Modeling the effect of unemployment augmented industrialization on the control of unemployment

Abstract

The increase in human population at large scale has given the birth to the problem of unemployment as industrialization has not taken place in the same proportion. Some developing countries are making efforts for sustainable industrial growth to overcome with this problem. To tackle this problem, in the present paper, we propose a nonlinear mathematical model to study the role of unemployment-dependent industrial growth (geared industrialization) on the control of unemployment. The study of the model is based on using the method of stability theory of differential equations. The study reveals that the sustainable growth of industrialization helps to control the problem of unemployment. Finally, certain numerical calculations have been carried out to support analytical findings.

Appendix

1.1 A Proof of Lemma 1

We add second and third equations of the model (1), and get

$$\begin{aligned} \frac{dU}{dt}+\frac{dR}{dt}&= A-\delta _{1}U-\delta _{2}R, \end{aligned}$$

this gives

$$\begin{aligned} \frac{d[U+R]}{dt}\le A-\delta [U+R], \end{aligned}$$

where $\delta =\min \{\delta _{1},\delta _{2}\}$.

Above implies

$$\begin{aligned} \lim _{t\rightarrow \infty }\sup [U+R]\le \frac{A}{\delta }. \end{aligned}$$

(15)

Now, using equation (15) in the first equation of model (1), we obtain

$$\begin{aligned} \frac{dI(t)}{dt}\le Q_{0}+\lambda \frac{A}{\delta }+\alpha _{0}I_{0}-\alpha _{0}I(t). \end{aligned}$$

This gives

$$\begin{aligned} \lim _{t\rightarrow \infty }\sup I(t)\le I_{m}, \end{aligned}$$

(16)

where

$$\begin{aligned} I_{m}=\frac{Q_{0}+\lambda \frac{A}{\delta } +\alpha _{0}I_{0}}{\alpha _{0}}. \end{aligned}$$

Finally, from the fourth equation of the model (1) and the equation (16), we have

$$\begin{aligned} \frac{dV(t)}{dt}\le \eta (I_{m}-I_{0})-\eta _{0}V(t). \end{aligned}$$

The above inequality implies,

$$\begin{aligned} \begin{aligned} \lim _{t\rightarrow \infty }\sup V(t)&\le \frac{\eta }{\eta _{0}}(I_{m}-I_{0})\\&= \frac{\eta }{\eta _{0}}\frac{Q_{0}+\lambda A/ \delta }{\alpha _{0}}. \end{aligned} \end{aligned}$$

(17)

Hence, this proves the lemma.

1.2 B Proof of Theorem 2

We use Lyapunov’s method for global stability to establish this theorem. For this we use the positive definite function:

$$\begin{aligned} W_{0}=\frac{1}{2}(I-I^{*})^{2}+\frac{1}{2}m_{1}(U-U^{*})^{2}+\frac{1}{2}m_{2}(R-R^{*})^{2}+\frac{1}{2}m_{3}(V-V^{*})^{2}, \end{aligned}$$

(18)

for some suitably chosen positive constants $m_{i}'s, i=1,2,3$.

Now differentiate equation (18) with respect to time t along the solution of system (1), we have

$$\begin{aligned} \frac{dW_{0}}{dt}&= -\alpha _{0}(I-I^{*})^{2}-m_{1}\left\{ k(R_{a}+V-R) +\delta _{1}\right\} (U-U^{*})^{2}\nonumber \\&{\mathrel -m_{2}}\left\{ kU+\delta _{2}+\gamma \right\} (R-R^{*})^{2}-m_{3}\eta _{0}(V-V^{*})^{2}+\lambda (U-U^{*})(I-I^{*})\nonumber \\&\mathrel +\left\{ m_{1}(k U^{*}+\gamma )+m_{2}k (R_{a}+V^{*}-R^{*})\right\} (U-U^{*})(R-R^{*})\nonumber \\&\mathrel -m_{1}k U^{*}(U-U^{*})(V-V^{*})+m_{2}k U(R-R^{*})(V-V^{*})\nonumber \\&\mathrel +m_{3}\eta (I-I^{*})(V-V^{*}). \end{aligned}$$

(19)

Now, $\frac{dW_{0}}{dt}$ becomes negative definite in region of asymptotic stability $\Omega _{0}$ provided the following inequalities hold:

$$\begin{aligned}&\lambda ^{2}<\frac{1}{2}m_{1}\alpha _{0}\delta _{1}, \end{aligned}$$

(20)

$$\begin{aligned}&m_{1}(k U^{*}+\gamma )^{2}<\frac{1}{3}m_{2}\delta _{1}(\delta _{2}+\gamma ), \end{aligned}$$

(21)

$$\begin{aligned}&m_{2}k^{2}(R_{a}+V^{*}-R^{*})^{2}<\frac{1}{3}m_{1}\delta _{1}(\delta _{2}+\gamma ), \end{aligned}$$

(22)

$$\begin{aligned}&m_{1}(kU^{*})^{2}<\frac{1}{3}m_{3}\eta _{0}\delta _{1}, \end{aligned}$$

(23)

$$\begin{aligned}&m_{2}\left( \frac{k A}{\delta }\right) ^{2}<\frac{4}{9}m_{3}\eta _{0}(\delta _{2}+\gamma ), \end{aligned}$$

(24)

$$\begin{aligned}&m_{3}\eta ^{2}<\frac{2}{3}\eta _{0}\alpha _{0}. \end{aligned}$$

(25)

From equations (20) and (25), we can choose the positive values of $m_1=\frac{4\lambda ^2}{\alpha _0\delta _1}$ and $m_3=\frac{\eta _0\alpha _0}{3\eta ^2},$ respectively. After choosing $m_{1}>0$ and $m_{3}>0$ and using inequalities (21), (22) and (24), we may choose the positive value of $m_2$ as

$$\begin{aligned} \left\{ \frac{12\lambda ^2( k U^*+\gamma )^2}{\alpha _0 \delta _1^2(\delta _2+\gamma )}\right\}< m_{2} < \min \left\{ \frac{1}{3}\frac{4\lambda ^2(\delta _2+\gamma )}{\alpha _0k^2(R_a+V^*-R^*)^2}, \frac{4}{9}\frac{\eta _0^2\alpha _0\delta ^2(\delta _2+\gamma )}{\eta ^2A^2k^2}\right\} \end{aligned}$$

Hence, $\frac{dW_{0}}{dt}$ becomes negative definite in the region $\Omega _{0}$, if the conditions (13) and (14) are fulfilled.