J. Nig. Soc. Phys. Sci. 5 (2023) 1464 Journal of the Nigerian Society of Physical Sciences Numerical Analysis of Fractional-Order Dynamic Dengue Disease Epidemic in Sudan Fathelrhman EL Gumaa,b, Ossama M. Badawyc, Mohammed Berira,d, Mohamed A.Abdoond,∗ aDepartment of Mathematics, Faculty of Science and Arts, Albaha University, Albaha, Saudi Arabia bDepartment of statistical Study, Peace University, Sudan cDepartment of biology, Faculty of Science and Arts, Albaha University, Albaha, Saudi Arabia dDepartment of Mathematics, Faculty of Science, Bakht Al-Ruda University, Duwaym, Sudan Abstract The main idea of this work is numerical simulation and stability analysis for the fractional-order dynamics of the dengue disease outbreak in Sudan. This research uses a computer technique based on the Adams-Bashforth approach to numerically resolve a fractional-order dengue epidemic in Sudan. Analyses of numerical and dynamic stability show that the fractional-order dengue fever model is sensitive to initial conditions for those parameters. Therefore, the parameters’ values are critical in establishing how many individuals will get better from their sickness and how many will become ill. The proposed method is effective in providing an illustration of the solution’s dynamics over a very long horizon of time, which is crucial for making accurate predictions about the spread of dengue in Sudan. In addition, this method can be utilized to assess the efficacy of various intervention strategies and inform public health policies aimed at reducing the burden of dengue fever in Sudan. It can also assist in identifying areas most susceptible to dengue infestations and prioritizing disease control resources. DOI:10.46481/jnsps.2023.1464 Keywords: A fractional, Adams-Bashforth, Prediction, simulation, Dengue. Article History : Received: 25 March 2023 Received in revised form: 06 May 2023 Accepted for publication: 10 May 2023 Published: 28 May 2023 c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: B. J. Falaye 1. Introduction Dengue fever is a problem in Sudan. In 2010, 2013, and 2017, there were several outbreaks that were recorded. There is no information on the serotypes of dengue virus that are present in Sudan. In this regard, further research on the dengue virus (DENV) is required. In addition to the chikungunya, rift valley fever, malaria, and cholera outbreaks that are still active, there is currently an outbreak of dengue fever. The public health ∗Corresponding author tel. no: 00966535828039 Email address: fguma@bu.edu.sa (Fathelrhman EL Guma) sector’s capability for epidemic control and response is con- strained; long-running political and civil disputes have lowered the nation’s ability to do so. Dengue has spread to seven states nationwide since the outbreak began on August 8th. A poor prognosis could result from the increased probability of co- infection with malaria and/or chikungunya, which complicates case management [1-8]. Fractional calculus and nonlinear equations have found ap- plications in a vast variety of seemingly unrelated sectors of sci- ence and engineering over the last decade. Epidemics, acous- tics, biology, electromagnetics, engineering, Dengue is a signif- icant public health concern in tropical and subtropical regions. 1 Guma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1464 2 It is transmitted by mosquitoes of the species Aedes aegypti and Aedes albopictus. Four distinct serotypes can induce dengue illness. When a person infected with one serotype recovers, he or she develops complete immunity to that serotype but only partial and temporary immunity to the other [9-11]. In this paper, the fractional-order dengue epidemic model is investi- gated. Examining the stability of equilibrium locations Given are the numerical solutions for this model. We like to con- tend that fractional-order equations are more appropriate than integer-order equations for modeling memory-sensitive biolog- ical, economic, and social systems (generally complex adaptive systems). The Adams-Bashforth algorithm was used to solve and simulate the differential equation system. The format is as follows: The fractional model is introduced in Section 1. Section 2 provides fundamental definitions; Sec- tion 3 is devoted to the formulation model. Section 4 focuses on the Caputo derivative of the model while Sections 5 and 6 include stability analysis and numerical analysis. Section 7 is the conclusion. 2. Preliminaries Definition 2.1. The Riemann-Liouville fractional integral op- erator of order α > 0 for a function y(τ) is given by [12] : Dαy(t) := 1 Γ(n −α) ∫ t 0 (t −τ)n−α−1yn(τ)dτ. = In−αyn(t), t > 0. (1) Definition 2.2. For y�H1(0, t), t > 0, T > 0,α�(0, 1] Then the CF fractional operator [12] is given by. CF 0 D α t y(t) := B(α) 1 −α d dt ∫ t 0 y(τ) exp ( −α t −τ 1 −α ) dτ.0 < α < 1 (2) In this expression B(α) satisfies the condition B(0) = B(1) = 1. Definition 2.3. The fractional integral of order α of a function f is defined as [13] Iγt y(t) := 2(1 −γ) (2 −γ)M(γ) y(t) + 2(1 −γ) (2 −γ)M(γ) ∫ t 0 y(τ)dτ.t ≥ 0, 0 < α < 1 (3) Definition 2.4. Caputo derivative of order 0 ≤ n − 1 < α < n with the lower limit zero for a function y(τ) is given by [14] : Iαy(t) := 1 Γ(α) ∫ t 0 (t −τ)α−1y(τ)dτ.t > 0 (4) Definition 2.5. The Mittag-Leffler function is a generalization of the exponential function. This function can be expressed as follows: Eα(t) = ∞∑ k=0 tk Γ(αk + 1) (5) Definition 2.6. For y�H1(0, t), t > 0, T > 0,α�(0, 1] then the AB fractional operator [12] y(t) in the Riemann-Liouville is given by: AB 0 D α t y(t) := B(α) 1 −α d dt ∫ t 0 y(τ)Eα ( α 1 −α (t −τ)α ) dτ.0 < α < 1. (6) In this expression B(α) satisfies the condition B(0) = B(1) = 1. Definition 2.7. For y�H1(0, t), t > 0, T > 0 Then the AB frac- tional operator [12] y(t) in the Caputo sense is given by: AB 0 D α t y(t) := B(α) 1 −α ∫ t 0 dy(τ) dτ Eα ( α 1 −α (t −τ)α ) dτ.0 < α < 1 (7) In this expression B(α) satisfies the condition B(0) = B(1) = 1. Definition 2.8. Let 0 < α < 1 and the fractional CF derivative is expressed as: CF 0 D α t y(t) = h(t). (8) 3. The Model Formulation In this part, we get the mathematical formulation of the dengue fever infectivity model, which is based on a number of propositions, including the following: The total number of hu- mans (Nh) and mosquitoes (Nm) is assumed to be constant, the birth and mortality rates are assumed to be equal, the births in mosquito and human populations in each class are assumed to be the same, each individual in the population is likely to have the same number of mosquito bites, and the infected mosquito is likely to bite each component of the data provided. The vari- ables used in the dengue fever illness model are listed in Table 1. The preceding model can be interpreted mathematically as a host-vector interaction model, which is the following fractional differential model:  dS h dt = µh Nh − ( Bβmh Ih Nh + µh ) S h, dIh dt = Bβmh S h Ih Nh − (γh + µh) Ih, dRh dt = γh Ih −µhRh, dS m dt = µm Nm − ( Bβmh Ih Nh + µm ) S m, dIm dt = Bβmh Ih Nh S m −µm Im (9) we extend the model (9) employing the newly proposed Caputo-Fabrizio-Caputo; fractional derivatives with variable order α(t). 4. The Caputo- derivative to the fractional model Introducing the notion of fractional derivative in the sense of Riemann-Liouville to reformulate the dynamics of the clas- sical model (9) in terms of fractional derivatives, we apply the Caputo derivative to the dengue fever model (9). The fractional 2 Guma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1464 3 Table 1. Explanation of the components of the infectivity model: Variable Description Nh(t) Total number of humans (constant ≈ 44 million) S h(t) Susceptible humans Ih(t) Infected humans Rh(t) Recovery humans S m(t) Susceptible female mosquitoes Im(t) Infected female mosquitoes Parameter Description µ(h) Human mortality rate for every person µ(m) Corresponding value for the mosquitoes γ(h) Recovery rate of the humans B The biting rate β(m) The likelihood that human to mosquito transfer may occur β(h) The likelihood that mosquito to human transfer may occur dengue fever is obtained by replacing the classical derivative by the operator C0 D α(t) t : C 0 D λ 0 S h = θ1 − (θ2 + θ3) S h, C 0 D λ 0 Ih = θ2S h − (θ4 + θ3) Ih, 0 D λ 0 Rh = θ4 Ih − θ3Rh, (10) C 0 D λ 0 S m = θ6 − (θ2 + θ5) S m, C 0 D λ 0 Im = θ2S m − θ5 Im, where θ1 = µh Nh,θ2 = Bβmh Ih Nh = Bβhm Ih Nh , θ3 = µh,θ4 = γh,θ5 = µm,θ6 = µm Nm, (11) with the initial conditions: S h(0) = c1, Ih(0) = c2, Rh(0) = c3, S m(0) = c4, Im(0) = c5. (12) 5. Stability Analysis Here, we discuss this epidemiological model stability. The equilibrium points for system (10) and the Jacobian matrix are given by: J =  −Bβmh Ih Nh −µh −Bβmh S h Nh 0 0 0 Bβmh Im Nh − (γh + µh) 0 0 Bβmh sh Nh 0 γh −µh 0 0 0 −Bβmh S m Nh 0 −µm − Bβhm Ih Nh 0 0 Bβmh S m Nh 0 Bβhm Ih Nh −µm  (13) The disease-free equilibrium point is given as ( 44909351, 0, 0, 168000, 0 ), and the endemic equilibrium Table 2. The values of the initial value of model and the parameters are given by real data: Variable Description values Nh(t) total number of humans constant ≈ 44 million S h(0) susceptible humans 3326 Ih(0) infected humans 482 Rh(0) recovery humans 0 S m(0) susceptible female mosquitoes 117600 Im(t) infected female mosquitoes Parameter Range of values References Nh 44909351 Constant Nm 168000 Estimated µh 0.0000031 Estimated µm 0.1 [16] γh 2-10 days – [1/3] [16] B 0.7 [16] βmh 0.36 [16] βhm 0.36 [16] points (−3533,−8499,−2833,−132, 30017). The Jacobian matrix at disease free equilibrium point is given as follows: J =  −0.0000038 0.000000026 0 0 0 0.000000045 −0.11 0 0 0.000000026 0 0.01 −0.1 0 0 0 −0.0000009 0 −0.100000026 0 0 0.0000009 0 0.000000038 −0.1  (14) The eigenvalues corresponding to matrix J are: σ1 = −0.0000003,σ2 = −0.0000003,σ3 = 0.000003,σ4 = −0.01,σ5 = −0.0000003. (15) The system is stable because all the eigenvalues are negative. The system is unstable at the endemic equilibrium point be- cause some eigenvalues are positive. 5.1. The Basic Reproduction Number: The expected value of the secondary infections rate per time unit is denoted by R0, and the basic reproduction number is a baseline metric in epidemiology. Based on the fractional model of equation (10), We have two infected classes Ih(t), Im(t). CF 0 D λ 0 Ih = θ2S h − (θ4 + θ3) Ih (16) CF 0 D λ 0 Im = θ2S m − θ5 Im. (17) We assume that the infection risk rate is a constant β(t) taking the maximum value β(0) = β0 and the minimum value β (t∗) = β∗. Let x = (Ih, Im).and rewrite the system of equation (7) for the susceptible and infected classes in the general from d x dt = f (x) − v(x), (18) 3 Guma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1464 4 where f (x) = [ 0.252 0.036 0.252 1.262 ] , v(x) = [ 0.00000026 0 0.000009 0 ] . (19) Now the Jacobian of f (x) and v(x) of the disease-free equilib- rium point is F = [ 0.000000026 0 0.0000009 0 ] (20) V = [ 160.66 0 0 0.1 ] . (21) Therefore V−1 = [ 0.006 0 0 0.1 ] . (22) By using Eq. (11), we have R0 = ρ ( FV−1 ) = 10, (23) 6. Numerical analysis: We do a numerical simulation of the fractional model of dengue fever outbreaks in Sudan, epidemiological numerical analysis, and stability. By using real data in Sudan and eval- uating various possibilities by increasing and/or decreasing the model parameters’ values, we obtained a numerical simulation of the fractional dengue fever model. The Euler approach is em- ployed to generate the numerical results for the model (10). We used the values of numerous parameters from different credible sources to compute numerical results. 6.1. Adams-Bashforth Applied on Fractional Order System This section presents an explanation of the numerical pro- cedure that will be used by us to get the phase pictures of the fractional-order system (10). The Adams-Bashforth approach described by Garrappa in his review paper [17] is the one that we use. The solution to the fractional differential system (10) can be described as follows: φ (t, x1) = θ1 − (θ2 + θ3) S h, ϕ (t, x2) = θ2S h − (θ4 + θ3) Ih, θ (t, x3) = θ4 Ih − θ3Rh, (24) β (t, x4) = θ6 − (θ2 + θ5) S m, ρ (t, x5) = θ2S m − θ5 Im, as well as point tn; Eq. (6) may be recast in the following ways in accordance with a numerical strategy known as the Adams- Bashforth method: x (tn) = x(0) + h n κ̄(a)n φ(0) + n−1∑ j=1 κ (a) n− jφ ( t j, x1 j ) + κ (a) 0 φ ( t j, x p 1v ) y (tn) = y(0) + h a ᾱ(a)n ϕ(0) + N−1∑ j=1 κ (a) n− jϕ ( t j, x1 j ) + x(a)0 ϕ ( t j, x p 1n ) Figure 1. The size of susceptible human with time. z (tn) = z(0) + h a κ̄(a)n θ(0) + n−1∑ j=1 x(a)n− jθ ( t j, x1 j ) + x(a)0 θ ( t j, x p 1n ) (25) w (tn) = z(0) + h a κ̄(a)n β(0) + n−1∑ j=1 x(a)n− jβ ( t j, x1 j ) + x(a)0 β ( t j, x p 1n ) v (tn) = z(0) + h a κ̄(a)n ρ(0) + n−1∑ j=1 x(a)n− jρ ( t j, x1 j ) + x(a)0 ρ ( t j, x p 1n ) . The discretization parameters are specified as follows ( h is the step size): π̄(a)n = (n − 1)n − na(n −α− 1) Γ(2 + α) . (26) When n = 1, 2, . . ., we set the parameters as follows: κ (a) 0 = 1 Γ(2 + α)n κ(a)n = (n − 1)n+1 − 2na+1 + (n + 1)a+1 Γ(2 + α) . (27) We get an approximation of the functions in our model: φ ( t, x1 j ) = θ1 − (θ2 + θ3) S h j φ ( t, x2 j ) = θ2S h j − (θ4 + θ3) Ih j φ ( t, x3 j ) = θ4 Ih j − θ3Rh j (28) φ (t, x4 j) = θ6 − (θ2 + θ5) S m j φ ( t, x5 j ) = θ2S m j − θ5 Im j. The discretization technique that is going to be discussed in this part provides a lot of benefits. the procedure is reliable and guarantees convergence, and the Matlab implementation is both practical and simple. 7. Results and Discussion We present numerical results assuming the initial values of the model are: S h(0) = 3326, Ih(0) = 482, Rh(0) = 0, S m(0) = 117600, Im(0) = 5600 4 Guma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1464 5 Figure 2. The size of infectious human with time. Figure 3. The size of recovered human with time. Figure 4. The size of mosquitoes with time. Figure 5. The size of infectious mosquitoes with time. Figure 1 show the determine the property of a fractional order models. Figure 2 demonstrate the behavior of susceptible hu- man with t. From Figure 2, it is clear that susceptible human group increases with t. Figure 3 shows the behavior of infec- tious human with t. From Figure 3, It is clear that infectious people group decreases with time. Figure 4 demonstrate the behavior of recovery human with t. From Figure 4, It is clear that recovery human group increases with t. Figure 5 demon- strate the behavior of susceptible mosquitoes with time. Figure 6 demonstrates the behavior of infectious mosquitoes with time. Based on the frequency of dengue fever in Sudan, the funda- mental reproduction number indicates that one sick individual can spread the disease to up to ten additional people. 8. Conclusion In order to numerically resolve a fractional-order outbreak of dengue disease in Sudan, a computational technique built on the Adams-Bashforth approach has been applied in this study. The illustrations in this paper demonstrate how parameter val- ues and fractional derivatives both continually affect the re- sult. From numerical and stability analysis, it is clear that the fractional-order dengue fever model depends on its parameters at a given time t. As a result, the parameter values are cru- cial in determining how many people recover from their illness and how many contract it. The suggested method works well to demonstrate the behavior of the solution over a lengthy time horizon, which is useful for precisely forecasting the dengue virus outbreak in Sudan. In the future, we want to solve several novel fractional models, such as those in [17,20], and compare them to other numerical approaches [21,25]. References [1] M. A. Abdoon & F. L. Hasan, “Advantages of the differential equations for solving problems in mathematical physics with symbolic computa- tion”, Mathematical Modelling of Engineering Problems 9 (2022) 268. [2] M. Abdoon, F. Hasan & N. Taha, “Computational Technique to Study An- alytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation”, Abstract and Applied Analysis 2022 (2022). [3] A. Ahmed, Y. Ali, B. Elmagboul, O. Mohamed, A. Elduma, H. Bashab, A. Mahamoud, H. Khogali, A. Elaagip & T. Higazi, “Dengue fever in the Darfur area, Western Sudan”. Emerging Infectious Diseases. 25 (2019) 2125. [4] A. Ahmed, A. Elduma, B. Magboul, T. Higazi & Y. Ali, “The first outbreak of dengue fever in Greater Darfur, Western Sudan”, Tropical Medicine And Infectious Disease. 4 (2019) 43. [5] A. Ahmed, I. Dietrich, A. LaBeaud, S. Lindsay, A. Musa & S. Weaver, “Risks and challenges of arboviral diseases in Sudan: the urgent need for actions”, Viruses 12 (2020) 81. [6] A. Atangana, & D. Baleanu, “New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model”, ArXiv Preprint ArXiv:1602.03408 (2016). [7] M. Caputo, & M. Fabrizio, “A new definition of fractional derivative with- out singular kernel”, Progress in Fractional Differentiation & Applica- tions 1 (2015) 73. [8] S. Fadugba, “Solution of fractional order equations in the domain of the Mellin transform” , Journal of the Nigerian Society Of Physical Sciences 1 (2019) 138 [9] R. Garrappa, “Numerical solution of fractional differential equations: A survey and a software tutorial”, Mathematics 6 (2018) 16 [10] F. Hasan, & M. Abdoon, “The generalized (2+ 1) and (3+ 1)-dimensional with advanced analytical wave solutions via computational applications”, International Journal Of Nonlinear Analysis And Applications, 12 (2021) 1213 [11] J. Losada & J. Nieto, “Properties of a new fractional derivative without singular kernel”, Progr. Fract. Differ. Appl, 1 (2015) 87 5 Guma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1464 6 [12] A. Malik, K.Earhart, E. Mohareb , M. Saad, M. Saeed, A. Ageep, & A.Soliman, “Dengue hemorrhagic fever outbreak in children in Port Su- dan” , Journal of Infection and Public Health 4 (2011) 1 [13] M. Abdoon, R. Saadeh, M. Berir & F. Guma, “Analysis, modeling and simulation of a fractional-order influenza model”, Alexandria Engineer- ing Journal 74 (2023) 231 [14] C. Milici, G. Drăgănescu & J. Machado, Introduction to fractional differ- ential equations, Springer, 2018. [15] M. Soghaier, S. Mahmood, O. Pasha, S. Azam, M. Karsani, M. Elman- gory, B. Elmagboul, S. Okoued, S. Shareef & H. Khogali, “Factors as- sociated with dengue fever IgG sero-prevalence in South Kordofan State, Sudan, in 2012: Reporting prevalence ratios”, Journal Of Infection And Public Health 7 (2014) 51 [16] O. Uwaheren, A. Adebisi & O. Taiwo, “Perturbed collocation method for solving singular multi-order fractional differential equations of Lane- Emden type”, Journal Of The Nigerian Society Of Physical Sciences 2 (2020) 141. [17] A. Yusuf, R. Dima, & S. Aina, “Optimized breast cancer classification using feature selection and outliers detection”, Journal of the Nigerian Society Of Physical Sciences 3 (2021) 298. [18] R. A. Saadeh, M. Abdoon, A. Qazza & M. Berir, “A Numerical Solution of Generalized Caputo Fractional Initial Value Problems”, Fractal And Fractional 7 (2023) 332. [19] M. Abdoon, F. Hasan & N. Taha, “Computational Technique to Study An- alytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation”, Abstract And Applied Analysis 2022 (2022). [20] S. Kumar, S.Ghosh, M. Jleli & S.Araci, “A fractional system of Cauchy- reaction diffusion equations by adopting Robotnov function”, Numerical Methods For Partial Differential Equations, 38 (2022) 470. [21] H.Ramos & M. Rufai, “A new one-step method with three intermediate points in a variable step-size mode for stiff differential systems”, Journal of Mathematical Chemistry (2022) 1. [22] M. Duromola, A. Momoh, M. Rufai & I. Animasaun, “Insight into 2- step continuous block method for solving mixture model and SIR model”, International Journal Of Computing Science And Mathematics 14 (2021) 347. [23] H. Ramos & M. Rufai, “An adaptive one-point second-derivative Lobatto- type hybrid method for solving efficiently differential systems”, Interna- tional Journal of Computer Mathematics, 99 (2022) 1687. [24] A. Qazza, M.Abdoon, R.Saadeh & M.Berir, “A New Scheme for Solv- ing a Fractional Differential Equation and a Chaotic System”, European Journal of Pure and Applied Mathematics , 16 (2023) 1128. [25] F. E. Guma, O. M. Badawy, A. G. Musa, B. O. Mohammed, M. A. Ab- doon, M. Berir, S. Y. Salih,“Risk factors for death among COVID-19 Pa- tients admitted to isolation Units in Gedaref state, Eastern Sudan: a retro- spective cohort study”, Journal of Survey in Fisheries Sciences 10 (2023) 712. 6