^{1}

^{*}

^{2}

^{2}

Unemployment is one of the major vices in our contemporal society, which weigh greatly on the economy of such nation. It is also, a fact that knowing ones enemy before battle gives 50 per cent chance of victory ; thus, this research aimed at providing understanding about dynamics of unemployment with consideration for retirement and possible control criterion. And the objectives are; formulation of mathematical model using the concept of deterministic model and mathematical epidemiology ; then, model analysis. The model analysis include s , a numerical semi-analytical scheme for investigating validity of analytical solutions. The result of the analysis were that: 1 ) the model was mathematically well-pose and biologically meaningful 2 ) two equilibria points exist, and 3 ) a threshold for recruitment from the pool of unemployment, assuring victory in the fight against unemployment was also, obtained. The threshold is required to be well managed in order to win the battle against the socio-vice (unemployment) in the contemporary society. In addition, variational iterative method (VIM) is the numerical semi-analytic scheme employed to solve the mode l; thus, the approximate solution gave a practical meaningful interpretation supporting the analytical results and proof of verdict of assumptions of the model. The article concluded with three points; everyone has roles to play to curtail the socio-menace, beseech government and policy makers to look kindly, and create policy(ies) to sustain population growth, and the retiree should also, plan live after service, because over dependence on pension scheme could be died before death because of corruption in the scheme.

Securing a job or gain employment in our contemporary society is problematic for an average people and below; not to talk of desire job and underemployment is not something to rule out. In reference to Bureau of Labour Statistics defines unemployment as people who do not have a job, have actively looked for work in the past 4 weeks, and are currently available for work but could not get one or people who were temporarily laid off and were waiting to be called back to that job, Kimberly [

The unemployment is calculated as a percentage by dividing the number of unemployed individuals by all individual currently in labour force. During period of recession, an economy usually experiences a relatively high unemployment rate. According to the International Labour Organization (ILO), more than 200 million people globally or 6% of the work force were without job in 2012.

Economy of every nation is characterized by both active and inactive populations. The economically active ones are referred to as population willing and able to work and include those actively engaged in the production of goods and services and those who are employed, Muogbo and John-Akameli [

Nigeria has a challenge of accurate unemployment rates (i.e. data) to access. However according to Oyebade [

From

expectations of the primary school graduates. The causes of unemployment are heavily debated. Classical economics, new classical economics and the Austrian school of Economics argued that the market mechanisms are the reliable means of resolving means of unemployment; these theories against intervention imposed on the labour market from the outside such as unionization, bureaucratic work rules, minimum wages laws, taxes and other regulations that can claim discourage in the hiring of workers.

The problem of youth unemployment is very evident and severe in Nigeria. Every year, thousands of graduates are turn out for whom there are no jobs to accommodate them. Some are youth hawkers and bike riders who ordinarily would have found gained employment in some enterprises, or would have demonstrated their skills and resourcefulness if there are enabling environment and reliable management structure on ground. Instead, today youth have now shifted attention to cyber crime, terrorism and other criminal acts. It is the large number of youths who are unemployed capable of undermining democratic practices, constitute a serious threat if engaged by a political classes for clandestine activities.

In the study of youth unemployment activities especially in Nigeria, Adebayo [

Uddin and Osemengbe [

Another cause of unemployment in Nigeria is rapid growth rate. According tk worldometers, the current population of Nigeria is 196,364,552 as of Saturday 11th August, 2018, which was said to be the latest United Nations estimates while country meters read as of Sunday, 12th August, 2018, Nigeria population to be approximate 197.8 million. With this population, Nigeria is the most populous nation in Africa, the high population has resulted to rapid growth of labour force which is far outstripping the supply of jobs.

About mathematical contribution to knowledge of unemployment and method of analysis include; Pathan and Bhathawala [

Galindro and Torres [

Edogbanya and Ibrahim [_{0} of the model and used it as a threshold condition, they also carried out the sensitivity analysis on the parameters and carry out numerical simulations with respect to the data.

Akinboro et al. [

In view of all above, it is obvious that unemployment is one of the identified menace of contemprory society, especially, in Nigeria. To combat this problem, everybody is required to put positive effort and here I propose mathematical model to study the dynamics in population and set threshold to contain the challenge.

The deterministic mathematical model was used to study the dynamics of unemployment. Individuals in the population are divided into compartments depending on the status of individual. The total human population at time t denoted by P ( t ) , is divided into four sub-population of Unemployment U ( t ) , and Unemployment compartment splitted into active, U 2 ( t ) and passive, U 1 ( t ) , E ( t ) means Employment compartment, R ( t ) denotes Retired compartment and lastly V ( t ) represent vacancy/position available for recruitment compartment.

We assumed that there is an invisible interaction among all the human compartments and the vacancy compartment (this is represented with dotted line in the model diagram). The recruitment rate into the passive unemployment compartment is π 1 , π 2 is the recruitment rate into the active unemployment, k 1 is the rate at which individual moves from passive unemployment state, k 2 is the rate at which individual moves from active unemployment, to employment compartment, ω is the rate by which individual moves from employment class to retired. It is also assumed that people can still move from retired to both active and passive unemployment and the rate of their movements are denoted respectively by β 1 and β 2 . ϕ is the vacancy creation of the employed people, γ 1 and γ 2 are the rate at which the employed people become unemployed probably, dissmisal resulting from misconduct or lack of productivity, δ are those that dies while working, η is unpublicized available vacancies and μ is the natural death rate.

The model flow diagram (

d U 1 d t = π 1 − μ U 1 − k 1 U 1 E + γ 1 E + β 1 R (1)

d U 2 d t = π 2 − μ U 2 − k 2 U 2 E + γ 2 E + β 2 R (2)

d E d t = ( k 1 U 1 + k 2 U 2 ) E − ( μ + δ + γ 1 + γ 2 − ω ) E (3)

d R d t = ω E − ( μ + β 1 + β 2 ) R (4)

d V d t = ϕ U 2 + γ E − η V (5)

1) Only people of average and/or below (in a population) seek for job;

2) It is assumed that some unemployed individuals are not employable, yet, they seek for employment;

3) It is also part of assumption that some employed individuals have an establishment and also recruit from the pool of unemployment;

4) Some retired individuals added or return to unemployment population in either passive or active way;

5) For as long as people retire from job or loosing their job, there will always be vacancy and recruitment except it may not be publicized;

6) Available vacancy is somehow reserve for special people;

7) It is also assumed that the model considered able human being and dynamics of vacant positions; and

8) It is assumed that not everyone in the active compartment is looking for job.

The validity of any mathematical model depends whether the system of equations has a solution and communicate practical sense. If yes, is the solution unique? This subsection is concerned with examining if the system of equations has a solution(s) and possesses uniqueness, feasibility and positivity of solution properties.

Theorem 2.1.

Let Ω denote the region 0 ≤ α ≤ R , the system Equations (1)-(5) with initial conditions U 1 ( 0 ) > 0 , U 2 ( 0 ) > 0 , E ( 0 ) > 0 , R ( 0 ) > 0 and V ( 0 ) ≥ 0 , exit, bounded, positively invariant and attracting for all t > 0 .

Proof.

1) For existence and uniqueness of solution, we show that

∂ f i ∂ x j , i , j = 1 , 2 , 3 , 4 , 5 ,

are continuous and bounded in Ω .

From (1), F 1 = π 1 − μ U 1 − k 1 U 1 E + γ 1 E + β 1 R

∂ f 1 ∂ U 1 = − μ − k 1 E , | ∂ f 1 ∂ U 1 | = | − μ − k 1 E | < ∞ , | ∂ f 1 ∂ U 2 | = | ∂ f 1 ∂ V | = 0 < ∞ ,

| ∂ f 1 ∂ E | = | − k 1 u 1 + γ 1 | < ∞ , | ∂ f 1 ∂ R | = | β 1 | < ∞

From (2), F 2 = π 2 − μ U 2 − k 2 U 2 E + γ 2 E + β 2 R

| ∂ f 2 ∂ U 1 | = 0 < ∞ , | ∂ f 2 ∂ U 2 | = | − μ − k 2 E | < ∞ , | ∂ f 2 ∂ E | = | γ 2 | < ∞ ,

| ∂ f 2 ∂ R | = | β 2 | < ∞ , | ∂ f 2 ∂ V | = 0 < ∞

Same approach for the remaining equations and consider

d P ( t ) d t = π 1 + π 2 − μ ( U 1 + U 2 + E + R ) − δ E

d P ( t ) d t ≤ π − μ P (t)

where π = π 1 + π 2

d P d t + μ P ≤ π (6)

Integrating (6)

P ( t ) ≤ e − μ t ( π μ e μ t + C )

Using initial condition (i.e. P ( t 0 ) = P 0 at t = 0 ), it implies that

P ( t ) ≤ P 0 e − μ t + π μ ( 1 − e − μ t ) (7)

⇒ lim t → ∞ P ( t ) = π μ

Hence, π μ is the upper bound of P ( t ) .

Thus, when P 0 > π μ , the solution of { U 1 ( t ) , U 2 ( t ) , E ( t ) , R ( t ) } is bounded in the region of Ω .

Clearly, the partial derivatives of the whole system of equations exists, finite and bounded.

2) For feasibility region of solution

Let P = U 1 + U 2 + E + R ∈ ℝ + 4

d P d t = d U 1 d t + d U 2 d t + d E d t + d R d t ≤ π 1 + π 2 μ

Therefore, 0 ≤ P ≤ π 1 + π 2 μ and conclude that

Ω = { ( U 1 ( t ) , U 2 ( t ) , E ( t ) , R ( t ) ) ⊆ ℝ + 4 : 0 ≤ P ≤ π 1 + π 2 μ }

is the feasible region.

3) For positivity of solution, from the proposition of the theorem, initial condition of the system is { ( U 1 ( 0 ) , U 2 ( 0 ) , E ( 0 ) , R ( 0 ) , V ( 0 ) ) > 0 } , then, we show that the solution set ( U 1 ( t ) , U 2 ( t ) , E ( t ) , R ( t ) , V ( t ) ) of the system remain non-negative for all t > 0

d U 1 d t = π 1 − U 1 ( μ + K 1 E ) + γ 1 E + β 1 R

d U 1 d t ≥ − ( μ + k 1 E ) U 1

∫ d U 1 U 1 ≥ ∫ − ( μ − k 1 E ) d t

Integrating both side

U 1 ( t ) ≥ U 1 ( 0 ) e − μ t ≥ 0

Also,

d U 2 d t = π 2 − μ U 2 − k 2 U 2 E + γ 2 E + β 2 R

d U 2 d t = π 2 − U 2 ( μ + k 2 E ) + γ 2 E + β 2 R

d U 2 d t ≥ − ( μ + K 2 E ) U 2

∫ d U 2 U 2 ≥ ∫ − ( μ + k 2 E ) d t

ln U 2 ≥ − ( μ + k 2 E ) t + C

U 2 ≥ e − μ t + C

U 2 ( t ) ≥ U 2 ( 0 ) e − μ t ≥ 0

Similarly, d R d t = ω E − ( μ + β 1 + β 2 ) R

d R d t ≥ − ( μ + β 1 + β 2 ) R

∫ d R R ≥ ∫ − ( μ + β 1 + β 2 ) R

R ( t ) ≥ R ( 0 ) e − ( μ + β 1 + β 2 ) t ≥ 0

By deducation,

V ( t ) ≥ V ( 0 ) e − η t > 0

Hence, for t = 0 , all the state variables are non-negative. And this completes the proof.

We proceed to investigate the equilibrium points for the model, at equilibrium Equations (1)-(5) become

0 = π 1 − μ U 1 − k 1 U 1 E + γ 1 E + β 1 R (8)

0 = π 2 − μ U 2 − k 2 U 2 E + γ 2 E + β 2 R (9)

0 = ( k 1 U 1 + k 2 U 2 ) E − ( μ + δ + γ 1 + γ 2 + ω ) E (10)

0 = ω E − ( μ + β 1 + β 2 ) R (11)

0 = ϕ U 2 + γ E − η V (12)

In the absent of recruitment, E = 0 , R = 0 , to have

U 1 = π 1 μ , U 2 = π 2 μ , V = ϕ U 2 η

Therefore, the absent of recruitment equilibrium ( E * ) is obtained as

E * = { ( U 1 , U 2 , E , R , V ) : ( π 1 μ , π 2 μ , 0 , 0 , ϕ π 2 η μ ) } (13)

And persistent recruitment equilibrium, we assume E = 1 , solving for the remaining variables, we get R = ω μ + β 1 + β 2 , U 1 = ( π 1 + γ 1 ) ( μ + β 1 + β 2 ) + β 1 ω ( K 1 + μ ) ( μ + β 1 + β 2 ) , U 2 = ( π 2 + γ 2 ) ( μ + β 1 + β 2 ) + β 1 ω ( K 2 + μ ) ( μ + β 1 + β 2 ) , 1, ϕ U 2 + γ η .

Therefore the equilibrium of persistent recruitment denoted by E ** obtained as

E ** = { ( U 1 , U 2 , E , R , V ) : ( π 1 + γ 1 ) ( μ + β 1 + β 2 ) + β 1 ω ( K 1 + μ ) ( μ + β 1 + β 2 ) , ( π 2 + γ 2 ) ( μ + β 1 + β 2 ) + β 1 ω ( K 2 + μ ) ( μ + β 1 + β 2 ) , 1 , ω μ + β 1 + β 2 , ϕ U 2 + γ η } (14)

To examine the local stability of E * given by (13), is is imperative to obtain threshold parameter called basic reproduction number ( R 0 ) in the sense of epidemiology, which is defined as the average number of secondary infections caused by a typical infectious individual during its period of infectiousness in a completely susceptible population, (Samson et al., [

We consider only the Employment and Retired compartments and with approach of next generation matrix (NGM)

d d t ( E R ) = ( ( k 1 U 1 + k 2 U 2 ) E 0 ) − ( ( μ + δ + γ 1 + γ 2 + ω ) E − ω E + ( μ + β 1 + β 2 ) R )

For which recruitment matrix (F) and transition matrix (V) respectively deduced as

F = ( k 1 U 1 + k 2 U 2 0 0 0 ) and V = ( μ + δ + γ 1 + γ 2 + ω 0 − ω μ + β 1 + β 2 )

to obtain threshold for recruitment, which is the spectral radius of the NGM, written mathematically as

R e = ρ ( F V − 1 ) = k 1 U 1 ∗ + k 2 U 2 ∗ μ + δ + γ 1 + γ 2 + ω (15)

R e is the parameter that governs unemployment pool. When R e < 1 ; the situation for which the vacancy and employment population is more than unemployed individuals and it is closely observed. Otherwise, the economy of such soceity will greatly be affected.

Here, we carry out the steady states analysis; Absent of Recruitment steady state and the Persistent Recruitment steady state.

Therorem 2.2.

In the absence of recruitment equilibrium point, Equations (1)-(5) is locally stable if R e < 1 and unstable if otherwise.

Proof.

To check for the stability analysis of the recruitment free equilibrium of the model, we have to obtain the Jacobian matrix of the given Equations (1)-(5) as follow

J = ( − ( μ + k 1 E ) 0 − k 1 U 1 + γ 1 β 1 0 0 − ( μ + k 2 E ) − k 2 U 2 + γ 2 β 2 0 k 1 k 2 k 1 U 1 + k 2 U 2 − ( μ + δ + γ 1 + γ 2 ) 0 0 0 0 ω − ( μ + β 1 + β 2 ) 0 0 ϕ γ 0 − η ) (16)

J E = 0 = ( − μ 0 − k 1 U 1 + γ 1 β 1 0 0 − μ − k 2 U 2 + γ 2 β 2 0 k 1 k 2 k 1 U 1 + k 2 U 2 − ( μ + δ + γ 1 + γ 2 + ω ) 0 0 0 0 ω − ( μ + β 1 + β 2 ) 0 0 ϕ γ 0 − η ) (17)

| J − λ I | = | − ( μ + λ ) 0 − k 1 U 1 + γ 1 β 1 0 0 − ( μ + λ ) − k 2 U 2 + γ 2 β 2 0 k 1 k 2 ( k 1 U 1 + k 2 U 2 − ( μ + δ + γ 1 + γ 2 + ω ) ) − λ 0 0 0 0 ω − ( μ + β 1 + β 2 + λ ) 0 0 ϕ γ 0 − ( η + λ ) |

| J − I λ | = ( μ + λ ) 2 [ ( k 1 U 1 + k 2 U 2 − ( μ + δ + γ 1 + γ 2 + ω ) ) − λ ] × ( μ + β 1 + β 2 + λ ) ( η + λ ) = 0

Solving the above expression to obtain values for the λ’s, we have;

λ 1 = − μ , λ 2 = − μ , λ 3 = k 1 U 1 ∗ + k 2 U 2 ∗ − ( μ + δ + γ 1 + γ 2 + ω ) , λ 4 = − ( μ + β 1 + β 2 ) , λ 5 = − η (18)

Condition for stability is that, all eigenvalues of the Jacobian matrix must be less than zero [

Conversely, if λ 3 > 0 , then, the case employment free equilibrium will be unstable and further analysis would be required, which it not the interest of this research.

Applying the Semi-analytic Numerical Scheme; VIM and employ concept used in [

U 1 , K + 1 ( t ) = U 1 ( K ) + ∫ 0 t λ 1 ( t ) ( d U 1 ( K ) d t − π 1 + ( μ + k 1 E ( K ) ) U 1 ( K ) − γ 1 E ( K ) + β 1 R ( K ) ) d t U 2 , K + 1 ( t ) = U 2 ( K ) + ∫ 0 t λ 2 ( t ) ( d U 2 ( K ) d t − π 2 + ( μ + k 2 E ( K ) ) U 2 ( K ) − γ 2 E ( K ) + β 2 R ( K ) ) d t

E K + 1 ( t ) = E ( K ) + ∫ 0 t λ 3 ( t ) ( d E ( K ) d t − ( k 1 U 1 ( K ) + k 2 U 2 ( K ) ) E ( K ) + ( μ + δ + γ 1 + γ 2 ) E ( K ) + β 2 R ( K ) ) d t R k + 1 ( t ) = R ( K ) + ∫ 0 t λ 4 ( t ) ( d R ( K ) d t − w E ( K ) + ( μ + β 1 + β 2 ) R ( K ) ) d t V K + 1 ( t ) = V ( K ) + ∫ 0 t λ 5 ( t ) ( d V ( K ) d t − ϕ U 2 ( K ) − γ E ( K ) + η V ( K ) ) d t (19)

it is shown from Akinboro et al., [

U 1 , K + 1 ( t ) = U 1 ( K ) − ∫ 0 t ( d U 1 ( K ) d t − π 1 + ( μ + k 1 E ( K ) ) U 1 ( K ) − γ 1 E ( K ) + β 1 R ( K ) ) d t

U 2 , K + 1 ( t ) = U 2 ( K ) − ∫ 0 t ( d U 2 ( K ) d t − π 2 + ( μ + k 2 E ( K ) ) U 2 ( K ) − γ 2 E ( K ) + β 2 R ( K ) ) d t

E K + 1 ( t ) = E ( K ) − ∫ 0 t ( d E ( K ) d t − ( k 1 U 1 ( K ) + k 2 U 2 ( K ) ) E ( K ) + ( μ + δ + γ 1 + γ 2 ) E ( K ) + β 2 R ( K ) ) d t R k + 1 ( t ) = R ( K ) − ∫ 0 t ( d R ( K ) d t − w E ( K ) + ( μ + β 1 + β 2 ) R ( K ) ) d t V K + 1 ( t ) = V ( K ) − ∫ 0 t ( d V ( K ) d t − ϕ U 2 ( K ) − γ E ( K ) + η V ( K ) ) d t (20)

Implementing (20) on maple 18 with initial conditions U 1 ( 0 ) , U 2 ( 0 ) , E ( 0 ) , R ( 0 ) and V ( 0 ) and values in

U 1 ( t ) = 60 − 1186.0 t − 20248.83 t 2 + 196082.19 t 3 + 12000399.09 t 4 + 74310736.32 t 5 + ⋯ U 2 ( t ) = 40 − 524.50 t − 10934.45 t 2 + 9122.34 t 3 + 4435444.89 t 4 + 58674358.19 t 5 + ⋯

E ( t ) = 35 + 1838.60 t + 32191.71 t 2 − 193491.25 t 3 − 16490718.82 t 4 − 136637622.6 t 5 + ⋯ R ( t ) = 5 + 27.50 t + 901.42 t 2 + 7753.30 t 3 − 60749.11 t 4 − 1798174.31 t 5 + ⋯ V ( t ) = 10 + 56.00 t + 568.01 t 2 + 4747.13 t 3 − 42286.075 t 4 − 1147347.94 t 5 + ⋯ (21)

This article recorded the following findings:

・ When there is no employment, it is practically sensible that there would be no retirement but vacancy exists due to active unemployment and may/maynot be publicize, Equation (13).

・ As employment exist, the result obtained illustrates retirement, vacancies and population of unemployment (both passive and active), all exist, Equation (14).

・ A threshold that is to establsih stability of unemployment was obtained and required to be well manage, Equation (15). When R e < 1 , implies people getting job are more than those without job, otherwise, the economy of such community will face hardish.

・ It could be seen that all eigenvalues of the characteristics equation give expression less than zero except λ 3 , which for stability, must be less than zero and this said λ 3 is the threshold (implies the assurance of resolving problem of unemployment in contemporal world depends on the threshold) obtained for unemployment, see Equation (18). This is in support of result obtained is (15) and it importance.

・ VIM was employed to obtain an approximate solution of the propose model and the solution was hypothetically fitted with values to produce results in

Here is summary of the research carried out on mathematical modeling of unemployment with retirement and conclusion made based on our findings.

Parameters | Description | Value |
---|---|---|

π 1 | Recruitment rate to the passive unemployment | 65 |

π 2 | Recruitment rate to the active unemployment | 35 |

k 1 | Rate at which unemployment (passive) becomes employed | 0.6 |

k 2 | Proportion of active unemployment becomes employed | 0.4 |

γ 1 | Rate of employed people returned to passive unemployed | 0.4 |

γ 2 | Proportion of employed individual returned to active unemployed | 0.2 |

β 1 | Rate at which retired people becomes unemployed (passive) | 0.4 |

β 2 | Rate at which retired people becomes unemployed (active) | 0.2 |

ϕ | Vacancy creation by active unemployed people | 0.15 |

μ | Natural death | 0.05 |

δ | Those that died in the process of working | 0.01 |

η | Unpublicized available vacancies | 0.18 |

γ | Vacancy created by employed people | 0.2 |

t | U 1 | U 2 | E ( t ) | R ( t ) | V ( t ) |
---|---|---|---|---|---|

1 | 1.7105 × 10^{33} | 4.4213 × 10^{35} | 2.4807 × 10^{34} | 1.0441 × 10^{17} | 7.5315 × 10^{16} |

2 | 1.0841 × 10^{41} | 1.3558 × 10^{45} | 3.4676 × 10^{42} | 3.5142 × 10^{21} | 2.5317 × 10^{21} |

3 | 4.0179 × 10^{45} | 4.3561 × 10^{50} | 1.9940 × 10^{47} | 1.5502 × 10^{24} | 1.1164 × 10^{24} |

4 | 7.0379 × 10^{48} | 3.4323 × 10^{54} | 4.7401 × 10^{50} | 1.1640 × 10^{26} | 8.3810 × 10^{25} |

5 | 2.3123 × 10^{51} | 3.5780 × 10^{57} | 1.9673 × 10^{53} | 3.3148 × 10^{27} | 2.3864 × 10^{27} |

In this research, we are able to modify the model of [

The cooperate environment is now very dependent on skills and technical know-how, western education and certificate(s) is/are no longer enough/unenthusiastic to provide food on the table. Hence, this article concluded that:

1) Every individual have contribution to end unemployment by developing self skills of entrepreneur or learn one (this can be view from Equation (13)).

2) Important threshold (see Equation (15)) to curtail challenges of unemployment was established and beseeched government and policy makers to introduce policy/strategy in proportion to exponential population growth.

3) It is imperative for the employed individuals to prepare for retirement as pension scheme is not sufficient to cater for life after 35years of service (Equation (18)).

This article studied dynamics in mathematical sense, unemployment and established important parameter, called threshold, unemployment is said to be stable for R e < 1 (when no recruitment except active unemployed created job) and R e > 1 will also stable when there is employment (i.e. when everybody put effort and create job) but we did not consider the case R e = 1 , which could be refered to as bifurcation analysis in the mathematical view (when total number of vacant position is equal to number of population). Also, the model can be modified based on asumptions different from those here considered.

The authors declare no conflicts of interest regarding the publication of this paper.

Kazeem, A.B., Alimi, S.A. and Ibrahim, M.O. (2018) Threshold Parameter for the Control of Unemployment in the Society: Mathematical Model and Analysis. Journal of Applied Mathematics and Physics, 6, 2563-2578. https://doi.org/10.4236/jamp.2018.612214