D) type: Introduced independently by Beddington et al. and DeAngelis et al. (1975) [8], this rate incorporates density dependence related to both susceptible and infectious populations. The form for mosquito-to-human transmission is
- Crowley–Martin type: Proposed by Crowley and Martin [38], this form models mutual interference among both interacting populations. The form for mosquito-to-human transmission is
Crossover Behavior
- is the crossover point, which presents a transition in the governing dynamics from classical to fractional.
- is the classical derivative of a function
- is MABC fractional derivative defined as
3. Mathematical Properties of the PMABC Model (1)
3.1. Lipschitz Property of the Kernels
3.2. Existence of Unique Solutions
3.3. Boundedness and Positivity of Model (1)
- For , the equation is . Since we have shown , the inflow term . This leads to the inequality , which, by the same argument as for , implies .
- For , the equation is . With , we get , which implies .
- Similarly, for the vector compartments, having established and :
- -
- The equation for gives the inequality , implying .
- -
- With , the equation for gives , implying .
4. Comparative Analysis of Nonlinear Incidence Models
4.1. Dengue Model with Harmonic Mean Type Incidence Rate
4.1.1. Equilibrium Points and Basic Reproduction Number
4.1.2. Stability of DFE Point
4.2. Dengue Model with Holling Type II Incidence Rate
4.2.1. Positivity and Boundedness of Model (11)
4.2.2. Equilibrium Points and Basic Reproduction Number
4.2.3. Stability of the DFE Point
4.3. Dengue Model with Beddington–DeAngelis Type Incidence Rate
4.3.1. Positivity and Boundedness
4.3.2. Equilibrium Points and Basic Reproduction Number
4.3.3. Stability of DFE Point
4.4. Dengue Model with Crowley–Martin Type Incidence Rate
4.4.1. Positivity and Boundedness
4.4.2. Stability of Disease-Free Equilibrium Point and Basic Reproduction Number
5. Sensitivity Analysis
5.1. Model 1: Harmonic Mean Incidence Rate
- For :
5.2. Model 2: Holling Type II Incidence Rate
5.3. Model 3: Beddington–DeAngelis Incidence Rate
5.4. Model 4: Crowley–Martin Incidence Rate
6. Numerical Scheme of PMABC Dengue Fractional Model
7. Numerical Simulations and Discussion
8. Conclusions
- Model Calibration and Validation: Applying the framework to real-world dengue incidence data from a specific region to estimate parameters and validate predictions.
- Multi-Strain Dynamics: Extending the model to include the co-circulation of multiple dengue serotypes, which is crucial for understanding antibody-dependent enhancement.
- Spatial Heterogeneity: Incorporating spatial dynamics to model transmission between different geographical areas (e.g., urban vs. rural).
- Stochastic Formulation: Developing a stochastic version of the model to account for random fluctuations and generate prediction confidence intervals.
- Sensitivity to : Performing a rigorous sensitivity analysis of the crossover point to quantify the impact of intervention timing.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Parameter | Numerical Estimation | Unit | Ref |
---|---|---|---|
per month | Estimated | ||
per month | [42] | ||
per month | [43] | ||
per month | [37] | ||
Dimensionless | Estimated | ||
Dimensionless | [37] | ||
per month | Estimated | ||
per month | [44] | ||
b | per month | Estimated |
Parameter (ℓ) | Model 1 (Harmonic Mean) | Model 2 (Holling II) | Model 3 (Beddington–D.) | Model 4 (Crowley–Martin) |
---|---|---|---|---|
b | ||||
0 |
Feature | Model 1: Harmonic Mean | Model 2: Holling Type II | Model 3: Beddington–DeAngelis | Model 4: Crowley–Martin |
---|---|---|---|---|
Incidence Rate Form | ||||
Conceptual Meaning | Assumes transmission rate is limited by the less abundant population (human or vector). Simple interaction form. | Transmission saturates as the number of infectious individuals increases (vectors or humans ). Represents limited biting capacity or contact rate. | Transmission affected by density of both susceptibles (e.g., prevention ) and infectives (e.g., interference ). | Transmission reduced by handling time/interference effects from both susceptible and infectious populations independently. |
BRN () | Formula involves and death/incubation rates. Simpler structure. Section 5.1) | Similar parameters but structurally simpler than B-D/C-M. No saturation terms () appear at DFE evaluation. ) | Includes terms and in denominator, reflecting inhibition/prevention effects at DFE state. | Includes terms and in denominator, structurally similar to B-D at DFE evaluation. |
Key Differences in Formula | Simplest form. Directly proportional to . | Identical formula to Harmonic Mean in this derivation (linearization at DFE removes saturation terms). | explicitly reduced by baseline prevention/interference coefficients () related to total susceptible populations at DFE. | explicitly reduced by baseline interference coefficients () related to total susceptible populations at DFE. Identical form to B-D’s . |
Sensitivity Analysis | Not sensitive to . High sensitivity to (−0.64). | Highest sensitivity to (−1.14). Sensitive to (0.5). | Highest sensitivity to (−1.14). Sensitive to (0.5). Includes sensitivity to (not shown in Table 2 but implied by ). | Highest sensitivity to (−1.14). Sensitive to (0.5). Includes sensitivity to (not shown in Table 2 but implied by ). |
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Damag, F.H.; Qurtam, A.A.; Almalahi, M.; Aldwoah, K.; Adel, M.; Abd El-Latif, A.M.; Hassan, E.I. A Comparative Analysis of Harmonic Mean, Holling Type II, Beddington–DeAngelis, and Crowley–Martin Incidence Rates of a Piecewise Dengue Fever Dynamics Model. Fractal Fract. 2025, 9, 400. https://doi.org/10.3390/fractalfract9070400
Damag FH, Qurtam AA, Almalahi M, Aldwoah K, Adel M, Abd El-Latif AM, Hassan EI. A Comparative Analysis of Harmonic Mean, Holling Type II, Beddington–DeAngelis, and Crowley–Martin Incidence Rates of a Piecewise Dengue Fever Dynamics Model. Fractal and Fractional. 2025; 9(7):400. https://doi.org/10.3390/fractalfract9070400
Chicago/Turabian StyleDamag, Faten H., Ashraf A. Qurtam, Mohammed Almalahi, Khaled Aldwoah, Mohamed Adel, Alaa M. Abd El-Latif, and E. I. Hassan. 2025. "A Comparative Analysis of Harmonic Mean, Holling Type II, Beddington–DeAngelis, and Crowley–Martin Incidence Rates of a Piecewise Dengue Fever Dynamics Model" Fractal and Fractional 9, no. 7: 400. https://doi.org/10.3390/fractalfract9070400
APA StyleDamag, F. H., Qurtam, A. A., Almalahi, M., Aldwoah, K., Adel, M., Abd El-Latif, A. M., & Hassan, E. I. (2025). A Comparative Analysis of Harmonic Mean, Holling Type II, Beddington–DeAngelis, and Crowley–Martin Incidence Rates of a Piecewise Dengue Fever Dynamics Model. Fractal and Fractional, 9(7), 400. https://doi.org/10.3390/fractalfract9070400