J. Nig. Soc. Phys. Sci. 5 (2023) 1453 Journal of the Nigerian Society of Physical Sciences Modeling and Analysis of a Fractional Visceral Leishmaniosis with Caputo and Caputo–Fabrizio derivatives Dalal Khalid Almutairia, Mohamed A.Abdoonb, Salih Yousuf Mohamed Salihb, Shahinaz A.Elsamanib, Fathelrhman EL Gumac, Mohammed Berirb,c,∗ aDepartment of Mathematics, College of Education (Majmaah), Majmaah University, P.O.Box 66, Al-Majmaah, 11952, Saudi Arabia. bDepartment of Mathematics, Faculty of Science, Bakht Al-Ruda University, Duwaym, Sudan. cDepartment of Mathematics, Faculty of Science and Arts in Baljurashi, Albaha University, Albaha, Saudi Arabia. Abstract Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better epidemic forecasts. By taking into account the Caputo and Caputo-Fabrizio derivatives, a frictional model of visceral leishmaniosis was mathematically examined based on real data from Gedaref State, Sudan. The stability analysis for Caputo and Caputo-Fabrizio derivatives is analyzed. The suggested ordinary and fractional differential mathematical models are then simulated numerically. Using the Adams-Bashforth method, numerical simulations are conducted. The results demonstrate that the Caputo-Fabrizio derivative yields more precise solutions for fractional differential equations. DOI:10.46481/jnsps.2023.1453 Keywords: Leishmaniosis, Modelling, Caputo, Caputo–Fabrizio, Sudan. Article History : Received: 16 March 2023 Received in revised form: 21 May 2023 Accepted for publication: 19 June 2023 Published: 26 July 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 Visceral leishmaniasis, or kala azar, is a lethal vector-borne illness. India, Bangladesh, and Nepal have achieved substan- tial headway in lowering VL cases. East Africa has made less progress, especially with South Sudan’s continuous endemic- ity and VL outbreaks during the past 40 years. Lack of in- frastructure, clinical staff, IDPs, and hunger have hampered VL management in, and longer-term hazards to diagnostic kits and medications. Pentavalent antimonials have been the backbone ∗Corresponding author Email address: midriss@bu.edu.sa (Mohammed Berir ) of VL treatment for decades, and resistance to them, as pre- viously demonstrated in the Indian subcontinent, provides an- other major barrier to VL treatment and management. To pre- vent monotherapy and minimize treatment duration, first-line 30-day sodium stibogluconate SS is substituted with a 17-day injectable combination regimen of SSG and PM in WHO rec- ommendations in 2010 and Sudan Ministry of Health guidelines in 2011. Since 2012, AmBisome has been donated to WHO for these purposes. In East Africa, SSG/PM combo treatment had a 5% recurrence rate. Relapse may be due to insufficient cellular immunity following therapy due to HIV, TB, or malnutrition, or inadequate treatment resulting in considerable chronic para- sitaemia after initial clinical cure. In places like Sudan, where 1 Almutairi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1453 2 active patient follow-up is difficult and not common, VL recur- rence rates are passively evaluated by VL re-treatment admis- sions as a proportion of overall VL admissions. Passive mon- itoring shows an increase in re-treatment rates in recent years [1-7]. In the last 30 years, fractional calculus and nonlinear equa- tions has become more well-known and important. Fractional differential equations are used in physics, chemical engineer- ing, mathematical biology, and finance [8-16]. Simulating a fractional model simultaneously with a Caputo derivative and a CF derivative. In addition, modeling and graphing with the fractional derivative is a highly effective technique for demon- strating leishmaniasis using MATLAB. This could be done to better comprehend the infection. Using fractional derivatives as a research strategy for natural occurrences may result in more precise findings than other methods. As a result of this model’s use of a non-singleton kernel, the CF derivative has signifi- cantly improved predictive abilities. 2. Preliminaries Definition 2.1. Riemann-Liouville fractional integral (RLI) op- erator of order α > 0 for a function y (τ) is given by [17]: 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 [17] is given by CF 0 D α t y (t) = B(α) 1 −α) d dt ∫ t 0 y(τ)e−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. Caputo derivative of order 0 ≤ n − 1 < α < n with the lower limit zero for a function y (τ) is given by [18]: Iαy (t) = 1 Γ(α) ∫ t 0 (t −τ)α−1y(τ)dτ, t > 0. (3) 3. Anthropologic Visceral Leishmaniosis model with Caputo derivative In this Section, we describe the leishmaniasis model, which includes four subpopulations:susceptible, infectious, Recovered, and Recovered with permanent immunity, for the human popu- lation, and two compartments for the reservoir population: sus- ceptible and infected. In addition to that, we have two com- partments for sandflies: susceptible and infected. The human population is the only population in the model that has perma- nent immunity. The positivity, reproduction number, and equi- librium solutions of the model that was established in this work have all been determined to be free of leishmaniasis. Addition- ally, the leishmaniasis cases, along with their respective locali- ties and global stability properties, have also been determined. We obtain the model formulation by using a new variable: sh (t) = S H NH , ih (t) = IH NH , ph (t) = PH NH , rh (t) = RH NH , sr (t) = S R NR , Ir (t) = IR NR , sV (t) = S V NV , iV (t) = IV NV , sh (t) = S H NH , m = NV NH and N = NV NR . The system of differential equations is given by: C 0 D α t ih = abmiv Nh − ( α1 + δ + AH NH −δih ) ih, C 0 D α t ph = (1 −σ)α1ih − ( α2 + β + AH NH −δih ) ph, C 0 D α t ir = abniv sr − AH NH ir, C 0 D α t iv = acihS v + acphS v + acir S v − AV NV iv, C 0 D α t sh = AH NH − [ abmiv + AH NH −δih ] sh, C 0 D α t rh = σα1ih + (α2 + β) Ph − [ AH NH −δih ] rh, C 0 D α t S r = AR NR − abniv sr − AH NH sr, C 0 D α t sv = AV NV − [ acih + acPh + AV NV ] sv, (4) with initial conditions: sh (0) = c1, ih (0) = c2, rh (0) = c3, sr (0) = c4, Ir (0) = c5, sV (0) = c6, iV (0) = c7 . 4. Anthropologic Visceral Leishmaniosis model with FC deriva- tive In this Section, we obtain the fractional model formula- tion under Caputo–Fabrizio derivatives: sh (t) = S H NH , ih (t) = IH NH , ph (t) = PH NH , rh (t) = RH NH , sr (t) = S R NR , Ir (t) = IR NR , sV (t) = S V NV , iV (t) = IV NV , sh (t) = S H NH , m = NVNH and N = NV NR The system of differential equations is given by: FC 0 D α t ih = abmiv Nh − ( α1 + δ + AH NH −δih ) ih, FC 0 D α t ph = (1 −σ)α1ih − ( α2 + β + AH NH −δih ) ph, FC 0 D α t ir = abniv sr − AH NH ir, FC 0 D α t iv = acihS v + acphS v + acir S v − AV NV iv, FC 0 D α t sh = AH NH − [ abmiv + AH NH −δih ] sh, FC 0 D α t rh = σα1ih + (α2 + β) Ph − [ AH NH −δih ] rh, FC 0 D α t S r = AR NR − abniv sr − AH NH sr, FC 0 D α t sv = AV NV − [ acih + acPh + AV NV ] sv, (5) with initial conditions: sh (0) = c1, ih (0) = c2, rh (0) = c3, sr (0) = c4, Ir (0) = c5, sV (0) = c6, iV (0) = c7 . 5. Stability analysis In this part we discuss the stability of epidemiological model, the equilibrium points, eigenvalues value and the Jacobian ma- trix for the model (1). 2 Almutairi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1453 3 Table 1: Description of the variables for model. Variable Description NH (t) Human host population NR(t) Reservoir host population NV (t) Vector population S H (t) Susceptible humans PH (t) Recovered and have permanent immunity IH (t) Infected humans RH (t) Recovery humans Rs(t) Susceptible reservoir IR(t) Infected reservoir S V (t) Susceptible sandflies IV (t) Infected sandflies Table 2: Parameters values of the leishmaniasis model. Parameter Description Value Source a Biting rate of sandflies 0.2856 day−1 [16] b Progression rate of VL in sandfly 0.22 day−1 [16] c Progression rate of VL in human and reservoir 0.0714 day−1 [16] AH Human recruitment rate 10.1009 day−1 Estimated AR Reservoir recruitment rate 19.7795 day−1 Estimated AV Vector recruitment rate 38858.62 day−1 Estimated µh Natural mortality rate of humans 4.341e − 6 day−1 [2] µr Natural mortality rate of reservoirs 0.0017 day−1 [1] µv Natural mortality rate of vectors 0.0668 day−1 [1] α1 Treatment rate of VL 0.02 [2] α2 PKDL recovery rate without treatment 0.033 [20] σ Recovery rate from VL infection after treatment 0.9 [1] 1 −σ Developing PKDL rate after treatment 0.1 [1] δ Death rate due to VL 0.011 [16] β PKDL recovery rate after treatment 0.9 [1] 5.1. Equilibria The equilibrium points of dynamics (5) are computed solv- ing the nonlinear system. Table 3: The equilibrium points of the system. Ei Equilibria E1 (0, 0, 0, 0, 1, 0, 711.58, 1) E2 (32.5436, 0.1132, 711.5801, 22.7897,−29.7254,−1.9314, 0, 0) E3 (−31.1459,−0.0488, 711.58,−21.8109, 33.9641,−1.7693, 0, 0) E4 (85.0303,−72.8759, 711.5801, 59.5451,−82.2121,−71.0578, 0, 0) E5 (2.8182, 0.0062,−2.8244, 0, 0,−1.8244, 714.4046, 0) E6 (2.8182, 0.0062, 711.5801, 252.9373, 0,−1.8244, 0, 0) E7 (0, 0, 0, 0,−7.2892e11, 7.2892e11, 0, 0) E8 (0, 0, 0, 5.1155e8, 0, 0, 0, 0) 5.2. The Jacobian matrix for the model: Here, we talk about this epidemiological model stability. The disease-free equilibrium point is given as E1 = (0, 0, 0, 0, 1, 0, 711.58, 1) and the endemic equilibrium points E8 = (0, 0, 0, 5.1155e8, 0, 0, 0, 0). J (E1) =  −0.031 0 0 0.0157 0 0 0 0 0.002 −0.933 0 0 0 0 0 0 0 0 −4.297e−8 15907.35 0 0 0 0 0.02 0.02 0.02 −3.438e−11 0 0 0 0 0.011 0 0 −0.0157 −4.297e−8 0 0 0 0.018 0.0933 0 0 0 −4.297e−8 0 0 0 0 −15907.35 0 0 −4.297e−8 0 −0.02 −0.02 0 0 0 0 0 −3.438e−11  (6) J (E2) =  −0.031 0 0 0 8035456.6 0 0 0 0.002 −0.933 0 0 0 0 0 0 0 0 −4.297e−8 −0.0006 0 0 11435742383.06 0 0.000002 0.00002 0.00002 −3.438e−11 0 0 0 14.225 0 0 0 0 −8035456.6 0 0 0 0.028 0.0933 0 0 0 −0.00000003 0 0 0 0 0 0.0006 0 0 −11435742383.06 0 −0.00002 −0.0002 0 0 0 0 0 −0.000000027  (7) Table 4: Variable values. Eigenvalues Stability λ (17.833, 0, 0, 0, 0, 0, 0,−17.833) Unstable λ∗ (−3.438e−11,−2.8e−8,−2.8e−8, −4.297e−8,−0.31,−0.933,−035456.6,−1.14e10) Stable 5.3. The Basic Reproduction Number The basic reproduction number is a baseline statistic in epi- demiology and is represented by R0, which stands for the pre- dicted value of the secondary infections rate per time unit. Us- ing the equation’s fractional model (1), we have fours infected classes, rewrite the system of Equation 1 for the susceptible and 3 Almutairi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1453 4 infected classes in the general form: d x dt = f (x) − v (x) , (8) where f (x) =  abmiv sh 0 abmiv sr ac(ih + ph + ir )sv)  , and v (x) =  (α1 + δ + µh) ih (α2 + β + µh)ph − (1 −σ)α1ih µr ir µviv  . (9) Figure 1: Systems of fractional orders model for α=0.99. Figure 2: Systems of fractional orders model for α=1 (First part) Now, the Jacobian of f (x) and v (x) of the disease free equi- librium point is: F =  0 0 0 abm 0 0 0 0 0 0 0 abm ac ac ac 0  , and V =  α1 + δ + µh 0 0 0 −(1 −σ)α1 α2 + β + µh 0 0 0 0 µr 0 0 0 0 µv  (10) we have R0 = ρ ( FV−1 ) = √√√√√√√√√ ac[µr abm (α2 + δ + µh + (1 −σ) α1) + abn(α1 + δ + µh)(α2 + δ + µh) ] µv µr (α1 + δ + µh)(α2 + δ + µh). (11) Lemma 5.1. The disease-free equilibrium E0 is locally asymp- totically stable if R0 < 1 and unstable if R0 > 1. 6. Numerical Simulation and Graphical Representations This section is devoted to finding the approximate solutions of the proposed models (4) and (5) under fractional operators of Caputo and Caputo-Fabrizio, respectively. We simulate our model using some highly reliable numerical techniques. The finite difference scheme for the initial value problem yields the following numerical techniques for the underlying operators: c xr+1 = x0 + (∆t)ω Γ(ω + 1) r∑ k=0 [ (r − k + 1)ω − (r − k)ω ] F (xk) + O ( ∆t2 ) , CF xr+1 = x0 + (1 −δ)F (xr ) + δ∆t r∑ k=0 F (xk) + O ( ∆t2 ) , (12) Table 1 shows a description of the variables in the model. Fig- ures 1 and 2 were obtained with the Caputo (4) and CF methods (5) using the parameters in Table 2. Tables 3 & 4 show a sum- mary of equilibrium points and the corresponding eigenValues of the Jacobian matrix. 7. Conclusion A fractional model was simulated by using a Caputo deriva- tive as well as a CF derivative simultaneously. In addition, mod- eling and graphing with the aid of the fractional derivative is a very effective approach that can be used to show leishmaniasis with the use of MATLAB. This may be done in order to bet- ter understand the infection. 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