# Modeling and Mathematical Analysis of Labor Force Evolution

Authors: Sanaa ElFadily; Abdelilah Kaddar
Journal: Modelling and Simulation in Engineering
 (2019)

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## Abstract

In this paper, we propose a delayed differential system to model labor force (occupied labor force and unemployed) evolution. The mathematical analysis of our model focuses on the local behavior of the labor force around a positive equilibrium position; the existence of a branch of periodic solutions bifurcated from the positive equilibrium is then analyzed according to the Hopf bifurcation theorem. Finally, we performed numerical simulations to illustrate our theoretical results.


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## Body

## 1. Introduction

Mathematical modeling has gained considerable importance in recent decades in all fields, particularly in the macroeconomic field. This branch of applied mathematics is interested in the representation, with enough loyalty, of the real phenomena by abstract mathematical objects (variables, parameters, and equations) accessible to analysis and calculation. Its main objective is the exploitation of the conditions of yesterday and/or today to predict those of tomorrow (for example, we can use ordinary differential equations or delayed differential equations).This work is an opportunity to take stock of the evolution of the labor force over time to explain the reasons for the fluctuations. To do this, we propose delayed differential equations to model labor force evolution (knowing the past and the present, foresee the future).The study of the labor force evolution (occupied labor force and unemployed) finds all its relevance in the economic and mathematical researches for two reasons, the first relating to the role of labor force in the national economy and the second linking up to the fluctuations in labor force, that occur at different times, with different durations and amplitudes [1–5]. These fluctuations are natural responses of population to the exogenous and/or endogenous causes: technological changes, school failure, vocational or technical training failure, or noncompatibility of the training with the labor market.The article is organized as follows. In Section2, we formulate our model of the labor force evolution. In Section 3, local stability of the positive equilibrium and existence of Hopf bifurcation are given. Numerical simulations are performed, in Section 4, to illustrate our theoretical results. Finally, a summary of the results and some suggestions for future research are presented in Section 5.


## 2. Modeling with Differential Equations

For the estimation of labor force evolution, one needs to know the employment rateγ, the rate of job loss σ, the mortality rate μ, the maximum population growth rate ρ, the carrying capacity Nc, and the time lag needed to contribute in the reproductive process of a new individual looking for a job, τ. Therefore, our labor force evolution model is schematized in Figure 1. In this scheme, the active population is divided into unemployed individuals (compartment U) and occupied labor force individuals (compartment L). The arrows specify the proportion of individuals passing from one compartment to another, and the arrows passing through the small circle indicate the process of reproduction that gives the number of new people looking for a job.Figure 1
Scheme of the labor force evolution.The temporal evolution of this scheme is governed by the following differential equations:(1)dLdt=γU−σ+μL,dUdt=ρ1−Lτ+UτNcLτ+σL−μ+γU,where the variable L and U can be interpreted as the number of occupied labor force and the number of unemployed, respectively.The initial conditions for system (1) are(2)L0>0,U0>0,Lθ,Uθ=φ1θ,φ2θ,∀θ∈−τ,0,where φi∈C−τ,0,ℝ+,i=1,2.The first equation models the evolution of occupied labor force population. In this equation,γ is the employment rate. Thus, the term γU can be interpreted as the number of new employees among people looking for a job. The term σL corresponds to the number of employees who have lost their job, and the term μL counts the number of deaths in the occupied labor force compartment. The second equation models the evolution of unemployed population. In this equation, the function ρ1−L+U/NcL indicates the number of new people looking for a job. It reflects both maximum growth of job seekers when the unemployment rate is very low and very low labor force growth when the unemployment rate is very high. In other words, population growth is limited by the sharp increase in the number of unemployed.


## 3. Equilibria, Local Stability, and Existence of Hopf Bifurcation

### 3.1. Equilibria

In the following proposition, we prove the existence of the two equilibria.Proposition 1.
System (1) has the following equilibria.(i)
Ifγρ/μγ+σ+μ≤1, then system (1) has a unique equilibrium E0=0,0.(ii)
Ifγρ/μγ+σ+μ>1, then system (1) has two equilibria: the trivial equilibrium E0=0,0 and the positive equilibrium E∗=L∗,U∗, where U∗=σ+μL∗/γ and L∗=γ/γ+σ+μ−μ/ρNc.Proof.
To find equilibria, we consider the following system:(3)γU−σ+μL=0,ρ1−L+UNcL+σL−γ+μU=0.
It is easy to verify that0,0 is a solution of the system (3), thus E0=0,0 is a trivial equilibrium of the system (1). In the following, we prove that E∗ is the unique positive equilibrium of the system (1).
L
,
U is a positive solution of the system (3) if and only if(4)U=σ+μLγ,ρ1−γ+σ+μγLNc−γ+σ+μμγ=0.
It is clear that ifγρ/μγ+σ+μ>1, then the second equation of (4) has a unique positive solution L∗=γ/γ+σ+μ−μ/ρNc. This concludes the proof.


### 3.2. Stability and Hopf Bifurcation

By analyzing the characteristic equation associated to (1), we prove the existence of Hopf bifurcation around the equilibrium E∗.Consider the change of variablesx=L−L∗ and y=U−U∗. Then by linearizing system (1) around L∗,U∗, we have(5)dxdt=γy−σ+μx,dydt=σx+μγ+σ+μγ−ρL∗Ncxτ−γ+μy−ρL∗Ncyτ.The characteristic equation associated to system (5) takes the following form:(6)λ2+aλ+bλe−λτ+c+de−λτ=0,where(7)a=γ+σ+2μ,b=ρL∗Nc,c=μγ+σ+μ,d=ργ+σ+μL∗Nc−μγ+σ+μ.To give our results, we recall the Routh–Hurwitz criterion and the result of Kuang [6] for the characteristic equations of the second degree.Lemma 1.
For a second-degree polynomial,s2+a1s+a0, all the roots are in the left half plane if and only if both coefficients satisfy a0>0 and a1>0.Lemma 2.
Assumec+d≠0, and a2+b2+d2≠0.(1)
If there is no such imaginary solution of the characteristic equation (6), then the stability of the equilibrium does not change for any τ≥0.(2)
If there is only one imaginary solution with positive imaginary part of the characteristic equation (6), an unstable equilibrium never becomes stable for any τ≥0. If the equilibrium is asymptotically stable for τ=0, then there exists a critical value, τ0 of time delay, such that the equilibrium is uniformly asymptotically stable for τ≤τ0, and it becomes unstable for τ>τ0.(3)
If there are two imaginary roots with positive imaginary part,iω+ and iω−, such that ω+>ω−>0, then the stability of the equilibrium can change a finite number of time at most as τ is increased, and eventually it becomes unstable.According to Lemma 1 (Routh–Hurwitz criterion for a second-degree polynomial) and Lemma 2, we have the following results.Proposition 2.
Forτ=0, the equilibrium L∗,U∗ is locally asymptotically stable.Proof.
Forτ=0, the characteristic equation (6) reads as(8)λ2+γ+σ+2μ+ρL∗Ncλ+ργ+σ+μL∗Nc=0.
Sinceγ+σ+2μ+ρL∗/Nc>0 and ργ+σ+μL∗/Nc>0, then, according to Lemma 1, all the roots of the characteristic equation (6) are in the left half plane, and hence, the equilibrium L∗,U∗ is locally asymptotically stable.Proposition 3.
Ifγρ/μγ+σ+μ>1 and μ is close enough to zero, then there exists τ0>0 such that(i)
forτ∈0,τ0, the steady state L∗,U∗ is locally asymptotically stable;(ii)
forτ>τ0, L∗,U∗ is unstable;(iii)
forτ=τ0, equation (6) has a pair of purely imaginary roots ±iω0, ω0>0,where(9)τ0=1ω0arccos−abω02−dc−ω02bω02+d2,ω0=12b2−2c−a2+b2−2c−a22−4c2−d2.Proof.
Assume thatγρ/μγ+σ+μ>1. We calculate(10)c2−d2=ργ−μγ+σ+μ3μγ+σ+μ−ργ.
Moreover, we have forμ=0,(11)c2−d2=−ργ.
Then by continuity, there existsμ0>0 such that for all μ<μ0, then c2−d2<0. Hence the characteristic equation (6) has only one imaginary solution iω0=1/2b2−2c−a2+b2−2c−a22−4c2−d2i, with positive imaginary part. Applying Lemma 3.3, we conclude that the statement of our proposition is true.Proposition 4.
Ifγρ/μγ+σ+μ≥3, then there exists τ0>0 such that(i)
forτ∈0,τ0, the steady state L∗,U∗ is locally asymptotically stable;(ii)
forτ>τ0, L∗,U∗ is unstable;(iii)
forτ=τ0, equation (6) has a pair of purely imaginary roots ±iω0, ω0>0,where(12)τ0=1ω0arccos−abω02−dc−ω02bω02+d2,ω0=12b2−2c−a2+b2−2c−a22−4c2−d2.Proof.
Ifγρ/μγ+σ+μ≥3, then(13)c2−d2=μ2γ+σ+μ2γρμγ+σ+μ−13−γρμγ+σ+μ<0,hence, the characteristic equation (6) has only one imaginary solution iω0=1/2b2−2c−a2+b2−2c−a22−4c2−d2i, with positive imaginary part. Applying Lemma 3.3, we conclude that the statement of our proposition is true.
Next, we establish sufficient conditions for the local existence of Hopf bifurcation at the positive equilibrium.Theorem 1.
Suppose one of the following situations:H1: γρ/μγ+σ+μ>1 and μ is close enough to zero;H2: γρ/μγ+σ+μ≥3.
Then, a Hopf bifurcation of periodic solutions of system (1) occurs at L∗,U∗ when τ=τ0.Proof.
For the proof of this theorem, we verify the conditions of Hopf bifurcation theorem (see, for example [7]). From Proposition 4, the characteristic equation (6) has a pair of imaginary roots ±iω0 at τ=τ0. It is easy to show that this root is simple.
For the transversality condition, we have(14)signdReλdττ0=signb2−2c−a2−4c2−d2.
After some calculations, we get(15)b2−2c−a2−4c2−d2=γ2ρ21−μ2+γρμ−γ+σ+μ22γ+σ+μ2−4μ2γ+σ+μ2γρμγ+σ+μ−13−γρμγ+σ+μ.
If one of the two hypothesesH1 or H2 is verified, then 4μ2γ+σ+μ2γρ/μγ+σ+μ−13−γρ/μγ+σ+μ<0. Moreover, we have 0<μ<1. Consequently,(16)dReλdττ0>0.
This concludes the proof.
From Theorem 1, the following form of Hopf bifurcation theorem can now be stated (see Theorem 2.7., page 291 in [8]).Theorem 2.
Under the assumptions of Theorem 1, there is a constantε0>0 such that, for each 0≤ε<ε0, system (1) has a family of periodic solutions pε with period T=Tε, for the parameter values τ=τε such that p0=0, T0=2π/ω0, and τ0=τ0, where τ0 and ω0 are stated in Proposition 4.


## 3.1. Equilibria

In the following proposition, we prove the existence of the two equilibria.Proposition 1.
System (1) has the following equilibria.(i)
Ifγρ/μγ+σ+μ≤1, then system (1) has a unique equilibrium E0=0,0.(ii)
Ifγρ/μγ+σ+μ>1, then system (1) has two equilibria: the trivial equilibrium E0=0,0 and the positive equilibrium E∗=L∗,U∗, where U∗=σ+μL∗/γ and L∗=γ/γ+σ+μ−μ/ρNc.Proof.
To find equilibria, we consider the following system:(3)γU−σ+μL=0,ρ1−L+UNcL+σL−γ+μU=0.
It is easy to verify that0,0 is a solution of the system (3), thus E0=0,0 is a trivial equilibrium of the system (1). In the following, we prove that E∗ is the unique positive equilibrium of the system (1).
L
,
U is a positive solution of the system (3) if and only if(4)U=σ+μLγ,ρ1−γ+σ+μγLNc−γ+σ+μμγ=0.
It is clear that ifγρ/μγ+σ+μ>1, then the second equation of (4) has a unique positive solution L∗=γ/γ+σ+μ−μ/ρNc. This concludes the proof.


## 3.2. Stability and Hopf Bifurcation

By analyzing the characteristic equation associated to (1), we prove the existence of Hopf bifurcation around the equilibrium E∗.Consider the change of variablesx=L−L∗ and y=U−U∗. Then by linearizing system (1) around L∗,U∗, we have(5)dxdt=γy−σ+μx,dydt=σx+μγ+σ+μγ−ρL∗Ncxτ−γ+μy−ρL∗Ncyτ.The characteristic equation associated to system (5) takes the following form:(6)λ2+aλ+bλe−λτ+c+de−λτ=0,where(7)a=γ+σ+2μ,b=ρL∗Nc,c=μγ+σ+μ,d=ργ+σ+μL∗Nc−μγ+σ+μ.To give our results, we recall the Routh–Hurwitz criterion and the result of Kuang [6] for the characteristic equations of the second degree.Lemma 1.
For a second-degree polynomial,s2+a1s+a0, all the roots are in the left half plane if and only if both coefficients satisfy a0>0 and a1>0.Lemma 2.
Assumec+d≠0, and a2+b2+d2≠0.(1)
If there is no such imaginary solution of the characteristic equation (6), then the stability of the equilibrium does not change for any τ≥0.(2)
If there is only one imaginary solution with positive imaginary part of the characteristic equation (6), an unstable equilibrium never becomes stable for any τ≥0. If the equilibrium is asymptotically stable for τ=0, then there exists a critical value, τ0 of time delay, such that the equilibrium is uniformly asymptotically stable for τ≤τ0, and it becomes unstable for τ>τ0.(3)
If there are two imaginary roots with positive imaginary part,iω+ and iω−, such that ω+>ω−>0, then the stability of the equilibrium can change a finite number of time at most as τ is increased, and eventually it becomes unstable.According to Lemma 1 (Routh–Hurwitz criterion for a second-degree polynomial) and Lemma 2, we have the following results.Proposition 2.
Forτ=0, the equilibrium L∗,U∗ is locally asymptotically stable.Proof.
Forτ=0, the characteristic equation (6) reads as(8)λ2+γ+σ+2μ+ρL∗Ncλ+ργ+σ+μL∗Nc=0.
Sinceγ+σ+2μ+ρL∗/Nc>0 and ργ+σ+μL∗/Nc>0, then, according to Lemma 1, all the roots of the characteristic equation (6) are in the left half plane, and hence, the equilibrium L∗,U∗ is locally asymptotically stable.Proposition 3.
Ifγρ/μγ+σ+μ>1 and μ is close enough to zero, then there exists τ0>0 such that(i)
forτ∈0,τ0, the steady state L∗,U∗ is locally asymptotically stable;(ii)
forτ>τ0, L∗,U∗ is unstable;(iii)
forτ=τ0, equation (6) has a pair of purely imaginary roots ±iω0, ω0>0,where(9)τ0=1ω0arccos−abω02−dc−ω02bω02+d2,ω0=12b2−2c−a2+b2−2c−a22−4c2−d2.Proof.
Assume thatγρ/μγ+σ+μ>1. We calculate(10)c2−d2=ργ−μγ+σ+μ3μγ+σ+μ−ργ.
Moreover, we have forμ=0,(11)c2−d2=−ργ.
Then by continuity, there existsμ0>0 such that for all μ<μ0, then c2−d2<0. Hence the characteristic equation (6) has only one imaginary solution iω0=1/2b2−2c−a2+b2−2c−a22−4c2−d2i, with positive imaginary part. Applying Lemma 3.3, we conclude that the statement of our proposition is true.Proposition 4.
Ifγρ/μγ+σ+μ≥3, then there exists τ0>0 such that(i)
forτ∈0,τ0, the steady state L∗,U∗ is locally asymptotically stable;(ii)
forτ>τ0, L∗,U∗ is unstable;(iii)
forτ=τ0, equation (6) has a pair of purely imaginary roots ±iω0, ω0>0,where(12)τ0=1ω0arccos−abω02−dc−ω02bω02+d2,ω0=12b2−2c−a2+b2−2c−a22−4c2−d2.Proof.
Ifγρ/μγ+σ+μ≥3, then(13)c2−d2=μ2γ+σ+μ2γρμγ+σ+μ−13−γρμγ+σ+μ<0,hence, the characteristic equation (6) has only one imaginary solution iω0=1/2b2−2c−a2+b2−2c−a22−4c2−d2i, with positive imaginary part. Applying Lemma 3.3, we conclude that the statement of our proposition is true.
Next, we establish sufficient conditions for the local existence of Hopf bifurcation at the positive equilibrium.Theorem 1.
Suppose one of the following situations:H1: γρ/μγ+σ+μ>1 and μ is close enough to zero;H2: γρ/μγ+σ+μ≥3.
Then, a Hopf bifurcation of periodic solutions of system (1) occurs at L∗,U∗ when τ=τ0.Proof.
For the proof of this theorem, we verify the conditions of Hopf bifurcation theorem (see, for example [7]). From Proposition 4, the characteristic equation (6) has a pair of imaginary roots ±iω0 at τ=τ0. It is easy to show that this root is simple.
For the transversality condition, we have(14)signdReλdττ0=signb2−2c−a2−4c2−d2.
After some calculations, we get(15)b2−2c−a2−4c2−d2=γ2ρ21−μ2+γρμ−γ+σ+μ22γ+σ+μ2−4μ2γ+σ+μ2γρμγ+σ+μ−13−γρμγ+σ+μ.
If one of the two hypothesesH1 or H2 is verified, then 4μ2γ+σ+μ2γρ/μγ+σ+μ−13−γρ/μγ+σ+μ<0. Moreover, we have 0<μ<1. Consequently,(16)dReλdττ0>0.
This concludes the proof.
From Theorem 1, the following form of Hopf bifurcation theorem can now be stated (see Theorem 2.7., page 291 in [8]).Theorem 2.
Under the assumptions of Theorem 1, there is a constantε0>0 such that, for each 0≤ε<ε0, system (1) has a family of periodic solutions pε with period T=Tε, for the parameter values τ=τε such that p0=0, T0=2π/ω0, and τ0=τ0, where τ0 and ω0 are stated in Proposition 4.


## 4. Numerical Simulations

The following numerical simulations are given for system (1), for time series at the top and at the bottom in Figure 2, σ=0.1, ρ=0.01054, τ=25 years, γ=0.7,μ=0.000481,Nc=11000000 and σ=0.1, ρ=0.01054, τ=25 years, γ=0.2,μ=0.00481,Nc=51000000, respectively.Figure 2
Examples of times series obtained by numerical integration of labor force evolution model (1).
(a)
(b)In view of Figure2, we observe an oscillatory regime on the labor market: the number of occupied labor force and the number of unemployed oscillate and converge towards equilibrium. This shows that time delay, in the reproduction process, is a key parameter for the existence of a branch of bifurcated periodic solutions from the positive equilibrium and therefore of the existence of oscillations when the delay crosses certain critical values.


## 5. Conclusion

In this paper, we presented a study on the dynamics of a model of the evolution of the employed labor force and the unemployed population. We have shown the existence of a positive equilibriumE∗, and we have studied the analytic behavior around this point. Specially, we have shown the existence of a branch of periodic solutions bifurcated from the positive equilibrium E∗.The existence of the limit cycles, for certain positive values of the parameters of the model, supports the phenomenon of coexistence of the two populations: active occupied and unemployed. This coexistence can be explained by the fact that job seekers occur relatively quickly because of population growth.As a result of this paper, we plan to study the direction of the bifurcating periodic solutions and resonant codimension-two bifurcation (see, for example [9]). Also, we plan to develop this work by introducing the economic growth effect. The resulting model determines the conditions of interaction between economic growth and labor market evolution.



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