A review on fuzzy fractional order modeling in health systems with application to cardiovascular disease

Kottakkaran Sooppy Nisar; Muhammad Farman

doi:10.2478/ijmce-2026-0014

$A review on fuzzy fractional order modeling in health systems with application to cardiovascular disease Cover$

.blurhash-client-img { display: none !important; }

A review on fuzzy fractional order modeling in health systems with application to cardiovascular disease

International Journal of Mathematics and Computer in Engineering

AHEAD OF PRINT

By: Kottakkaran Sooppy Nisar and Muhammad Farman

Open Access

|Mar 2026

Figures & Tables

Simulation of the Model Compartments at fractional order γ = 1.00.

Simulation of the Model Compartments at fractional order γ = 0.95.

Simulation of the Model Compartments at fractional order γ = 0.90.

Simulation of the Model Compartments at fractional order γ = 0.85.

Simulation of the Model Compartments at fractional order γ = 0.80.

Simulation results of the model at different time points at γ = 0_80_

Time	S	H_A	H_C	I	T	R
0	500.0000	35.0000	20.0000	41.0000	15.0000	15.0000
10	4.2840	79.6565	29.7862	35.8800	22.7665	4.7281
15	4.5781	55.6059	21.4699	24.7168	18.1625	3.3692
20	4.7406	42.6288	17.4538	18.8152	14.6362	2.5637
25	4.8432	34.8770	15.0662	15.3228	12.0838	2.0453
30	4.9140	29.8506	13.4723	13.0666	10.2338	1.6909
35	4.9659	26.3744	12.3266	11.5074	8.8693	1.4370
40	5.0056	23.8453	11.4602	10.3720	7.8408	1.2480
45	5.0371	21.9297	10.7800	9.5105	7.0481	1.1028
50	5.0626	20.4309	10.2306	8.8352	6.4242	0.9884

Simulation results of the model at different time points at γ = 0_85_

Time	S	H_A	H_C	I	T	R
0	500.0000	35.0000	20.0000	41.0000	15.0000	15.0000
10	4.2201	68.5668	21.6898	30.3474	21.8069	4.0770
15	4.5978	44.1806	15.5613	19.2695	16.1936	2.6871
20	4.7872	32.3705	12.7564	14.0184	12.2355	1.9164
25	4.8998	25.9176	11.1314	11.1731	9.5961	1.4518
30	4.9743	22.0151	10.0638	9.4543	7.8252	1.1527
35	5.0270	19.4550	9.3054	8.3239	6.6074	0.9494
40	5.0663	17.6646	8.7370	7.5302	5.7442	0.8050
45	5.0967	16.3479	8.2941	6.9439	5.1132	0.6982
50	5.1208	15.3405	7.9387	6.4935	4.6384	0.6166

Parameters’ descriptions_

Parameter	Description
P	Rate at which individuals enter the population
β˜1{{\tilde \beta }_1}	Natural death rate in each compartment
β˜2{{\tilde \beta }_2}	Transition rate from chronic cases to heart-attack cases
β˜3{{\tilde \beta }_3}	Contact rate of S(t) with H_C(t)
β˜4{{\tilde \beta }_4}	Contact rate of S(t) with I(t)
β˜5{{\tilde \beta }_5}	Rate at which recovered individuals become susceptible again
β˜6{{\tilde \beta }_6}	H_A(t)’s treatment rate
β˜7{{\tilde \beta }_7}	H_C(t)’s treatment rate
β˜8{{\tilde \beta }_8}	I(t)’s treatment rate
β˜9{{\tilde \beta }_9}	Disease-induced death rate for infected individuals
β˜10{{\tilde \beta }_{10}}	Transition rate from chronic cases to infected cases
β˜11{{\tilde \beta }_{11}}	Rate at which treated individuals move to the recovered class
β˜12{{\tilde \beta }_{12}}	Disease-induced death rate for heart-attack individuals

Initial conditions and parameters’ values [100]_

Variable	Initial condition	Parameter	Value	Parameter	Value
S (0)	500	β˜1{\widetilde \beta _1}	0.2	β˜7{\widetilde \beta _7}	0.01
Ha (0)	35	β˜2{\widetilde \beta _2}	0.486	β˜8{\widetilde \beta _8}	0.09
Hc (0)	20	β˜3{\widetilde \beta _3}	0.1679	β˜9{\widetilde \beta _9}	0.019
I (0)	41	β˜4{\widetilde \beta _4}	0.0003	β˜10{\widetilde \beta _{10}}	0.1981
T (0)	15	β˜5{\widetilde \beta _5}	0.003	β˜11{\widetilde \beta _{11}}	0.001
P (0)	15	β˜6{\widetilde \beta _6}	0.001	β˜12{\widetilde \beta _{12}}	0.09
		P	5

Crisp and fuzzy bounds of all compartments for different fractional orders_

Fractional Order	Compartment	Crisp	Fuzzy Min	Fuzzy Max
1	S	3.9185 × 10¹⁰	1.6155 × 10¹⁰	8.0882 × 10¹⁰
1	H_A	3.2297 × 10⁶	2.0615 × 10⁶	4.6590 × 10⁶
1	H_C	0	0	0
1	I	1.3390 × 10⁶	8.8027 × 10⁵	1.8464 × 10⁶
1	T	65006	40905	94657
1	R	1812.4	1449.9	2174.9
0.95	S	2.4488 × 10¹⁰	1.0091 × 10¹⁰	5.0562 × 10¹⁰
0.95	H_A	2.3017 × 10⁶	1.4691 × 10⁶	3.3202 × 10⁶
0.95	H_C	0	0	0
0.95	I	9.4477 × 10⁵	6.2001 × 10⁵	1.3074 × 10⁶
0.95	T	46352	29171	67488
0.95	R	1226.2	980.94	1471.4
0.9	S	1.5186 × 10¹⁰	6.2546 × 10⁹	3.1366 × 10¹⁰
0.9	H_A	1.6308 × 10⁶	1.0410 × 10⁶	2.3526 × 10⁶
0.9	H_C	0	0	0
0.9	I	6.6342 × 10⁵	4.3469 × 10⁵	9.2108 × 10⁵
0.9	T	32865	20687	47845
0.9	R	815.41	652.33	978.49
0.85	S	9.3413 × 10⁹	3.8451 × 10⁹	1.9302 × 10¹⁰
0.85	H_A	1.1486 × 10⁶	7.3316× 10⁵	1.6569 × 10⁶
0.85	H_C	0	0	0
0.85	I	4.6350 × 10⁵	3.0326 × 10⁵	6.4540 × 10⁵
0.85	T	23166	14585	33720
0.85	R	531	424.8	637.2
0.8	S	5.6979 × 10⁹	2.3438 × 10⁹	1.1779 × 10¹⁰
0.8	H_A	8.0394 × 10⁵	5.1316× 10⁵	1.1597 × 10⁶
0.8	H_C	0	0	0
0.8	I	3.2208 × 10⁵	2.1045 × 10⁵	4.4967 × 10⁵
0.8	T	16230	10221	23621
0.8	R	336.82	269.46	404.19

Simulation results for the model compartments over time at γ = 1_

Time	S	H_A	H_C	I	T	R
0	500.0000	35.0000	20.0000	41.0000	15.0000	15.0000
10	5.5083	31.0891	4.0791	12.4312	17.2906	2.0702
15	5.3049	12.8700	4.4243	4.8926	8.5515	0.7879
20	5.3277	8.6353	4.3934	3.2770	4.3704	0.3041
25	5.3249	7.6447	4.3965	2.9288	2.6253	0.1204
30	5.3252	7.4128	4.3961	2.8537	1.9401	0.0505
35	5.3252	7.3585	4.3961	2.8374	1.6790	0.0239
40	5.3252	7.3457	4.3960	2.8339	1.5812	0.0137
45	5.3252	7.3427	4.3960	2.8331	1.5449	0.0099
50	5.3252	7.3420	4.3960	2.8330	1.5315	0.0084

Simulation results of the model compartments over time at γ = 0_90_

Time	S	H_A	H_C	I	T	R
0	500.0000	35.0000	20.0000	41.0000	15.0000	15.0000
10	4.2601	56.7490	14.7438	24.5743	20.5965	3.4152
15	4.7013	33.0526	10.8143	14.0625	13.9289	2.0238
20	4.8947	23.1164	9.1128	9.7642	9.6698	1.3170
25	5.0016	18.3131	8.1476	7.7023	7.1129	0.9273
30	5.0687	15.6690	7.5216	6.5637	5.5649	0.6969
35	5.1143	14.0470	7.0812	5.8597	4.5964	0.5516
40	5.1471	12.9635	6.7537	5.3851	3.9646	0.4546
45	5.1717	12.1910	6.5003	5.0438	3.5339	0.3865
50	5.1907	11.6123	6.2981	4.7864	3.2277	0.3366

Simulation results of the model compartments over time γ = 0_95_

Time	S	H_A	H_C	I	T	R
0	500.0000	35.0000	20.0000	41.0000	15.0000	15.0000
10	4.5025	44.2874	8.9331	18.6090	19.1061	2.7453
15	4.9252	22.5143	7.1270	9.2256	11.3788	1.3878
20	5.0725	15.1313	6.3753	6.1570	7.0169	0.7761
25	5.1470	12.2051	5.9528	4.9537	4.7488	0.4813
30	5.1906	10.8292	5.6811	4.3821	3.5689	0.3289
35	5.2188	10.0704	5.4914	4.0606	2.9322	0.2443
40	5.2381	9.5947	5.3513	3.8550	2.5688	0.1935
45	5.2521	9.2674	5.2436	3.7112	2.3472	0.1608
50	5.2626	9.0269	5.1581	3.6045	2.2029	0.1383

References

Authors

Metrics

Articles in this issue

DOI: https://doi.org/10.2478/ijmce-2026-0014 | Journal eISSN: 2956-7068

Journal RSS Feed

Language: English

Submitted on: Jan 15, 2026

Accepted on: Feb 20, 2026

Published on: Mar 18, 2026

Published by: Harran University

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Human health,

heart diseases,

mathematical modeling,

health and wellness,

myocardial infarction

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Introductions and overviews,

Physics,

© 2026 Kottakkaran Sooppy Nisar, Muhammad Farman, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

AHEAD OF PRINT

A review on fuzzy fractional order modeling in health systems with application to cardiovascular disease

Figures & Tables

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Fig. 16

Fig. 17

Fig. 18

Fig. 19

Simulation results of the model at different time points at γ = 0_80_

Simulation results of the model at different time points at γ = 0_85_

Parameters’ descriptions_

Initial conditions and parameters’ values [100]_

Crisp and fuzzy bounds of all compartments for different fractional orders_

Simulation results for the model compartments over time at γ = 1_

Simulation results of the model compartments over time at γ = 0_90_

Simulation results of the model compartments over time γ = 0_95_

Paradigm

My account