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# A Study on the Nonlinear Caputo-Type Snakebite Envenoming Model with Memory

1 Department of Mathematics, National Institute of Technology Puducherry, Karaikal, 609609, India

2 Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Atakum, Samsun, 55200, Turkey

3 Department of Mathematics, Cankaya University, Ankara, 06530, Turkey

4 Institute of Space Sciences, Magurele, Bucharest, R76900, Romania

5 Department of Computer Science and Mathematics, Lebanese American University, Beirut, 11022801, Lebanon

* Corresponding Author: Pushpendra Kumar. Email:

(This article belongs to the Special Issue: Advanced Numerical Methods for Fractional Differential Equations)

*Computer Modeling in Engineering & Sciences* **2023**, *136*(3), 2487-2506. https://doi.org/10.32604/cmes.2023.026009

**Received** 09 August 2022; **Accepted** 10 November 2022; **Issue published** 09 March 2023

## Abstract

In this article, we introduce a nonlinear Caputo-type snakebite envenoming model with memory. The well-known Caputo fractional derivative is used to generalize the previously presented integer-order model into a fractional-order sense. The numerical solution of the model is derived from a novel implementation of a finite-difference predictor-corrector (L1-PC) scheme with error estimation and stability analysis. The proof of the existence and positivity of the solution is given by using the fixed point theory. From the necessary simulations, we justify that the first-time implementation of the proposed method on an epidemic model shows that the scheme is fully suitable and time-efficient for solving epidemic models. This work aims to show the novel application of the given scheme as well as to check how the proposed snakebite envenoming model behaves in the presence of the Caputo fractional derivative, including memory effects.## Keywords

Nowadays, fractional calculus [1–3] is being applied to solve various real-world problems in terms of mathematical modeling. Different fractional derivatives [4] have been successfully used to model various problems. More specifically, several deadly epidemics have been modeled by using mathematical models in a fractional-order sense. It is a well-known fact that the fractional-order operators are non-local in nature and may be more effective for modeling history (memory)-dependent systems. Moreover, a fractional order can be fixed as any positive real number that better fits a real-data. So, by using such an operator, an accurate adjustment can be done in a model to fit with real data for better predicting the outbreaks of an epidemic. Recently, several applications of fractional derivatives have been recorded in epidemiology. In [5–7], the authors have studied the dynamics of the COVID-19 epidemic by using fractional-order mathematical models. In [8], the authors used non-singular Caputo-Fabrizio and Atangana-Baleanu fractional derivatives to study the dynamical nature of a malaria epidemic model. Kumar et al. in [9] have solved canine distemper virus and rabies epidemic models in the sense of generalized Caputo derivative. Kumar et al. [10] defined two different types of fractional-order models to study the dynamics of mosaic disease. A novel application of the generalized Caputo derivative in environmental infection dynamics can be seen in [11]. Sinan et al. [12] proposed the mathematical modeling of typhoid fever in terms of fractional derivatives. In [13], some novel analyses on the numerical modeling of biological systems using fractional derivatives have been given. Rihan et al. [14] studied fractional-order predator-prey models, including delay with Holling type-II functional response. In [15], some novel applications of delay differential equations were proposed. Work on the application of fractional derivatives in ecology and psychology can be understood from [16,17], respectively.

To analyze the various types of fractional-order systems, several computational schemes have been proposed by researchers. Odibat et al. [18] derived a new generalized form of the predictor-corrector (PC) scheme to investigate fractional initial value problems. Kumar et al. [19] introduced a new method to simulate fractional-order systems with various examples. In [20], the PC method was derived to simulate delay fractional differential equations. A modified form of the PC scheme in terms of the generalized Caputo derivative to solve delay-type systems has been introduced in reference [21]. Odibat et al. [22] have derived the generalized differential transform method for solving fractional impulsive differential equations. In [23], some computational schemes to solve delayed parabolic and time-fractional partial differential equations have been proposed. Shah et al. [24] simulated the dynamics of some important fractional order differential equations. In [25], some novel analyses of the Cauchy-type fractional-order dynamical system in terms of piecewise equations have been given. Some more applications of fractional derivatives can be seen in [26–28].

Jhinga et al. [29] introduced a novel finite-difference predictor-corrector (L1-PC) scheme to solve fractional-order systems in the sense of the Caputo derivative. That L1-PC scheme is not yet applied to any epidemic model to check how it will work and show the disease dynamics. In this paper, we fill this research gap by implementing the given L1-PC method on a Caputo-type snakebite envenoming (SBE) mathematical model, which was previously proposed in the integer-order sense in reference [30]. Currently, SBE is a deathly neglected disease, mainly in developing nations. The World Health Organization (WHO) recognized SBE as a fatal disease that generally results from the injection of a combination of various toxins (venom) following a venomous snakebite. Mainly poor, rural societies in tropical and subtropical nations all over the world are typically affected by SBE, which is a big threat to the health and well-being of about 5.8 billion population [31]. More information on SBE can be collected from the references [32–34]. To date, only a few research studies have gone into the mathematical modeling of SBE. In [35], the authors introduced a model using the law of mass action to analyze snakebite incidence. In [36], a mathematical model considering the socio-demographic components that influence the death risk from SBE in India was proposed.

The motivation of this study is to justify the application of the aforementioned L1-PC scheme in epidemiology by solving a mathematical model of SBE. Also, the aim is to check how the proposed SBE model will behave in the presence of the Caputo fractional derivative, which is a non-local differential operator that allows memory effects in the system. The given study is designed using the following systems: Section 2 recalls some preliminaries of fractional calculus. The description of the considered model of SBE in terms of the Caputo derivative is given in Section 3. The necessary numerical analysis, like the solution algorithm, error estimation, and method stability, is performed in Section 4. A brief discussion of the proposed methodology, its novelty, and outputs is given in Section 5 with supporting conclusions.

Here we recall some preliminaries of fractional calculus.

Definition 1. A real function

a)

b)

Definition 2. [2] The Riemann-Liouville fractional integral of

Definition 3. [2] The Caputo fractional derivative of

Theorem 1. [37] Consider the function f such that f and

Therefore, from Theorem 1, we say that if

Lemma 1. [38] Let

where

with the same initial constraints, where j is a positive integer. If

Here we introduce the generalized form of the previously published integer-order model [30] into Caputo-type fractional-order sense (

The total snake population is defined by

where,

The definitions of the model parameters alongwith their numerical values are given in Table 1.

Theorem 2. There exists a unique solution for the model (3) and (4) which belongs to

Proof. By using the Theorem 3.1 and Remark 3.2 of [39], the global existence of the unique solution can easily be proved. Now to prove the non-negativity of the solution, let us write the following auxiliary system of fractional differential equations:

with

by continuity of

Now before performing the further numerical simulations on the proposed fractional-order model (3), we rewrite the model into a compact form by representing it in terms of an initial value problem, defined as follows:

Let us consider

By using (6), we have

where

4 Numerical Analysis on the Model

In this section, we perform the necessary numerical simulations (solution derivation, error estimation and stability) to derive the solution of the proposed fractional-order model (3) by using the L1-predictor-corrector scheme [29].

Consider the above given initial value problem (IVP) for

where

4.1 Derivation of the Solution

According to the L1-PC method, the Caputo fractional derivative is numerically defined by

where,

We approximate

where

(11) can be rewritten as

After rewriting the terms (12), we get the following from:

Substituting

in (12), we get

Define

Remark that

•

•

•

Given Eqs. (14) and (15), the following form can be obtained:

We can see that Eq. (16) is of the form

and

Hence using the scheme of the DGJ method gives approximate value of

The three term approximation of

where

Using the above given methodology, the approximation equations of the proposed model (3) in terms of L1-PC method are derived as follows:

where,

and

The above given algorithm is coded in Mathematica and the bunch of Figs. 1 and 2 plotted to explore the outputs of the scheme at various fractional-order values. From the Figs. 1c–1f, we identify when the fractional order decreases, the population of the SBE humans;

The brief analysis on the error estimation of L1-PC scheme has been given in the studies [29,40,41] and now investigated below. The error estimate is given by

here C is a positive constant depends on

Derive

In view of (21)

To derive the error estimation, we will use the lemmas given below:

Lemma 2. [42] For

Lemma 3. [29] Consider

where

Lemma 4. [29] Consider

where

Theorem 3. Consider

where

Proof. Let

Further observe that

Using Lemma 4 in Eq. (25), we get

Therefore

where

The behaviour of the absolute remainder error is plotted in the bundle of Figs. 3 and 4 at the previously used fractional-order values

The term stability means that small deviations in the initial values do not make the big changes in the numerical solutions. Consider that

here h is the step size given in Eq. (9).

Theorem 4. Suppose

Proof. We have to prove that

Denote by

Further observe that

Using discrete form of Gronwalls inequality and Eq. (15), we obtain

where c is a constant and

Using (27) and (28) in (26), we get

where C is a constant.

In this research work, we have performed a novel implementation of a finite-difference predictor-corrector scheme on a fractional-order nonlinear snakebite envenoming model in terms of the Caputo fractional derivative. The numerical solution of the model has been plotted by using several graphs to justify the behavior of the model at various fractional-order values. The analysis related to the error estimation and stability of the scheme has also been derived from exploring method’s accuracy. In the error estimation, the fractional order

Funding Statement: The authors received no specific funding for this study.

Author Contributions: P. Kumar: Conceptualization, Visualization, Resources, Formal analysis, Investigation, Writing-original draft. V.S. Erturk: Conceptualization, Investigation, Visualization, Software, Writing-review & editing. V. Govindaraj: Investigation, Resources, Formal analysis, Writing-review & editing. D. Baleanu: Investigation, Resources, Formal analysis, Writing-review & editing.

Availability of Data and Materials: The data used in this research is available/mentioned in the manuscript.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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*Computer Modeling in Engineering & Sciences*,

*136*

*(3)*, 2487-2506. https://doi.org/10.32604/cmes.2023.026009

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