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Research article

A mathematical model with control strategies for marijuana smoking prevention

  • Received: 21 December 2023 Revised: 05 March 2024 Accepted: 07 March 2024 Published: 22 March 2024
  • Our goal of this study is to prevent marijuana smoking in the human population. In this manuscript, an updated mathematical model was established by incorporating two additional compartments: The hospitalized class and the prisoner's class. The updated model was validated, and it was shown to be novel compared to the non-user, experimental, recreational, and addicted (NERA) users' model. This distinction was crucial as it was challenging to prevent marijuana usage without these realistic classes. The entire population was split into six primary groups, including these new classes: non-users, experimental, recreational, addicted, hospitalized, and prisoners' class. Additionally, control techniques for marijuana prevention in the population were addressed with the aid of sensitivity analysis. The important point at which we may have determined the preliminary transmission rate of marijuana smoking was the basic reproductive number R0. Utilizing MATLAB, the Runge-Kutta method of order four was employed for the numerical simulation of the updated model to investigate the impact of control measures on marijuana smoking prevention.

    Citation: Atta Ullah, Hamzah Sakidin, Kamal Shah, Yaman Hamed, Thabet Abdeljawad. A mathematical model with control strategies for marijuana smoking prevention[J]. Electronic Research Archive, 2024, 32(4): 2342-2362. doi: 10.3934/era.2024107

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  • Our goal of this study is to prevent marijuana smoking in the human population. In this manuscript, an updated mathematical model was established by incorporating two additional compartments: The hospitalized class and the prisoner's class. The updated model was validated, and it was shown to be novel compared to the non-user, experimental, recreational, and addicted (NERA) users' model. This distinction was crucial as it was challenging to prevent marijuana usage without these realistic classes. The entire population was split into six primary groups, including these new classes: non-users, experimental, recreational, addicted, hospitalized, and prisoners' class. Additionally, control techniques for marijuana prevention in the population were addressed with the aid of sensitivity analysis. The important point at which we may have determined the preliminary transmission rate of marijuana smoking was the basic reproductive number R0. Utilizing MATLAB, the Runge-Kutta method of order four was employed for the numerical simulation of the updated model to investigate the impact of control measures on marijuana smoking prevention.



    We are exploring the initial value problem (IVP) defined as:

    z=f(t,z),z(t0)=z0,z(t0)=z0, (1.1)

    where f:R×RmRm and z0,z0Rm. The above equation is widely applicable in various scientific and engineering contexts. Notably, Eq (1.1) lacks z.

    The Numerov method facilitates the numerical advancement of the solution from tk to tk+1=h+tk, a well-established approach for solving Eq (1.1). It is expressed as:

    zk+1=2zkzk1+h212(fk+1+10fk+fk1),

    where zkz(tk) and fkzn=f(tk,zk). Note that fk,zkRm.

    Hairer [1], Cash [2] and Chawla [3] introduced hybrid implicit Numerov-type methods (i.e., using non-mesh points) approximately 40–45 years ago. Addressing the P-stability property, crucial for handling stiff oscillatory problems, was the primary challenge then. Chawla [4] proposed the modified Numerov scheme, evaluated explicitly as follows:

    v1=zk1,v2=zk,v3=2zkzk1+h2f(tk,v2),zk+12zk+zk1=112h2(f(tk+1,v3)+10f(tk,v2)+f(tk1,v1)), (1.2)

    where h is a constant step length:

    h=tktk1=tk+1tk==t1t0.

    The vectors zk+1,zk, and zk1 approximate z(tk+h),z(tk), and z(tkh) respectively, while v1Rm,v2Rm, and v3Rm represent the stages (alternatively named: function evaluations) used by the method.

    We utilize the information known at the mesh:

    v1=zk1,v2=zk.

    Since f(tk1,v1) is computed in the previous step, only f(tk+1,v3) and f(tk,v2) need evaluation each step, resulting in only two function evaluations per step.

    Tsitouras then introduced a Runge–Kutta–Nyström (RKN)-style method [5], significantly reducing the cost. Consequently, only four steps are required to create a sixth-order method, whereas previous implementations required six function evaluations (see [6]).

    Subsequent to this, our group extensively investigated the topic. Tsitouras developed eighth-order methods with nine steps per step in [7]. Ninth-order methods were studied in [8]. Concurrently, a group of Spanish researchers conducted highly interesting work on the same topic [9,10,11].

    In this study, we aim to present a new method for better addressing problems with periodic solutions. Traditionally, various properties from a simple test equation are fulfilled for this purpose. The novelty lies in training the available free parameters across a wide set of relevant problems. Differential evolution is employed for this training. It is anticipated that this methodology will yield a method better tuned for oscillatory problems.

    For the numerical treatment of Eq (1.1) with higher-order algebraic methods, there exists a considerable demand. We can represent the independent variable t as one of the components of z (if necessary, add t=0 see [12, pg. 286] for details). Consequently, our focus, without loss of generality, lies on the autonomous system z=f(z). Subsequently, a hybrid Numerov method with s stages, as delineated in [7], may be expressed as:

    zk+1=2zkzk1+h2(wIs)f(v), v=(1+a)zkazk1+h2(DIs)f(v) (2.1)

    where IsRs×s represents the identity matrix, DRs×s,wTRs,aRs denote the coefficient matrices of the method, and

    1=[111]TRs.

    To present the coefficients, we utilize the Butcher tableau [13,14],

    aDw.

    The method described in (1.2) can be represented using matrices [8]. As the function evaluations are computed sequentially, these methods are explicit. Therefore, D represents a strictly lower triangular matrix. For the case when s=8, the method takes the following structure:

    fk1=f(tk1,zk1)fk=f(tk,zk)zα=a3zk1+(1a3)zk+h2(d31fk1+ad2fk), fα=f(tka3h,zα),zβ=a4zk1+(1a4)zk+h2(d41fk1+d42fk+d43fα),fβ=f(tka4h,zβ),zc=a5zk1+(1a5)zk+h2(d51fk1+d52fk+d53fα+d54fβ),fc=f(tka5h,zc),zδ=a6zk1+(1a6)zk+h2(d61fk1+d62fk+d63fα+d64fβ+d65fc),fδ=f(tka6h,zδ),ze=a7zk1+(1a7)zk+h2(d71fk1+d72fk+d73fα+d74fβ+d75fc+d76fδ),fe=f(tka7h,ze),zg=a8zk1+(1a8)zk+h2(d81fk1+d82fk+d83fα+d84fβ+d85fc+d86fδ+d87fe),zk+1=2zkzk1+h2(w1fk1+w2fk+w3fα+w4fβ+w5fc+w6fδ+w7fe+w8fg).

    After assuming [15]

    w3=0,w5=w4,w7=w6,w8=w1,a5=a4,a6=a7,a8=1,

    the associated matrices take the form

    D=[0000000000000000d31d32000000d41d42d4300000d51d52d53d540000d61d62d63d64d65000d71d72d73d74d75d7600d81d82d83d84d85d86d870],
    w=[w1w20w4w4w6w6w1]anda=[10a3a4a4a5a51]T.

    Given that fk1 is determined from the preceding stage, seven function assessments are performed per step. To achieve an algebraic order eight, it is imperative to nullify the corresponding error truncation components (refer to [16]).

    Our technique encompasses a total of 34 parameters. As noted earlier, there exist 27 coefficients for matrix D, denoted as

    d31,d32,d41,d42,d43,,d87.

    Moreover, there are 4 coefficients associated with vector w and 3 elements pertaining to vector a. The quantity of condition equations for various orders matches those of the RKN methods [17,18], as presented in Table 1. To attain an eighth order, a cumulative total of 1+1+2+3+6+10+20+36=79 equations must be fulfilled. The equations up to the ninth order can be found in assorted tables within [16].

    Table 1.  Number of order conditions.
    Order p 1 2 3 4 5 6 7 8 9 10 11
    No of conditions 1 1 2 3 6 10 20 36 72 137 275

     | Show Table
    DownLoad: CSV

    The parameters are fewer than the equations, presenting a comparable challenge encountered in devising Runge-Kutta (RK) techniques. Hence, we are compelled to employ simplifying assumptions that diminish the quantity of conditions, thereby also decreasing the number of coefficients. The most prevalent options include

    (D1)(38)=12(a2+a)(38)(Da)(38)=16(a3a)(38)(Da2)(48)=112(a4+a)(48) (2.2)

    with

    ai=[(1)i0ai3ai4(a4)i(a5)iai51]T,

    and for κ1<κ2

    (v)(κ1κ2)=[vκ1vκ1+1vκ2]T.

    The remaining order conditions are presented in Table 2. In this table, the symbol "*" can be interpreted as element-wise multiplication:

    [u1u2un]T[v1v2vn]T=[u1v1u2v2unnn]T.
    Table 2.  Equations of condition up to eighth order, under assumptions (2.2).
    w1=1, wa2=16, wa4=115, wa6=128,
    wD2a=0, wD31=120160, wD(aDc)=1115120,
    wD3a=0, wD(aD21)=17560, w(aD2c)=1710080,
    w(aD(aDa))=1720, w(aD31)=2360480, w(D1D2a)=1720160.

     | Show Table
    DownLoad: CSV

    This operation holds lower precedence. Parentheses, exponents, and dot products are always computed prior to "*".

    Given the thirteen order conditions outlined in Table 2 and the fulfillment of 17 assumptions (2.2), we determine that only thirty equations are necessary. This results in four coefficients remaining as variables. Let's consider a3,a4,a5, and d64. The issue can be resolved explicitly, and the corresponding efficient Mathematica [19] module is depicted in Table 3.

    Table 3.  Mathematica listing for the derivation of the coefficients with respect to a3,a4,a5 and d64.
    BeginPackage["Numerov8'"];
    Clear["Numerov8'*"]
    Numerov8::usage = " Numerov8[x1, x2, x3, x4] for 7-stages 8-order explicit Numerov"
    Begin["'Private'"];
    Clear["Numerov8'Private'*"];
    Numerov8[aa3_?NumericQ, aa4_?NumericQ, aa5_?NumericQ, dd64_?NumericQ] :=
    Module[{a3, a4, a5, w, w1, w2, w4, w6, w7, a, d, d31, d32, d41, d42, d43,
         d85, d54, d61, d63, d72, d74, d53, d51, d84, d62, d52, e, so,
         d87, d75, d64, d71, d81, d83, d85, d65, d73, d82, d86, d76},
     w = {w1, w2, 0, w4, w4, w6, w6, w1};
     a = {-1, 0, a3, a4, -a4, -a5, a5, 1};
     d = {{0, 0, 0, 0, 0, 0, 0, 0},
        {0, 0, 0, 0, 0, 0, 0, 0},
        {d31, d32, 0, 0, 0, 0, 0, 0},
        {d41, d42, d43, 0, 0, 0, 0, 0},
        {d51, d52, d53, d54, 0, 0, 0, 0},
        {d61, d62, d63, d64, d65, 0, 0, 0},
        {d71, d72, d73, d74, d75, d76, 0, 0},
        {d81, d82, d83, d84, d85, d86, d87, 0}};
     e = {1, 1, 1, 1, 1, 1, 1, 1};
     a3 = Rationalize[aa3, 10^-17]; a4 = Rationalize[aa4, 10^-17];
     a5 = Rationalize[aa5, 10^-17]; d64 = Rationalize[dd64, 10^-17];
     so = Solve[{-1 + w. e, -(1/12) + w. a^2/2, -(1/360) + w. a^4/24,
           -(1/20160) + w. a^6/720} == {0, 0, 0, 0}, {w1, w2, w4, w6}];
     w = Simplify[w /. so[[1]]];
     so = Solve[
      Join[(d. e - 1/2*(a^2 + a))[[3;; 8]], (d. a - 1/6*(a^3 - a))[[3;; 8]],
         (d. a^2 - 1/12*(a^4 + a))[[4;; 8]], {w. d. d. a,
         w. d. d. d. e - 1/20160, w. d. (a d. a) + 11/15120,
         - w. d. d. d. a, w. d. (a d. d. e) + 1/7560,
         w. (a d. d. a) - 17/10080, w. (a d. (a d. a)) + 1/720,
         w. (a d. d. d. e) - 23/60480, w. (d. e d. d. a) - 17/20160}]
                                  == Array[0 &, 26],
         {d32, d31, d42, d41, d52, d51, d62, d61, d72, d71, d82, d81, d43,
         d53, d63, d73, d83, d54, d65, d74, d75, d76, d84, d85, d86, d87}];
     d = Simplify[d /. so][[1]];
     Return[{a, w, d}]]
    End[];
    EndPackage[];

     | Show Table
    DownLoad: CSV

    For comprehensive details regarding the computation of truncation error coefficients, refer to the comprehensive overview in [16]. Coleman [20] emphasized the utilization of the B2 series representation of the local truncation error, drawing connections with the T2 rooted trees.

    In [21], the scalar test problem

    z=ω2z,ωR, (3.1)

    was introduced to examine the periodic characteristics of techniques applied to solve (1.1).

    Upon employing a Numerov-style approach akin to (2.1) to tackle problem (3.1), a discrete equation is formulated, taking the form

    zk+1+S(ψ2)zk+P(ψ2)zk1=0, (3.2)

    where ψ=ωh, and S(ψ2),P(ψ2) represent polynomials in ψ2.

    The periodicity interval (0,ψ0) encompasses all 0<ψ<ψ0 with P(ψ2)1 and 0<|S(ψ2)|<2. A method deemed P-stable exhibits ψ0=.

    The fulfillment of the zero dissipation property necessitates that

    P(ψ2)=1ψ2w(Is+ψ2D)1a1,

    ensuring that the numerical method approximating (3.1) remains within its cyclic orbit.

    The dissipation order ρ of a method is characterized by the number for which 1P(ψ2)=O(ψρ). It is worth noting that

    P(ψ2)=1+j=0ψ2j+1wDja=1+ψq1+ψ3q3+.

    A method with algebraic order 2i satisfies the terms in the aforementioned series for j=0,1,,i1. Consequently, for an eighth order method, it is advantageous to address

    q9=wD4a=0,q11=wD5a=0,etc.,

    to enhance the dissipation order. In the case of a zero-dissipative method, only q9=z11=q13=q15=q17=0 is necessary, and as for the lower triangular matrix D, all other q-s vanish,

    q2i+1=wDia=0,fori>8.

    The difference in angles between the numerical and theoretical cyclic solution of (3.1) is called phase-lag. Since the solution of (3.1) is

    z(t)=eωt1,

    we may write Eq (3.2) as

    Λ=e2ψ1+S(ψ2)eψ1+P(ψ2)=O(ψτ), (3.3)

    with the number τ the phase-lag order of the method. Since

    S(ψ2)=2ψ2w(I+ψ2D)1(1+a),

    we observe that expression (3.3) is a series of the form

    Λ=i=1ψ2i(1)i+1(ij=11(2(ij))!wDj1(1+a)wDi1a2ij=11(2j)!(2(ij))!), (3.4)

    or in a compact form

    Λ=ψ2λ2+ψ4λ4+ψ6λ6+O(ψ8).

    In this series, λ2=λ4==λ2i=0 for i=1,2,,p12+1, where p denotes the algebraic order of the method. For eighth order methods, the order conditions yield λ2=λ4=λ6=λ8=0. Given that p=8, and for i=3, we infer from (3.4):

    λ6=1(2(31))!w(1+a)+12!wD(1+a)+wD212(12!4!+14!2!+16!0!)=0.

    In case of i=4, we get (observe already that w1=1,wc=0,wDc=0, etc.),

    λ8=12720160+1720(wa+w1)+124(wDa+wD1)
    +12(wD2a+wD21)+wD31=0.

    Further we have that,

    λ10=wD4111814400,
    λ12=12wD4a+wD511239500800,
    λ14=124wD4c+12wD5a+12wD51+wD612310897286400,
    λ16=1720wD4c+124wd5c+124wD51
    +12wD6c+12wD61+wD716473487131648000.

    Then we may ask for simultaneous satisfaction of phase-lag order conditions:

    λ10=0,λ12=0,λ14=0,λ16=0. (3.5)

    The set of four nonlinear equations (3.5) can be resolved to determine the four independent parameters. Our analysis reveals that the method exhibits a phase error on the order of O(ψ18), whereas the amplification error is O(ψ9). Consequently, the newly devised method demonstrates dissipative characteristics and lacks a periodicity interval.

    The free parameters satisfying (3.5) in double precision are the following [15],

    a3=0.870495922977052833,a4=0.265579060733883584,
    a5=1.11694341482497459,d64=2.43624015403357971,

    and form the method N8ph18 that outperforms other methods in oscillatory problems.

    Another noteworthy characteristic is P-stability [2,3]. In this context, it is essential to ensure σ1, while also meeting the condition

    2(2ψ2w(Isψ2D)1(1+a))2.

    Only implicit methods are capable of fulfilling these two criteria simultaneously.

    From the aforementioned set, our aim is to create a specific hybrid Numerov-style approach. The resultant technique should excel when applied to challenges exhibiting oscillatory solutions. Therefore, we opt to evaluate the following scenarios for testing purposes.

    z(x)=μ2z(t),z(0)=1,z(0)=0,t[0,10π],

    with the analytical solution z(t)=cos(μx). This scenario was tested using five distinct values of μ: specifically, μ=1,3,5,7,9. These numbers were chosen arbitrarily. Different choices will produce slightly different coefficients. Anyway, Differential Evolution is a metaheuristic method that produces random results in (hopefully) the direction of desired solutions. We may get thousands of results extremely close to each other. Consequently, we have five scenarios denoted as 1–5.

    Our current project's primary framework is rooted in [22]. Upon selection of the independent parameters a3,a4,a5,d64, we establish a method termed NEW8. Each scenario undergoes four runs with varying step counts. For each run we evaluated the maximum global error geproblem,steps observed and we record the "accurate digits" i.e., log10(geproblem,steps). The mean value r, computed over these 20 problems, serves as an efficacy metric to be optimized. To facilitate this optimization, we employ the differential evolution technique [23].

    DE operates through iterative steps, where each iteration, or generation g, involves a "population" of individuals (a(g)3,i,a(g)4,i,a(g)5,i,d(g)64,i), i=1,2,,N, with N denoting the population size. The initial population (a(0)3,i,a(0)4,i,a(0)5,i,d(0)64,i), i=1,2,,N is randomly generated in the first step. Furthermore, we designate r as the fitness function, computed as the average precision over the 20 aforementioned runs. This fitness function is then assessed for each individual within the initial population. In every generation (iteration) g, a three-step sequential process updates all individuals involved, consisting of Differentiation, Crossover, and Selection.

    We utilized MATLAB [24] software DeMat [25] for the implementation of the aforementioned technique. Indeed, notable enhancements were achieved through selection:

    a3=0.9442042052877105,a4=0.4611624530665672,a5=0.8575664014828354,d64=12.56127525577038.

    The coefficients of the new method in matrix forms are given below, which are suitable for double precision computations.

    D=(000000000000000013073231765816835116242593112904488730000001550430130344630420454652969150362054685615481128700000046437667980424298950918151054858556137848999469113561290636998791667000023846431619519460335551854734301857693862090392293777917897911057714824761030425807485391861900027674601510373998991131050235688407688186740362103009932511393142131212843869150014782512313274896071398108981421100348994066631129559599291110149698123715587744963114191118307167862726461355397241959142749796241969555741123970163167188440919931375841240640),
    w=[1341579392312827461160043108495844203060712891258074655306071289125807465542607879894937316426078798949373161341579392312827],

    and

    a=[101987811512105277124336150294026523433615029402652396673439112729975966734391127299751]T.

    With this approach, we achieved a value of approximately r9.24, which demonstrates remarkable performance. In fact, numerous methods yielding r>9.1 were obtained, indicating the presence of a narrow range of parameter combinations a3,a4,a5,d64 where r reaches elevated levels. It is noteworthy that in the current configuration, the amplification differs from unity (σ1), and the phase lag is on the order of O(v8), implying ρ=O(v8), where ρ80. Moreover, no specific property is satisfied under these conditions.

    In Table 4 we present the results for the new method and the method N8ph18 presented in [15] that was especially formed for addressing oscillatory problems. For this latter method we observe a performance ρ7.82 which is much smaller.

    Table 4.  Training phase. Accurate digits delivered after using various steps by NEW8 and N8ph18 in the interval [0,10π].
    Problem Steps NEW8 N8ph18
    1 20 7.5 6.6
    40 11.2 9.4
    60 12.3 11.0
    80 13.3 12.1
    2 50 6.0 5.4
    100 10.1 8.2
    150 11.2 9.8
    200 12.0 10.9
    3 80 5.6 5.0
    130 8.2 7.0
    180 10.8 8.3
    230 12.0 9.2
    4 100 4.7 4.4
    150 7.0 6.0
    200 8.6 7.2
    250 11.0 8.1
    5 150 5.5 4.9
    225 7.7 6.6
    300 9.6 7.7
    375 10.3 8.6

     | Show Table
    DownLoad: CSV

    The NEW8 method was designed to excel following multiple iterations on model scenarios. In the assessments outlined in Table 4, it was anticipated to outperform alternative methods for the specified intervals and step counts.

    Hence, we aim to subject NEW8 to a distinct array of challenges, encompassing varying intervals and step counts. To this end, we re-evaluate problems 1–5 over an extended interval [0,20π]. These problems are now labeled as 1,2,,5. Additionally, we introduce two additional nonlinear problems and a wave equation to broaden the scope of evaluation. Specifically, we consider:

    z(t)=100z(t)+99sint,z(0)=1,z(0)=11,t[0,20π],

    with the theoretical solution z(t)=cos(10t)+sin(10t)+sint.

    Next, we choose the equation

    z(t)=1500cos(1.01t)z(t)z(t)3,z(0)=0.2004267280699011,z(0)=0,

    with an approximate analytical solution given in [16],

    z(t){61016cos(11.11t)+4.6091013cos(9.09t)+3.7434951010cos(7.07t)+3.040149839107cos(5.05t)+2.469461432611104cos(3.03t)+0.2001794775368452cos(1.01t)}.

    Finally, we consider the linearized wave equation, which is a rather large-scale problem [16],

    ϑ2uϑt2=4ϑ2uϑx2+sintcos(πxb), 0xb=100, t[0,20π],ϑuϑx(t,0)=ϑuϑx(t,b)=0,u(0,x)0, ϑuϑt(0,x)=b24π2b2cosπxb,

    with the theoretical solution

    u(t,x)=b24π2b2sintcosπxb. (5.1)

    We discretize ϑ2uϑx2 using fourth-order symmetric differences for internal points, while boundary points utilize one-sided differences of the same order (while considering the information about ϑuϑx at those points). This results in the following system:

    [z0z1zN]=4(Δx)2[41572838918257144103742914801124352431121124352431120148297410325714418893841572][z0z1zN]+sint[cos(0Δxbπ)cos(1Δxbπ)cos(NΔxbπ)].

    Here, z0,z1zN may be understood as coordinates of zRN+1, and not as time steps. Upon selecting Δx=5, we establish a system with constant coefficients and N=20. The outcomes for this scenario were primarily influenced by the errors arising from the semi-discretization process. As a consequence, an error of about 106.1 is added constantly to the theoretical solution (5.1). Thus, no method can have a true error smaller than this. But, as shown in Table 5, our new method even though it has limited accuracy, is faster (i.e., uses fewer time steps) than N8ph18.

    Table 5.  Numerical tests phase. Accurate digits delivered after using various steps by NEW8 and N8ph18 in the interval [0,20π].
    Problem Steps NEW8 N8ph18
    1 40 7.2 6.3
    80 10.9 9.1
    120 12.0 10.7
    160 12.9 11.8
    2 100 5.7 5.1
    200 9.8 7.9
    300 10.9 9.5
    400 11.7 10.6
    3 160 5.2 4.7
    260 7.9 6.7
    360 10.5 8.0
    460 12.0 8.9
    4 200 4.4 4.1
    300 6.7 5.7
    400 8.3 6.9
    500 10.7 7.8
    5 300 5.2 4.6
    450 7.4 6.2
    600 9.3 7.4
    750 10.0 8.3
    6 240 2.9 3.0
    480 7.0 5.9
    720 10.1 7.5
    960 10.1 8.6
    7 100 4.8 4.9
    200 7.7 7.3
    300 9.3 8.7
    400 10.4 9.7
    8 60 6.0 5.0
    70 6.1 5.4
    80 6.1 5.8
    90 6.1 5.9

     | Show Table
    DownLoad: CSV

    We execute these 8 scenarios with varying step counts and present the outcomes in Table 5. Notably, we also incorporate results obtained using the N8ph18 method. For economy and ease of reading the results, only the best methods of eighth order were tested on oscillatory problems. i.e., NEW8 and N8ph18. N8ph18 has already proven to outperform other 8th order methods [15,16]. It becomes evident from the table that NEW8 significantly outperforms all other methods documented in the literature. Overall, an improvement of nearly one decimal digit in accuracy was achieved.

    The proposed method is constructed for application to second order Ordinary Differential Equations (ODEs) with oscillatory solutions. However, this is a rather wide category of problems that is constantly under the interest of respected scholars. As seen from problem 8 (wave equation), our method may also apply to a certain kind of partial differential equations sharing periodic solutions after proper transformation to system of ODEs.

    The key aspects of our investigation were as follows:

    ● We explored a family of eighth-order hybrid two-step techniques characterized by minimal stage counts, with a notable innovation being the proposal of a methodology for selecting appropriate independent parameters.

    ● The parameters of the novel technique were determined following extensive evaluation of their performance across a diverse array of periodic scenarios.

    ● Optimal parameter selection was achieved through the application of the differential evolution approach. Across a broad spectrum of challenges featuring oscillatory solutions, the devised approach demonstrated significant superiority over methods belonging to both similar and disparate families.

    ● The method we introduced is finely calibrated for scenarios with periodic solutions, particularly those featuring substantial linear components.

    Both authors of this article have been contributed equally. Both authors have read and approved the final version of the manuscript for publication.

    The authors declare that no Artificial Intelligence (AI) tools were used in the creation of this article.

    This work does not have any conflicts of interest.



    [1] D. S. Timberlake, A comparison of drug use and dependence between blunt smokers and other cannabis users, Subst. Use Misuse, 44 (2009), 401−415. https://doi.org/10.1080/10826080802347651 doi: 10.1080/10826080802347651
    [2] L. E. Zimmer, J. P. Morgan, Marijuana myths, marijuana facts: A review of the scientific evidence, Lindes Center NY, (1997).
    [3] A. Berenson, Tell your children: The truth about marijuana, mental illness, and violence, Free Press, (2020).
    [4] C. L. Hart, W. van Gorp, M. Haney, R. W. Foltin, M. W. Fischman, Effects of acute smoked marijuana on complex cognitive performance, Neuropsychopharmacol, 25 (2001), 757−765. https://doi.org/10.1016/S0893-133X(01)00273-1 doi: 10.1016/S0893-133X(01)00273-1
    [5] R. L. Pacula, J. Ringel, Does marijuana use impair human capital formation?, NBERC, (2003). https://doi.org/10.3386/w9963 doi: 10.3386/w9963
    [6] W. H. Jeynes, Adolescent religious commitment and their consumption of marijuana, cocaine, and alcohol, J. Health Soc. Policy, 21 (2006), 1−20. https://doi.org/10.1300/J045v21n04_01 doi: 10.1300/J045v21n04_01
    [7] S. F. Tapert, A. D. Schweinsburg, S. A. Brown, The influence of marijuana use on neurocognitive functioning in adolescents, Curr. Drug Abuse Rev., 1 (2008), 99−111. https://doi.org/10.2174/1874473710801010099 doi: 10.2174/1874473710801010099
    [8] G. Thomas, R. A. Kloner, S. Rezkalla, Adverse cardiovascular, cerebrovascular, and peripheral vascular effects of marijuana inhalation: What cardiologists need to know, AM. J. Cardiol., 113 (2014), 187−190. https://doi.org/10.1016/j.amjcard.2013.09.042 doi: 10.1016/j.amjcard.2013.09.042
    [9] H. Wen, J. Hockenberry, J. R. Cummings, The effect of medical marijuana laws on marijuana, alcohol, and hard drug use, NBER, (2014). https://doi.org/10.1016/j.amjcard.2013.09.042 doi: 10.1016/j.amjcard.2013.09.042
    [10] M. A. Alsherbiny, C. G. Li, Medicinal cannabis—potential drug interactions, Medicines, 6 (2018), 3. https://doi.org/10.3390/medicines6010003 doi: 10.3390/medicines6010003
    [11] A. Bhatnagar, W. Maziak, T. Eissenberg, K. D. Ward, G.Thurston, B. A. King, et al., Water pipe (hookah) smoking and cardiovascular disease risk: A scientific statement from the American Heart Association, Circulation, 139 (2019), e917−e936. https://doi.org/10.1161/CIR.0000000000000671 doi: 10.1161/CIR.0000000000000671
    [12] Aruni Bhatnagar, L. P. Whitsel, K. M. Ribisl, C. Bullen, F. Chaloupka, M. R. Piano, et al., Electronic cigarettes: A policy statement from the American Heart Association, Circulation, 130 (2014), 1418−1436. https://doi.org/10.1161/CIR.0000000000000107 doi: 10.1161/CIR.0000000000000107
    [13] J. E. Layden, I. Ghinai, I. Pray, A. Kimball, M. Layer, M. W. Tenforde, et al., Pulmonary illness related to e-cigarette use in Illinois and Wisconsin, N. Eng. J. Med., 382 (2020), 903−916. https://doi.org/10.1056/NEJMoa1911614 doi: 10.1056/NEJMoa1911614
    [14] T. M. Kaufman, S. Fazio, M.D. Shapiro, Brief commentary: Marijuana and cardiovascular disease—what should we tell patients? Ann. Intern. Med., 170 (2019), 119−120. https://doi.org/10.7326/M18-3009 doi: 10.7326/M18-3009
    [15] S. Mushayabasa, G. Tapedzesa, Modeling illicit drug use dynamics and its optimal control analysis, Comput. Math. Methods Med., 2015 (2015).
    [16] A. L. Bretteville-Jensen, H. O. Melberg, A. M. Jones, Sequential patterns of drug use initiation-Can we believe in the gateway theory?, BE. J. Econ. Anal. Policy, 8 (2008). https://doi.org/10.2202/1935-1682.1846 doi: 10.2202/1935-1682.1846
    [17] A. Ghaffar, M. Iqbal, M. Bari, S. M. Hussain, R. Manzoor, K. S. Nisar, et al., Construction and application of nine-tic B-spline tensor product SS, Math, 7 (2019), 675. https://doi.org/10.3390/math7080675 doi: 10.3390/math7080675
    [18] M. E. Passey, R. W. Sanson-Fisher, C. A. D'Este, J. M. Stirling, Tobacco, alcohol and cannabis use during pregnancy: clustering of risks, Drug. Alcohol. Dep., 134 (2014), 44−50. https://doi.org/10.1016/j.drugalcdep.2013.09.008 doi: 10.1016/j.drugalcdep.2013.09.008
    [19] L. Cristino, V. Di Marzo, Fetal cannabinoid receptors and the "dis‐joint‐ed" brain, EMBO J., 33 (2014), 665−667. https://doi.org/10.1002/embj.201488086 doi: 10.1002/embj.201488086
    [20] A. Shah, H. Khan, M. De la Sen, J. Alzabut, S. Etemad, C. T. Deressa, et al., On non-symmetric fractal-fractional modeling for ice smoking: Mathematical analysis of solutions, Symmetry, 15 (2022), 87. https://doi.org/10.3390/sym15010087 doi: 10.3390/sym15010087
    [21] T. D. Metz, E. H. Stickrath, Marijuana use in pregnancy and lactation: A review of the evidence, Am. J. Obstet. Gynecol., 213 (2015), 761−778. https://doi.org/10.1016/j.ajog.2015.05.025 doi: 10.1016/j.ajog.2015.05.025
    [22] A. Alkhazzan, J. Wang, Y. Nie, H. Khan, J. Alzabut, A stochastic SIRS modeling of transport-related infection with three types of noises, Alexandria Eng. J., 76 (2023), 557−572. https://doi.org/10.1016/j.aej.2023.06.049 doi: 10.1016/j.aej.2023.06.049
    [23] J. D. Ellis, S. M. Resko, K. Szechy, R. Smith, T. J. Early, Characteristics associated with attitudes toward marijuana legalization in Michigan, J. Psychoact. Drugs, 51 (2019), 335−342. https://doi.org/10.1080/02791072.2019.1610199 doi: 10.1080/02791072.2019.1610199
    [24] R. Dapari, M. H. Mahfot, A. M. Nazan, M. R.Hassan, Na. C. Dom, S. S. S. A. Rahim, Acceptance towards decriminalization of medical marijuana among adults in Selangor, Malaysia, Plos One, 17 (2022), e0262819. https://doi.org/10.1371/journal.pone.0262819 doi: 10.1371/journal.pone.0262819
    [25] H. Khan, J. Alzabut, A. Shah, S. Etemad, S. Rezapour, C. Park, A study on the fractal-fractional tobacco smoking model, AIMS Math., 7 (2022), 13887−13909. https://doi.org/10.3934/math.2022767 doi: 10.3934/math.2022767
    [26] A. Khan, J. F. Gómez-Aguilar, T. S. Khan, H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos Solitons Fractal, 122 (2019), 119−128. https://doi.org/10.1016/j.chaos.2019.03.022 doi: 10.1016/j.chaos.2019.03.022
    [27] W. Shatanawi, A. Raza, M. Shoaib Arif, M. Rafiq, M. Bibi, M. Mohsin, Essential features preserving dynamics of stochastic Dengue model, Model Eng. Sci., 126 (2021), 201−215. https://doi.org/10.32604/cmes.2021.012111 doi: 10.32604/cmes.2021.012111
    [28] R. J. Evans-Polce, M. E. Patrick, S. E. McCabe, R. A. Miech, Prospective associations of e-cigarette use with cigarette, alcohol, marijuana, and nonmedical prescription drug use among US adolescents, Drug Alcohol. Dep., 216 (2020), 108303. https://doi.org/10.1016/j.drugalcdep.2020.108303 doi: 10.1016/j.drugalcdep.2020.108303
    [29] S. Pisanti, M. Bifulco, Medical cannabis: A plurimillennial history of an evergreen, J. Cell Physiol., 234 (2019), 8342−8351. https://doi.org/10.1002/jcp.27725 doi: 10.1002/jcp.27725
    [30] E. Krediet, D. G. A. Janssen, E. R. Heerdink, T. C. G. Egberts, E. Vermetten, Experiences with medical cannabis in the treatment of veterans with PTSD: Results from a focus group discussion, Eur. Neuropsychopharmacol., 36 (2020), 244−254. https://doi.org/10.1016/j.euroneuro.2020.04.009 doi: 10.1016/j.euroneuro.2020.04.009
    [31] J. Rehm, T. Elton-Marshall, B. Sornpaisarn, J. Manthey, Medical marijuana. What can we learn from the experiences in Canada, Germany and Thailand?, Int. J. Drug Policy, 74 (2019), 47−51. https://doi.org/10.1016/j.drugpo.2019.09.001 doi: 10.1016/j.drugpo.2019.09.001
    [32] N. R. Dash, G. Khoder, A. M. Nada, M. T. Al-Bataineh, Correction: Exploring the impact of Helicobacter pylori on gut microbiome composition, Plos One, 16 (2021), e0256274. https://doi.org/10.1371/journal.pone.0256274 doi: 10.1371/journal.pone.0256274
    [33] A. Elmazny, S. M. Hamdy, M. Abdel-Naseer, N. M. Shalaby, H. S. Shehata, N. A. Kishk, et al., Interferon-beta-induced headache in patients with multiple sclerosis: Frequency and characterization, J. Pain. Res., (2020), 537−545. https://doi.org/10.2147/JPR.S230680 doi: 10.2147/JPR.S230680
    [34] D. C. Oshi, T. Ricketts-Roomes, S. N. Oshi, K. A. Campbell-Williams, T. Ricketts-Roomes, S. N. Oshi, et al., Factors associated with awareness of decriminalization of marijuana in Jamaica, J. Subst. Use, 25 (2020), 152−156. https://doi.org/10.1080/14659891.2019.1664671 doi: 10.1080/14659891.2019.1664671
    [35] N. R. A. Rahman, F. Mohmad, M. S. Isuan, M. R. Isa, Towards a higher percentage of recreational drugs-free among patients on methadone maintenance treatment in Larkin Health Clinic, Q. Bull., 1 (2021), 60−71.
    [36] E. Cook, State of the death penalty in Southeast Asia, Eureka Street, 28 (2018), 13−15.
    [37] J. M. Ginoux, R. Naeck, Y. B. Ruhomally, M. Z. Dauhoo, M. Perc, Chaos in a predator–prey-based mathematical model for illicit drug consumption, Appl. Math. Comput., 347 (2019), 502−513. https://doi.org/10.1016/j.amc.2018.10.089 doi: 10.1016/j.amc.2018.10.089
    [38] M. Z. Dauhoo, B. S. N. Korimboccus, S. B. Issack, On the dynamics of illicit drug consumption in a given population IMA J. Appl. Math., 78 (2013), 432−448. https://doi.org/10.1093/imamat/hxr058 doi: 10.1093/imamat/hxr058
    [39] A. Ullah, H. Sakidin, S. Gul, K. Shah, M. S. Muthuvalu, T. Abdeljawad, et al., Sensitivity analysis-based validation of the modified NERA model for improved performance, J. Adv. Res. Appl. Sci. Eng. Technol., 32 (2023), 1−11. https://doi.org/10.37934/araset.32.3.111 doi: 10.37934/araset.32.3.111
    [40] M. Zamir, G. Zaman, A. S. Alshomrani, Sensitivity analysis and optimal control of anthroponotic cutaneous leishmania, PloS One, 11 (2016), e0160513. https://doi.org/10.1371/journal.pone.0160513 doi: 10.1371/journal.pone.0160513
    [41] A. Ullah, H. Sakidin, S. Gul, K. Shah, Y. Hamed, M. Aphane, et al., Sensitivity analysis-based control strategies of a mathematical model for reducing marijuana smoking, AIMS Bioeng, 10 (2023), 491−510. https://doi.org/10.3934/bioeng.2023028 doi: 10.3934/bioeng.2023028
    [42] W. A. Khan, R. Zarin, A. Zeb, Y. Khan, A. Khan, Navigating food allergy dynamics via a novel fractional mathematical model for antacid-induced allergies, J. Math. Technol. Model., 1 (2024), 25−51. https://orcid.org/0009-0004-0810-280X
    [43] M. Zamir, T. Abdeljawad, F. Nadeem, A. Wahid, A. Yousef, An optimal control analysis of a COVID-19 model, Alexandria Eng. J., 60 (2021), 2875−2884. https://doi.org/10.1016/j.aej.2021.01.022 doi: 10.1016/j.aej.2021.01.022
    [44] P. Liu, A. Din, R. Zarin, Numerical dynamics and fractional modeling of hepatitis B virus model with non-singular and non-local kernels, Results Phys., 39 (2022), 105757. https://doi.org/10.1016/j.rinp.2022.105757 doi: 10.1016/j.rinp.2022.105757
    [45] F. M. Khan, Z. U. Khan, Numerical analysis of fractional order drinking mathematical model, J. Math. Technol. Model., 1 (2024), 11−24. https://doi.org/10.56868/jmtm.v1i1.4 doi: 10.56868/jmtm.v1i1.4
    [46] J. Yang, M. Gong, G. Q. Sun, Asymptotical profiles of an age-structured foot-and-mouth disease with nonlocal diffusion on a spatially heterogeneous environment, J. Differ. Equations, 377 (2023), 71−112. https://doi.org/10.1016/j.jde.2023.09.001 doi: 10.1016/j.jde.2023.09.001
    [47] M. Zamir, R. Sultana, R. Ali, W. A. Panhwar, S. Kumar, Study on the threshold conditions for infection of visceral leishmaniasis, Surj-Sci Ser, 47 (2015), 619−622.
    [48] M. Zamir, F. Nadeem, M. A. Alqudah, T. Abdeljawad, Future implications of covid-19 through mathematical modeling, Result. Phys., 33 (2022), 105097. https://doi.org/10.1016/j.rinp.2021.105097 doi: 10.1016/j.rinp.2021.105097
    [49] V. Madhusudanan, M. N. Srinivas, B. S. N. Murthy, K. J. Ansari, A. Zeb, A. Althobaiti, et al., The influence of time delay and Gaussian white noise on the dynamics of tobacco smoking model, Chaos Soliton Fractal, 173 (2023), 113616. https://doi.org/10.1016/j.chaos.2023.113616 doi: 10.1016/j.chaos.2023.113616
    [50] A. Din, Y. Li, Q. Liu, Viral dynamics and control of hepatitis B virus (HBV) using an epidemic model, Alexandria Eng. J., 59 (2020), 667−679. https://doi.org/10.1016/j.aej.2020.01.034 doi: 10.1016/j.aej.2020.01.034
    [51] A. Din, Y. Li, T. Khan, G. Zaman, Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China, Alexandria Eng. J., 141 (2020), 110286. https://doi.org/10.1016/j.chaos.2020.110286 doi: 10.1016/j.chaos.2020.110286
    [52] S. Bushnaq, K. Shah, S. Tahir, K. J. Ansari, M.Sarwar, T. Abdeljawad, Computation of numerical solutions to variable order fractional differential equations by using non-orthogonal basis, AIMS Math., 7 (2022), 10917−10938. https://doi.org/10.3934/math.2022610 doi: 10.3934/math.2022610
    [53] L. Rarità, C. D'Apice, B. Piccoli, D. Helbing, Sensitivity analysis of permeability parameters for flows on Barcelona networks, J. Differ. Equations, 249 (2010), 3110−3131. https://doi.org/10.3934/math.2022610 doi: 10.3934/math.2022610
    [54] N. Ahmad, S. Charan, V. P. Singh, Study of numerical accuracy of Runge-Kutta second, third and fourth order method, Int. J. Comput. Math. Sci., 4 (2015), 111.
    [55] M. A. Islam, Accurate solutions of initial value problems for ordinary differential equations with the fourth order Runge Kutta method, J. Math. Res., 7 (2015), 41. https://doi.org/10.5539/jmr.v7n3p41 doi: 10.5539/jmr.v7n3p41
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