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

1  Introduction

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.

2  Preliminaries

Here we recall some preliminaries of fractional calculus.

Definition 1. A real function f(s), s>0 belongs to the space

a) Cη,η∈R if there exists a real number q>η, such that f(s) = sqf1(s), f1∈C[0,∞). Clearly,   Cη⊂Cα if α≤η.

b) Cηm,m∈N∪{0} if fm∈Cη.

Definition 2. [2] The Riemann-Liouville fractional integral of f(t)∈Cη(η≥−1) is defined as follows:

Jγf(t)=1Γ(γ)∫0t(t−s)γ−1f(s)ds,J0f(t)=f(t).

Definition 3. [2] The Caputo fractional derivative of f∈C−1m is written by

Dtγf(t)={dmf(t)dtm,ifγ=m∈N1Γ(m−γ)∫0t(t−ϑ)m−γ−1f(m)(ϑ)dϑ,ifm−1<γ<m,m∈N.(1)

Theorem 1. [37] Consider the function f such that f and CDc+γf are continuous for γ∈(0,1]. Then, ∀t∈(c,d], there exists some k∈(c,t) following the constraint

f(t)=f(c)+1Γ(γ+1)CDc+γf(k)(t−c)γ.

Therefore, from Theorem 1, we say that if CDc+γf(t)>0, for allt∈[c,d], then f is strictly increasing, and ifCDc+γf(t)<0, for all t∈[c,d], then the function f is strictly decreasing.

Lemma 1. [38] Let γ∈(0,1),n∈N and define the vectors X:=(x1,x2,…,xn) and Y:=(y1,y2,…,yn). For each i=1,2,…,n, let us consider Gi:[c,d]×Rn→R be a continuous function fulfils the Lipschitz condition with respect to the second variable such that

|Gi(t,X)−Gi(t,Y)|≤Li‖X−Y‖,

whereLi is a constant. Let us write G:=(G1,G2,…,Gn) and define the two fractional differential equations

CDc+γX(t)=G(t,X)+1jandCDc+γX(t)=G(t,X),(2)

with the same initial constraints, where j is a positive integer. If jX∗:=(jx1∗,…,jxn∗) andX∗:=(x1∗,x2∗,…,xn∗) are the solutions of (2), simultaneously, thenjX∗(t)→X∗(t) asj→∞ for all t∈[c,d].

3  Model Description

Here we introduce the generalized form of the previously published integer-order model [30] into Caputo-type fractional-order sense (CDγ). In our model, the total human population NH(t) at time t is split out into the following mutual exclusive classes: unaware susceptible humans, (SU(t)), aware susceptible humans, (SE(t)), SBE population, (I(t)), humans getting early remedy with antivenom, (TE(t)), peoples getting late remedy with antivenom, (TL(t)), peoples suffering from early adverse reaction (EAR) at the time of early remedy, (VE(t)), peoples facing EAR at the time of late remedy, (VL(t)), recovered peoples with disabilities, (RD(t)), and recovered peoples without disabilities, (RW(t)). Therefore, the total population size is defined as

N(t)=SU(t)+SE(t)+I(t)+TE(t)+TL(t)+VE(t)+VL(t)+RD(t)+RW(t).

The total snake population is defined by (NS(t)) and D(t) shows the total deaths caused by snakebite. For taking the equal time-dimension day−γ on the both sides of the fractional-order model, the power γ is applied to the parameters in time unit day−1 in the classical case. Therefore, the fractional-order nonlinear model for SBE is given as follows:

CDγSU=ΛHγ−(λ+K1)SU,CDγSE=ϵγSU+ϕ1γRD+ϕ2γRW−(Π1λ+K2)SE,CDγI=(Π1SE+SU)λ−K3I,CDγTE=τγkI−K4TE,CDγTL=τγΠ2I−K5TL,CDγVE=α1γTE−K6VE,CDγVL=α2γTL−K7VL,CDγRD=σ1γρ1TL+σ2γρ2VL−K8RD,CDγRW=r1γTE+r2γVE+σ1γΠ3TL+σ2γΠ4VL−K9RW,CDγNS=ΛSγNS(1−NSKS)−μSγNS,CDγD=δ1γI+(TL+VL)δ2γ,(3)

λ(t)=βγNSNH+NS,

where,

K1=ϵγ+μHγ,K2=μHγ,K3=τγ+δ1γ+μHγ,K4=α1γ+r1γ+μHγ,K5=α2γ+σ1γ+δ2γ+μHγ,K6=r2γ+μHγ,K7=σ2γ+δ2γ+μHγ,K8=ϕ1γ+μHγ,K9=ϕ2γ+μHγ,Π1=1−θ,Π2=1−k,Π3=1−ρ1,Π4=1−ρ2, with the initial conditions

SU(0)>0,SE(0)≥0,I(0)≥0,TE(0)≥0,TL(0)≥0,VE(0)≥0,VL(0)≥0,RD(0)≥0,RW(0)≥0,NS(0)≥0,D(0)≥0.(4)

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(R0+)11:={(SU∗,SE∗,I∗,TE∗,TL∗,VE∗,VL∗,RD∗,RW∗,NS∗,D∗)∈R+11}.

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:

CDγSU=ΛHγ−(λ+K1)SU+1k,CDγSE=ϵγSU+ϕ1γRD+ϕ2γRW−(Π1λ+K2)SE+1k,CDγI=(Π1SE+SU)λ−K3I+1k,CDγTE=τγkI−K4TE+1k,CDγTL=τγΠ2I−K5TL+1k,CDγVE=α1γTE−K6VE+1k,CDγVL=α2γTL−K7VL+1k,CDγRD=σ1γρ1TL+σ2γρ2VL−K8RD+1k,CDγRW=r1γTE+r2γVE+σ1γΠ3TL+σ2γΠ4VL−K9RW+1k,CDγNS=ΛSγNS(1−NSKS)−μSγNS+1k,CDγD=δ1γI+(TL+VL)δ2γ+1k,(5)

with k∈N. We will show that solution of (5) (SUk∗(t),SEk∗(t),Ik∗(t),TEk∗(t),TLk∗(t),VEk∗(t), VLk∗(t),RDk∗(t),RWk∗(t),NSk∗(t),Dk∗(t)) belongs to (R0+)11, ∀t≥0. For obtaining a contradiction, we opine that there exists a point of time where the condition fails. Let t0:=inf{t>0|(SUk∗(t),SEk∗(t),Ik∗(t),TEk∗(t),TLk∗(t),VEk∗(t), VLk∗(t),RDk∗(t),RWk∗(t),NSk∗(t), Dk∗(t))∈(R0+)11}. Thus, (SUk∗(t0),SEk∗(t0),Ik∗(t0),TEk∗(t0),TLk∗(t0),VEk∗(t0),VLk∗(t0),RDk∗(t0), RWk∗(t0),NSk∗(t0), Dk∗(t0))∈(R0+)11 and one of the quantities (SUk∗(0),SEk∗(0),Ik∗(0),TEk∗(0),TLk∗(0),VEk∗(0), VLk∗(0),RDk∗(0), RWk∗(0),NSk∗(0),Dk∗(0)) is zero. Suppose that Ik∗(t0)=0. Since

CDγIk∗(t0)=(Π1SUk∗(t0)+SEk∗(t0))λ+1k>0

by continuity of CD0+γIk∗, we conclude that CD0+γIk∗([t0,t0+ξ)⊂R+, for some ξ>0. By Theorem 1, Ik∗([t0,t0+ξ)⊂R0+ and so Ik∗ is non negative. In an analogous way we can justify that the remaining functions SUk∗,SEk∗,TEk∗,TLk∗,VEk∗, VLk∗,RDk∗,RWk∗,NSk∗, and Dk∗ are non-negative, establishing a contradiction. Using Lemma 1 as k→∞, we get that the solution (SUk∗(t),SEk∗(t),Ik∗(t),TEk∗(t),TLk∗(t),VEk∗(t), VLk∗(t),RDk∗(t),RWk∗(t),NSk∗(t),Dk∗(t)) of (5) belongs to (R0+)11, ∀t≥0, giving the required result.

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

{f1(t,SU,…,D)=ΛHγ−(λ+K1)SU,f2(t,SU,…,D)=ϵγSU+ϕ1γRD+ϕ2γRW−(Π1λ+K2)SE,f3(t,SU,…,D)=(Π1SE+SU)λ−K3I,f4(t,SU,…,D)=τγkI−K4TE,f5(t,SU,…,D)=τγΠ2I−K5TL,f6(t,SU,…,D)=α1γTE−K6VE,f7(t,SU,…,D)=α2γTL−K7VL,f8(t,SU,…,D)=σ1γρ1TL+σ2γρ2VL−K8RD,f9(t,SU,…,D)=r1γTE+r2γVE+σ1γΠ3TL+σ2γΠ4VL−K9RW,f10(t,SU,…,D)=ΛSγNS(1−NSKS)−μSγNS,f11(t,SU,…,D)=δ1γI+(TL+VL)δ2γ.(6)

By using (6), we have

CDγ𝒜(t)=Φ(t,𝒜(t)),t∈[0,T],0<γ≤1,𝒜(0)=𝒜0,(7)

where

𝒜(t)={SU(t)SE(t)I(t)TE(t)TL(t)VE(t)VL(t)RD(t)RW(t)NS(t)D(t),𝒜0(t)={SU0(t)SE0(t)I0(t)TE0(t)TL0(t)VE0(t)VL0(t)RD0(t)RW0(t)NS0(t)D0(t),Φ(t,𝒜(t))={f1(t,SU,…,D)f2(t,SU,…,D)f3(t,SU,…,D)f4(t,SU,…,D)f5(t,SU,…,D)f6(t,SU,…,D)f7(t,SU,…,D)f8(t,SU,…,D)f9(t,SU,…,D)f10(t,SU,…,D)f11(t,SU,…,D).(8)

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 0<γ<1,

CDγ𝒜(t)=Φ(t,𝒜(t)),t∈[0,T],𝒜(0)=𝒜0.(9)

where CDγ represents the Caputo derivatives and Φ:[0,T]×D→R,D⊂R. Split the time span [0, T] into N subintervals. Take an uniform grid with step size of h=TN with tk=kh,k=0,1,…,N.

4.1 Derivation of the Solution

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

[CDγ𝒜(t)]t=tn=1Γ(1−γ)∫0tk(tn−s)−γ𝒜′(s)ds=1Γ(1−γ)∑k=0n−1∫tktk+1(tn−s)−γ𝒜′(s)ds≈1Γ(1−γ)∑k=0n−1∫tktk+1(tn−s)−γ𝒜(tk+1)−𝒜(tk)hds=∑k=0n−1bn−k−1(𝒜(tk+1)−𝒜(tk)),(10)

where,

bk=h−γΓ(2−γ)[(k+1)1−γ−k1−γ].

We approximate CDγ𝒜(t) by the Eq. (10) and put it into (9) to get

[CDγ𝒜(t)]t=tn=∑k=0n−1bn−k−1(𝒜(tk+1)−𝒜(tk))=Φ(tn,𝒜n),(11)

where 𝒜k defines the approximate value of the solution of (9) at t=tk and

bn−k−1=h−γΓ(2−γ)[(n−k)1−γ−(n−k−)1−γ].

(11) can be rewritten as

bn−1(𝒜1−𝒜0)+bn−2(𝒜2−𝒜1)+⋯+b0(𝒜n−𝒜n−1)=Φ(tn,𝒜n).(12)

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

b0𝒜n=b0𝒜n−1−∑k=0n−2bk+1𝒜n−1−k+∑k=1n−1bk𝒜n−1−k+Φ(tn,𝒜n).(13)

Substituting

b0=h−γΓ(2−γ) and bk=h−γΓ(2−γ)[(n−k)1−γ−(n−k−)1−γ].

in (12), we get

𝒜n=𝒜n−1−(21−γ−11−γ)𝒜n−1−∑k=1n−2((2+k)1−γ−(1+k)(1−γ))𝒜n−1−k+∑k=1n−2((1+k)1−γ−(k)(1−γ))𝒜n−1−k+(n1−γ−(n−1)1−γ)𝒜0+Γ(2−γ)hγΦ(tn,𝒜n)=(n1−γ−(n−1)1−γ)𝒜0+∑k=1n−1[2(n−k)1−γ−(n+1−k)(1−γ)−(n−1−k)(1−γ)]𝒜k+Γ(2−γ)hγΦ(tn,𝒜n).(14)

Define

ak:=k+1(1−γ)−k(1−γ).(15)

Remark that ak′s have the following characteristics:

•   ak>0,k=0,1,…,n−1.

•   a0=1>a1>⋯>ak, and ak→0 as k→∞.

•   ∑k=0n−1(ak−ak+1)+an=(1−a1)+∑k=1n−2(ak−ak+1)+an−1=1.

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

𝒜n=an−1𝒜0+∑k=1n−1(an−1−k−an−k)𝒜k+Γ(2−γ)hγΦ(tn,𝒜n).(16)

We can see that Eq. (16) is of the form 𝒜n=g+N(𝒜n), if we identify

g=an−1𝒜0+∑k=1n−1(an−1−k−an−k)𝒜k

and

N(𝒜n)=Γ(2−γ)hγΦ(tn,𝒜n).

Hence using the scheme of the DGJ method gives approximate value of 𝒜n, given by

𝒜n,0=g=an−1𝒜0+∑k=1n−1(an−1−k−an−k)𝒜k,𝒜n,0=N(𝒜n,0)=Γ(2−γ)hγΦ(tn,𝒜n),𝒜n,2=N(𝒜n,0+𝒜n,1−N(𝒜n,0).

The three term approximation of 𝒜n≈𝒜n,0+𝒜n,0+𝒜n,2. Therefore, this approximated solution of DGJ scheme gives the following predictor-corrector algorithm called as L1-PCM.

𝒜np=an−1𝒜0+∑k=1n−1(an−1−k−an−k)𝒜k,znp=N(𝒜n)=Γ(2−γ)hγΦ(tn,𝒜np),𝒜nc=𝒜np+Γ(2−γ)hγΦ(tn,𝒜np+znp),(17)

where 𝒜np and znp are predictors and 𝒜nc is the corrector.

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

{SUnc=SUnp+Γ(2−γ)hγf1(tn,SUnp+z1np,…,Dnp+z11np),SEnc=SEnp+Γ(2−γ)hγf2(tn,SUnp+z1np,…,Dnp+z11np),Inc=Inp+Γ(2−γ)hγf3(tn,SUnp+z1np,…,Dnp+z11np),TEnc=TEnp+Γ(2−γ)hγf4(tn,SUnp+z1np,…,Dnp+z11np),TLnc=TLnp+Γ(2−γ)hγf5(tn,SUnp+z1np,…,Dnp+z11np),VEnc=VEnp+Γ(2−γ)hγf6(tn,SUnp+z1np,…,Dnp+z11np),VLnc=VLnp+Γ(2−γ)hγf7(tn,SUnp+z1np,…,Dnp+z11np),RDnc=RDnp+Γ(2−γ)hγf8(tn,SUnp+z1np,…,Dnp+z11np),RWnc=RWnp+Γ(2−γ)hγf9(tn,SUnp+z1np,…,Dnp+z11np),NSnc=NSnp+Γ(2−γ)hγf10(tn,SUnp+z1np,…,Dnp+z11np),Dnc=Dnp+Γ(2−γ)hγf11(tn,SUnp+z1np,…,Dnp+z11np),(18)