Journal Details
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Journal
eISSN
2444-8656
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01 Jan 2016
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Languages
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Open Access

# Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model

###### Accepted: 08 May 2022
Journal Details
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Innovation model analysis of current network education in colleges and universities
Definition 1

After social and economic development has entered the knowledge age, human development attaches more importance to educational innovation. [1]Using the theoretical method of differential equation, the researchers conducted a preliminary study based on the mathematical model under the condition that there were only two kinds of innovation generation relations, and mainly discussed whether the explanation with the initial value far from the equilibrium point could be inclined to the equilibrium point. The differential equation model of educational innovation competition is as follows[2]: ${dxdt=k1x(t)(1−x(t)M1−b1y(t)M2)dydt=k2y(t)(1−b2x(t)M1−y(t)M2)$ \left\{\matrix{{{dx} \over {dt}} = {k_1}x\left(t \right)\left({1 - {{x\left(t \right)} \over {{M_1}}} - {b_1}{{y\left(t \right)} \over {{M_2}}}} \right) \hfill \cr {{dy} \over {dt}} = {k_2}y\left(t \right)\left({1 - {b_2}{{x\left(t \right)} \over {{M_1}}} - {{y\left(t \right)} \over {{M_2}}}} \right) \hfill \cr} \right.

Theorem 1

In the above formula, K1 and K2 represent the penetration rate of educational innovation X and Y without considering the environmental impact; M1 and M2 represent the maximum possible number of X and Y; X (t) and y (t) represent the cumulative number of recipients of X and Y; B1 >0 and B2 >0 represent the interaction coefficients when X and Y coexist in the system.

Proposition 2

According to the above formula, the positive equilibrium point $P1(M1(1−b1)1−b1b2,M2(1−b2)1−b1b2)$ {P_1}\left({{{{M_1}\left({1 - {b_1}} \right)} \over {1 - {b_1}{b_2}}},{{{M_2}\left({1 - {b_2}} \right)} \over {1 - {b_1}{b_2}}}} \right) is globally uniformly asymptotic and tends to be stable if the conditions of $0 0 < {b_2} < {{\root 2 \of {{M_1}{M_2}} - {b_1}{M_1}} \over {{M_2}}} and $0 0 < {{\root 2 \of {{M_1}{M_2}} - {M_2}} \over {{M_1}}} < {b_1} < 1 are met in the case of coexistence of X and Y. It is proved that the equilibrium point of the above formula $p1(M1(1−b1)1−b1b2,M2(1−b2)1−b1b2)$ {p_1}\left({{{{M_1}\left({1 - {b_1}} \right)} \over {1 - {b_1}{b_2}}},{{{M_2}\left({1 - {b_2}} \right)} \over {1 - {b_1}{b_2}}}} \right) , p2(M1,0), p3(0, M2), is p4 (0,0). In the case of coexistence of X and Y, research and analysis can be carried out under the condition of $R+2={(x,y)|x>0,y>0}$ R_ + ^2 = \left\{{\left({x,y} \right)|x > 0,y > 0} \right\} , and only the equilibrium point P1 can be discussed.

Suppose p1(x*, y*), where $x*=M1(1−b1)1−b1b2>0$ {x^*} = {{{M_1}\left({1 - {b_1}} \right)} \over {1 - {b_1}{b_2}}} > 0 , $y*=M2(1−b2)1−b1b2>0$ {y^*} = {{{M_2}\left({1 - {b_2}} \right)} \over {1 - {b_1}{b_2}}} > 0 , 0 < b1 < b2 < 1, then the following equation is satisfied: ${1M1x*+b1M2y*=1b2M1x*+1M2y*=1$ \left\{\matrix{{1 \over {{M_1}}}{x^*} + {{{b_1}} \over {{M_2}}}{y^*} = 1 \hfill \cr {{{b_2}} \over {{M_1}}}{x^*} + {1 \over {{M_2}}}{y^*} = 1 \hfill \cr} \right.

Lemma 3

And the above model can be transformed into: ${dxdt=−k1x(t)(x−x*M1+b1M2(y−y*))dydt=−k2y(t)(b2M1(x−x*)−1M2(y−y*))$ \left\{\matrix{{{dx} \over {dt}} = - {k_1}x\left(t \right)\left({{{x - {x^*}} \over {{M_1}}} - {{{b_1}} \over {{M_2}}}\left({y - {y^*}} \right)} \right) \hfill \cr {{dy} \over {dt}} = - {k_2}y\left(t \right)\left({{{{b_2}} \over {{M_1}}}\left({x - {x^*}} \right) - {1 \over {{M_2}}}\left({y - {y^*}} \right)} \right) \hfill \cr} \right.

The corresponding function formula is: $V(x,y)=1k1(x−x*−x*lnxx*)+1k2(y−y*−y*lnyy*)$ V\left({x,y} \right) = {1 \over {{k_1}}}\left({x - {x^*} - {x^*}\ln {x \over {{x^*}}}} \right) + {1 \over {{k_2}}}\left({y - {y^*} - {y^*}\ln {y \over {{y^*}}}} \right)

The final calculation is as follows: $V(x*,y*)=0,∂V(x,y)∂x=1k1(1−x*x),∂V(x,y)∂y=1k2(1−y*y),∂2V(x,y)∂x2=x*k1x2,∂2V(x,y)∂y2=y*k2y2,∂2V(x,y)∂x∂y=0.$ \eqalign{& V\left({{x^*},{y^*}} \right) = 0,{{\partial V\left({x,y} \right)} \over {\partial x}} = {1 \over {{k_1}}}\left({1 - {{{x^*}} \over x}} \right),{{\partial V\left({x,y} \right)} \over {\partial y}} = {1 \over {{k_2}}}\left({1 - {{{y^*}} \over y}} \right), \cr & {{{\partial ^2}V\left({x,y} \right)} \over {\partial {x^2}}} = {{{x^*}} \over {{k_1}{x^2}}},{{{\partial ^2}V\left({x,y} \right)} \over {\partial {y^2}}} = {{{y^*}} \over {{k_2}{y^2}}},{{{\partial ^2}V\left({x,y} \right)} \over {\partial x\partial y}} = 0. \cr}

Corollary 4

It follows that V (x, y) is a positive definite function inside R2, and as it approaches the x or y axis, it satisfies the condition V(x, y) → +∞, so it is a function with infinity inside. By analyzing it in accordance with the solution ideas of the above formula, the total derivative can be obtained as follows: $V*(x,y)=−(x−x*)(x−x*M1+b1(y−y*)M2)−(y−y*)(b2(x−x*)M1+y−y*M2)=−[1M1(x−x*)2+(b2M1+b1M2)(x−x*)(y−y*)+1M2(y−y*)2]$ \eqalign{& \mathop V\limits^* \left({x,y} \right) = - \left({x - {x^*}} \right)\left({{{x - {x^*}} \over {{M_1}}} + {{{b_1}\left({y - {y^*}} \right)} \over {{M_2}}}} \right) - \left({y - {y^*}} \right)\left({{{{b_2}\left({x - {x^*}} \right)} \over {{M_1}}} + {{y - {y^*}} \over {{M_2}}}} \right) \cr & = - \left[{{1 \over {{M_1}}}{{\left({x - {x^*}} \right)}^2} + \left({{{{b_2}} \over {{M_1}}} + {{{b_1}} \over {{M_2}}}} \right)\left({x - {x^*}} \right)\left({y - {y^*}} \right) + {1 \over {{M_2}}}{{\left({y - {y^*}} \right)}^2}} \right] \cr}

$V*(x,y)$ \mathop V\limits^* \left({x,y} \right) stands for quadratic homogeneous functions of (x-x*) and (y-y*). And in accordance with the condition of $Δ=(b2M1+b1M2)2−4M1M2<0$ \Delta = {\left({{{{b_2}} \over {{M_1}}} + {{{b_1}} \over {{M_2}}}} \right)^2} - {4 \over {{M_1}{M_2}}} < 0 , $V*(x,y)$ \mathop V\limits^* \left({x,y} \right) is negative definite, and the simplified formula is as follows: $Δ=(b1M+b2M2)2−4M1M2M12M22<0$ \Delta = {{{{\left({{b_1}M + {b_2}{M_2}} \right)}^2} - 4{M_1}{M_2}} \over {M_1^2M_2^2}} < 0

Thus, it can be obtained: $(b1M1+b2M2)2−4M1M2<0$ {\left({{b_1}{M_1} + {b_2}{M_2}} \right)^2} - 4{M_1}{M_2} < 0

Conjecture 5. There exists a unique positive equilibrium solution which is globally uniformly asymptotically stable.

At the same time, because X axis and Y axis belong to the internal track of the system, there will be no closed track outside the three equilibrium points p2 (M1, 0), P3 (0, M2) and P4 (0, 0), otherwise there will be conflicts with the intersection of X axis and Y axis.

Under the condition that B1B2 =1, there is a balance between the axial line L1: $1−xM1−b1yM2=0$ 1 - {x \over {{M_1}}} - {b_1}{y \over {{M_2}}} = 0 and the line L2: $1−b2xM1−yM2=0$ 1 - {b_2}{x \over {{M_1}}} - {y \over {{M_2}}} = 0 . At this time, there is no other equilibrium point in the system, and the closed y-rail contains a singularity, so the system does not have a closed rail.

Under the condition of b1b2 ≠ 1, limit cycles will not appear outside the equilibrium point $p1(M1(1−b1)1−b1b2,M2(1−b2)1−b1b2)$ {p_1}\left({{{{M_1}\left({1 - {b_1}} \right)} \over {1 - {b_1}{b_2}}},{{{M_2}\left({1 - {b_2}} \right)} \over {1 - {b_1}{b_2}}}} \right) inside the system.

Assuming ${dxdt=k1x(1−xM1−b1yM2)Δ__P(x,y)dydt=k2y(1−b2xM1−yM2)Δ__Q(x,y)$ \left\{\matrix{{{dx} \over {dt}} = {k_1}x\left({1 - {x \over {{M_1}}} - {b_1}{y \over {{M_2}}}} \right)\underline{\underline \Delta} P\left({x,y} \right) \hfill \cr {{dy} \over {dt}} = {k_2}y\left({1 - {b_2}{x \over {{M_1}}} - {y \over {{M_2}}}} \right)\underline{\underline \Delta} Q\left({x,y} \right) \hfill \cr} \right. , and the function B(x, y) = xα−1yβ−1 is obtained inside the first quadrant, where α, β represents a specific constant, it can be calculated as follows: $DΔ__∂(BP)∂x+∂(BQ)∂y=k1xα−1yβ−1(α−αM1x−b1αM2−1M1x)+k2xα−1yβ−1(β−b2βM1x−βM2y−1M2y)=xα−1yβ−1[−1M1(k1α+k2b2β+k1)x−1M2(k1b1α+k2β+k2)y+k1α+k2β]$ \eqalign{& D\underline{\underline \Delta} {{\partial \left({BP} \right)} \over {\partial x}} + {{\partial \left({BQ} \right)} \over {\partial y}} \cr & = {k_1}{x^{\alpha - 1}}{y^{\beta - 1}}\left({\alpha - {\alpha \over {{M_1}}}x - {{{b_1}\alpha} \over {{M_2}}} - {1 \over {{M_1}}}x} \right) + {k_2}{x^{\alpha - 1}}{y^{\beta - 1}}\left({\beta - {{{b_2}\beta} \over {{M_1}}}x - {\beta \over {{M_2}}}y - {1 \over {{M_2}}}y} \right) \cr & = {x^{\alpha - 1}}{y^{\beta - 1}}\left[{- {1 \over {{M_1}}}\left({{k_1}\alpha + {k_2}{b_2}\beta + {k_1}} \right)x - {1 \over {{M_2}}}\left({{k_1}{b_1}\alpha + {k_2}\beta + {k_2}} \right)y + {k_1}\alpha + {k_2}\beta} \right] \cr}

Example 6

Since the corresponding symbols of X and y have been identified in the study, the following requirements should be met in order to ensure the invariable sign of D: ${k1α+k2b2β+k1=0k1b1α+k2β+k2=0$ \left\{\matrix{{k_1}\alpha + {k_2}{b_2}\beta + {k_1} = 0 \hfill \cr {k_1}{b_1}\alpha + {k_2}\beta + {k_2} = 0 \hfill \cr} \right.

From this, it can be calculated: $α=k2b2−k1k1(1−b2b2),β=k1b1−k2k2(1−b1b2),$ \alpha = {{{k_2}{b_2} - {k_1}} \over {{k_1}\left({1 - {b_2}{b_2}} \right)}},\beta = {{{k_1}{b_1} - {k_2}} \over {{k_2}\left({1 - {b_1}{b_2}} \right)}},

And: $D=k1(b1−1)+k2(b2−1)1−b1b2xα−1yβ−1$ D = {{{k_1}\left({{b_1} - 1} \right) + {k_2}\left({{b_2} - 1} \right)} \over {1 - {b_1}{b_2}}}{x^{\alpha - 1}}{y^{\beta - 1}}

Assuming constant k1(b1−1)+k2 (b2−1) ≠ 0 meets this condition, it is proved that the system does not have closed orbits. If the constant k1 (b1 −1)+ k2 (b2 −1) = 0 satisfies this condition, then the proof system also has no limit cycles.

Researchers proposed a complementary differential equation model of educational innovation with the Logistic equation as the core, as shown below[3]: ${dxdt=k1x(t)(1−x(t)M1+b1y(t)M2)dydt=k2y(t)(1−x(t)M2+b2y(t)M1)$ \left\{\matrix{{{dx} \over {dt}} = {k_1}x\left(t \right)\left({1 - {{x\left(t \right)} \over {{M_1}}} + {b_1}{{y\left(t \right)} \over {{M_2}}}} \right) \hfill \cr {{dy} \over {dt}} = {k_2}y\left(t \right)\left({1 - {{x\left(t \right)} \over {{M_2}}} + {b_2}{{y\left(t \right)} \over {{M_1}}}} \right) \hfill \cr} \right.

In the above model, the positive equilibrium point $P1(M1(1+b1)1−b1b2,M2(1+b2)1−b1b2)$ {P_1}\left({{{{M_1}\left({1 + {b_1}} \right)} \over {1 - {b_1}{b_2}}},{{{M_2}\left({1 + {b_2}} \right)} \over {1 - {b_1}{b_2}}}} \right) is globally uniformly asymptotically stable where X and Y coexist and satisfy the conditions $0 0 < {b_2} < {{\root 2 \of {{M_1}{M_2}} - {b_1}{M_1}} \over {{M_2}}} and $0 0 < {{\root 2 \of {{M_1}{M_2}} - {M_2}} \over {{M_1}}} < {b_1} < 1 .

The equilibrium point of the above model is $p1(M1(1+b1)1−b1b2,M2(1+b2)1−b1b2)$ {p_1}\left({{{{M_1}\left({1 + {b_1}} \right)} \over {1 - {b_1}{b_2}}},{{{M_2}\left({1 + {b_2}} \right)} \over {1 - {b_1}{b_2}}}} \right) , p2 (M1,0), p3 (0, M2) p4 (0,0). In the case of coexistence of X and Y, research and analysis can be carried out under the condition of $R+2={(x,y)|x>0,y>0}$ R_ + ^2 = \left\{{\left({x,y} \right)|x > 0,y > 0} \right\} , and only the equilibrium point P1 can be discussed.

Suppose p1(x* y*), where $x*=M1(1−b1)1−b1b2>0$ {x^*} = {{{M_1}\left({1 - {b_1}} \right)} \over {1 - {b_1}{b_2}}} > 0 , $y*=M2(1−b2)1−b1b2>0$ {y^*} = {{{M_2}\left({1 - {b_2}} \right)} \over {1 - {b_1}{b_2}}} > 0 , 0 < b1, b2 < 1, then the following equation is satisfied: ${1M1x*−b1M2y*=1−b2M1x*+1M2y*=1$ \left\{\matrix{{1 \over {{M_1}}}{x^*} - {{{b_1}} \over {{M_2}}}{y^*} = 1 \hfill \cr - {{{b_2}} \over {{M_1}}}{x^*} + {1 \over {{M_2}}}{y^*} = 1 \hfill \cr} \right.

Convert the above formula to: ${dxdt=−k1x(t)(x−x*M1−b1M2(y−y*))dydt=−k2y(t)(−b2M1(x−x*)+1M2(y−y*))$ \left\{\matrix{{{dx} \over {dt}} = - {k_1}x\left(t \right)\left({{{x - {x^*}} \over {{M_1}}} + {{{b_1}} \over {{M_2}}}\left({y - {y^*}} \right)} \right) \hfill \cr {{dy} \over {dt}} = - {k_2}y\left(t \right)\left({{{{b_2}} \over {{M_1}}}\left({x - {x^*}} \right) - {1 \over {{M_2}}}\left({y - {y^*}} \right)} \right) \hfill \cr} \right.

And get: $V(x,y)=1k1(x−x*−x*lnxx*)+1k2(y−y*−y*lnyy*)$ V\left({x,y} \right) = {1 \over {{k_1}}}\left({x - {x^*} - {x^*}\ln {x \over {{x^*}}}} \right) + {1 \over {{k_2}}}\left({y - {y^*} - {y^*}\ln {y \over {{y^*}}}} \right)

The actual calculation results are:

So V of x, y is a positive definite function inside R2, and as (x, y) approaches the x and y axes, we get V(x, y) → +∞.

The total derivative of V (x, y) according to the above model is: $V*(x,y)=−(x−x*)(x−x*M1−b1(y−y*)M2)−(y−y*)(−b2(x+x*)M1+y−y*M2)=−[1M1(x−x*)2−(b2M1+b1M2)(x−x*)(y−y*)+1M2(y−y*)2]$ \eqalign{& \mathop V\limits^* \left({x,y} \right) = - \left({x - {x^*}} \right)\left({{{x - {x^*}} \over {{M_1}}} - {{{b_1}\left({y - {y^*}} \right)} \over {{M_2}}}} \right) - \left({y - {y^*}} \right)\left({- {{{b_2}\left({x - {x^*}} \right)} \over {{M_1}}} + {{y - {y^*}} \over {{M_2}}}} \right) \cr & = - \left[{{1 \over {{M_1}}}{{\left({x - {x^*}} \right)}^2} - \left({{{{b_2}} \over {{M_1}}} + {{{b_1}} \over {{M_2}}}} \right)\left({x - {x^*}} \right)\left({y - {y^*}} \right) + {1 \over {{M_2}}}{{\left({y - {y^*}} \right)}^2}} \right] \cr}

Under the condition of b1b2 ≠ 1, limit cycles will not appear outside the equilibrium point $p1(M1(1−b1)1−b1b2,M2(1−b2)1−b1b2)$ {p_1}\left({{{{M_1}\left({1 - {b_1}} \right)} \over {1 - {b_1}{b_2}}},{{{M_2}\left({1 - {b_2}} \right)} \over {1 - {b_1}{b_2}}}} \right) inside the system.

Note 7. Suppose ${dxdt=k1x(1−xM1−b1yM2)Δ__P(x,y)dydt=k2y(1−b2xM1−yM2)Δ__Q(x,y)$ \left\{\matrix{{{dx} \over {dt}} = {k_1}x\left({1 - {x \over {{M_1}}} - {b_1}{y \over {{M_2}}}} \right)\underline{\underline \Delta} P\left({x,y} \right) \hfill \cr {{dy} \over {dt}} = {k_2}y\left({1 - {b_2}{x \over {{M_1}}} - {y \over {{M_2}}}} \right)\underline{\underline \Delta} Q\left({x,y} \right) \hfill \cr} \right. , and inside the first quadrant we get the function B(x, y) = xα−1 yβ−1, where α, β represents a particular constant, from which we can calculate $DΔ__∂(BP)∂x+∂(BQ)∂y=k1xα−1yβ−1(α−αM1x+b1αM2y−1M1x)+k2xα+1yβ−1(β+b2βM1x−βM2y−1M2y)=xα−1yβ−1[−1M1(k1α−k2b2β+k1)x−1M2(k2β−k1b1α)y+k1α+k2β]$ \eqalign{& D\underline{\underline \Delta} {{\partial \left({BP} \right)} \over {\partial x}} + {{\partial \left({BQ} \right)} \over {\partial y}} \cr & = {k_1}{x^{\alpha - 1}}{y^{\beta - 1}}\left({\alpha - {\alpha \over {{M_1}}}x + {{{b_1}\alpha} \over {{M_2}}}y - {1 \over {{M_1}}}x} \right) + {k_2}{x^{\alpha - 1}}{y^{\beta - 1}}\left({\beta + {{{b_2}\beta} \over {{M_1}}}x - {\beta \over {{M_2}}}y - {1 \over {{M_2}}}y} \right) \cr & = {x^{\alpha - 1}}{y^{\beta - 1}}\left[{- {1 \over {{M_1}}}\left({{k_1}\alpha - {k_2}{b_2}\beta + {k_1}} \right)x - {1 \over {{M_2}}}\left({{k_2}\beta - {k_1}{b_1}\alpha} \right)y + {k_1}\alpha + {k_2}\beta} \right] \cr}

Open Problem 8

After the symbols of x and y are specified, in order to keep D unchanged, the following requirements must be met: ${k1α−k2b2β+k1=0k2β−k1b1α+k2=0$ \left\{\matrix{{k_1}\alpha - {k_2}{b_2}\beta + {k_1} = 0 \hfill \cr {k_2}\beta - {k_1}{b_1}\alpha + {k_2} = 0 \hfill \cr} \right.

It can be calculated as follows: $α=k1+k2b2k1(b1b2−1),β=k1b1+k2k2(b1b2−1)$ \alpha = {{{k_1} + {k_2}{b_2}} \over {{k_1}\left({{b_1}{b_2} - 1} \right)}},\beta = {{{k_1}{b_1} + {k_2}} \over {{k_2}\left({{b_1}{b_2} - 1} \right)}}

And: $D=k1(1+b1)+k2(1+b2)1−b1b2xα−1yβ−1$ D = {{{k_1}\left({1 + {b_1}} \right) + {k_2}\left({1 + {b_2}} \right)} \over {1 - {b_1}{b_2}}}{x^{\alpha - 1}}{y^{\beta - 1}}

Assuming constant k1(b1 −1)+ k2 (b2 −1) ≠ 0 meets this condition, combined with the above studies, it is proved that the system does not have closed orbits.

Evaluation and analysis of the comprehensive management efficiency of Online education in Chinese colleges and universities
Efficiency Evaluation

To put it simply, the evaluation of network education comprehensive management efficiency is to study the relationship between output and input during teaching management, so as to better understand the ability of network education in colleges and universities to allocate resources, as well as the sustainable development and competition level in the market competition. Starting from different research perspectives, the efficiency of online education in colleges and universities can be divided into two situations, one is technical efficiency, the other is scale efficiency, the former refers to the highest output level of online education under given input conditions, while the latter refers to whether online education can be operated under standard investment. According to the analysis of the single input and single output efficiency curve shown in FIG. 1 below, the production function y=f (x) represents the maximum output y that can be obtained when the production is in the optimal state.[4]

The technical efficiency studied in this paper mainly analyzes the distance between network education teaching and production function. Assuming that the teaching center of A university operates at point A, the actual technical efficiency can be calculated as follows:

Technical efficiency of A = Bd/bA

Because the production function represents the relationship between input and output under the ideal state, all teaching centers in colleges and universities will operate below the production function curve, with the technical efficiency ≤1. The technical efficiency of online education on the production function curve =1, also known as technical efficiency.

Point E divides the production function into two parts. The function on the left keeps rising, which means that increasing the amount of input can increase output, and online education has the initiative of investment. This range can be regarded as the stage of increasing returns to scale. Glaze function is in a continuous decline stage. In the case that the investment amount is X, if it continues to increase, the efficiency of output improvement is not high. At this time, online education has no enthusiasm to continue investment. It can be seen that the input scale represented by point E is the most appropriate. Generally speaking, the calculation formula of scale efficiency of online education in colleges and universities is as follows: The scale efficiency of A is equal to B C over B D

In general, scale efficiency ≤1 is regarded as scale effective when it is equal to 1. The online education represented by point E is scale effective, while the online education represented by A, D and F is not effective.

Empirical Research

At present, online education in Colleges and universities in China has begun to cultivate talents needed for national construction in large quantities. This study selected the education center of online education college built in a certain place for empirical analysis, including 13 teaching centers as the evaluation target, and evaluated their output orientation and management effectiveness. The specific results are shown as follows[5]:

First, data sets. The index data of the teaching center in 2020 are selected to form the input-output data set, as shown in Table 1 below:

Input-output data sets of online education institutes in different regions

The teaching center The evaluation index Input indicators Output indicators
The average number of (X1) Number of receiving devices (X2) Number of multimedia classrooms (X3) Number of networked computers (X4) Number of students graduating in the year (Y1) Employment rate (Y2) Average Grade (Y3)
Tianjin First Teaching Center 41 5 5 150 54 32.3 78
Tianjin Second Teaching Center 71 1 7 180 247 20.5 76
Tianjin Third Teaching Center 44 1 6 40 75 60.8 72
Tianjin Fourth Teaching Center 90 2 5 150 243 33.4 79
Tianjin Fifth Teaching Center 55 1 5 248 102 46.2 78
Tianjin Sixth Teaching Center 19 1 3 80 86 50.6 75
Tanggu Teaching Center 34.5 1 9 250 270 61.8 85
Dagang branch 9.5 1 2 40 117 42 72
Tertiary Teaching Center 59 1 10 85 356 68.9 73
Shenzhen Teaching Center 39.5 1 15 300 146 76.2 82
Jinzhou Teaching Center 51 1 4 200 107 80 66
Huludao Teaching Center 33 2 11 150 301 83.6 54
Zhengzhou Teaching Center 42 1 1 60 43 75 69

Second, evaluate the results. Based on the linear gauge construction of the above education centers, the corresponding efficiency values can be calculated, as shown in Table 2 below:

Efficiency evaluation results of online education institutes in different regions

Tianjin First Teaching Center Efficiency value (C3R)
Tianjin Second Teaching Center 0.40
Tianjin Third Teaching Center 0.99
Tianjin Fourth Teaching Center 1.00
Tianjin Fifth Teaching Center 0.92
Tianjin Sixth Teaching Center 1.00
Tanggu Teaching Center 1.00
Dagang branch 1.00
Tertiary Teaching Center 1.00
Shenzhen Teaching Center 1.00
Jinzhou Teaching Center 1.00
Huludao Teaching Center 1.00
Zhengzhou Teaching Center 0.97
Tianjin First Teaching Center 1.00

Third, result analysis. Efficiency is 1 of the teaching center's comprehensive management efficiency has effectiveness. Combined with the results obtained from the above research, it is proved that the teaching center with efficiency, quality and effectiveness has cultivated a large number of graduates with a higher actual employment rate, although it has invested less manpower and equipment. However, other teaching centers with relatively ineffective management efficiency need to be further promoted and eradicated, only in this way can the management efficiency be guaranteed to be effective.

Fourthly, study on evaluation of super efficiency. Combined with the above research results, the Education Center of College of Network Education is selected as the DMU and its indicators in 2020 are selected, as shown in Table 3 below:

Evaluation results of central efficiency of online education colleges in different regions

Tianjin First Teaching Center Efficiency value (C3R) Super efficiency value (SE-DEA)
Tianjin Second Teaching Center 0.40 0.40
Tianjin Third Teaching Center 0.99 0.99
Tianjin Fourth Teaching Center 1.00 1.00
Tianjin Fifth Teaching Center 0.92 0.92
Tianjin Sixth Teaching Center 1.00 1.00
Tanggu Teaching Center 1.00 1.02
Dagang branch 1.00 1.16
Tertiary Teaching Center 1.00 2.21
Shenzhen Teaching Center 1.00 2.33
Jinzhou Teaching Center 1.00 1.09
Huludao Teaching Center 1.00 1.08
Zhengzhou Teaching Center 0.97 0.97
Tianjin First Teaching Center 1.00 3.57

According to the above numerical calculation and analysis, the final super-efficiency evaluation results prove that the management effectiveness of online education is higher if the value is greater than or equal to 1. Research integrated management effectiveness of university network education, compared with traditional assessment methods, this paper studies on the basis of the construction of ordinary differential equation model for the empirical analysis, is on the basis of the related numerical carried on deep discussion, finally it is concluded that the evaluation results more objective and fair, to an invalid network education put forward effective improvement scheme, And provide scientific basis for educational administrators to make relevant policies[6].

Development countermeasures of network education in colleges and universities in the future
Realizing resource sharing based on university cooperation

In recent years, China's university network education has begun to take shape, especially after the construction of modern distance education cooperation group, gradually expand the scope and level of university network education institutions of cooperation. For example, in the development of curriculum resources, based on cooperative discussion to master more educational courseware; In the construction of learning sites and face-to-face guidance, the construction of resource sharing platform. However, nowadays network education in colleges and universities is still promoting relatively independent teaching plans and management strategies, so it is difficult to build a good cooperation platform. Therefore, on the basis of clarifying the importance of cooperation among colleges and universities, the educational administrative departments should construct the cooperation platform of network education according to the educational nature of colleges and universities in different regions, and clarify the standardized courses and educational services shared by colleges and universities. In the case of mature conditions, we can also draw lessons from western virtual universities, develop similar educational programs and teaching courses, and uniformly issue academic degrees, so as to show the advantages of network education of colleges and universities in different regions.[7]

Create diversified teaching resources based on network technology

Network technology makes university education work more interesting, the actual education resources more abundant, the teaching content more vivid and dynamic, can carry out all-round and diversified teaching guidance to accept educators. According to the characteristics analysis of online education as shown in the figure below, current investigations and studies show that when humans receive external information, 30% of them rely on vision, 11% on hearing, 1% on taste, 1.5% on touch and 3.5% on smell. Therefore, multi-channel transmission of educational knowledge is more effective than single channel. Network education in colleges and universities to help students to obtain access to information from various angles, there are a lot of common channel, for instance in network channels to collect readings, download the music platform audio clip, in a web page to watch 2 d or 3 d images, etc., can participate in teaching activities, fully mobilize students' various senses and can focus on the students' attention, Enable them to master the knowledge in accelerated comprehension and full expression.[8]

Attach importance to the organization and management of network education in colleges and universities

In the design and promotion of institutions, network education belongs to the basic content of educational management in colleges and universities. Based on the analysis of the requirements put forward by the current education department, it contains the organizational structure and responsibilities and functions as shown in the following table. Although the overall design of institutions and departments is very perfect, the communication and cooperation between various departments should be strengthened in the development of practice. Only in this way can the management level of practice education be improved and the cultivation of professional talents be done well.

Organizational structure, responsibilities and functions of online education in colleges and universities

Set up agencies Organizational functions and responsibilities
Administration department Responsible for the daily affairs and management of the Network College; Responsible for the personnel management of the Network College; Responsible for the coordination of various departments and rooms within the college, the external liaison work of the network college and the internal financial management of the college.
Academic affairs department Responsible for the recruitment of network college; Responsible for the development of various types of education programs, the development of teaching plans, teaching programs; Organize and hire teachers to implement network teaching, including teaching, online tutoring and answering questions, homework correction, experiment and practice teaching, graduation defense, etc., the development of network courses and various teaching course assessment, credit recognition, certificate issuance; Student course selection management.
Technical department Responsible for the construction and management of off-campus teaching stations; Network college all teaching equipment management and maintenance; Research and development of network teaching platform; Network college website construction and maintenance, update and daily management.
Attach importance to the construction of teachers for network education in colleges and universities

In the network education of colleges and universities, the continuous optimization of teachers' educational ideas and the enrichment of teachers' source channels are conducive to the rational allocation of teachers' resources while sharing them. First of all, diversity of teachers should be guaranteed. As more of a college network education to guide teachers in the social, so colleges can directly to the social demand positioned and open recruitment, so don't need to worry about the teacher's geographical location, no longer depend on the unified textbooks, but on the basis of widely collecting network data access to the most practical and most innovative knowledge technology. Secondly, the intellectual resources of teachers should be integrated. The lack of sufficient teachers is a major obstacle to the implementation of higher education in China, and online education has no limitations in age and other aspects. As long as individuals can meet the needs of online education, they can find suitable positions. It can not only change the collective behavior of teachers, but also optimize the organizational form and professional division of the existing network education by allowing outstanding talents from all fields of society to participate in the network education work, comprehensively integrating the governance resources of the society and enhancing the cooperative and interactive relationship between education. Finally, it is helpful for professional teachers to participate in scientific research and ability training. The large-scale promotion of network education in colleges and universities has put forward higher requirements for professional teachers. On the one hand, they need to expand their horizons, update their knowledge structure and technical ability in time, and quickly grasp the core achievements of the new direction of modern science and technology development. On the other hand, according to the changing trend of the growth of human science and technology knowledge, we should integrate the most advanced and novel knowledge and technology into the teaching work, keep the enthusiasm of learning and discussion for a long time, pay attention to the construction and promotion of knowledge system, so as to improve our comprehensive education management level.[9, 10, 11]

Conclusion

To sum up, in the steady development of online education in Colleges and universities in China, the implementation quality and management efficiency are the main direction of empirical research. At present, theoretical analysis and empirical research on network education in colleges and universities are in the stage of development to be improved, and there are many problems that have not been solved. Therefore, researchers and scholars should enrich the research results in this field in continuous exploration and pay attention to the rational use of ordinary differential equation model for review and research. Only in this way can network education in colleges and universities become the main direction of educational innovation in the future and truly meet the increasing personalized learning needs of the masses of citizens.

#### Efficiency evaluation results of online education institutes in different regions

Tianjin First Teaching Center Efficiency value (C3R)
Tianjin Second Teaching Center 0.40
Tianjin Third Teaching Center 0.99
Tianjin Fourth Teaching Center 1.00
Tianjin Fifth Teaching Center 0.92
Tianjin Sixth Teaching Center 1.00
Tanggu Teaching Center 1.00
Dagang branch 1.00
Tertiary Teaching Center 1.00
Shenzhen Teaching Center 1.00
Jinzhou Teaching Center 1.00
Huludao Teaching Center 1.00
Zhengzhou Teaching Center 0.97
Tianjin First Teaching Center 1.00

#### Evaluation results of central efficiency of online education colleges in different regions

Tianjin First Teaching Center Efficiency value (C3R) Super efficiency value (SE-DEA)
Tianjin Second Teaching Center 0.40 0.40
Tianjin Third Teaching Center 0.99 0.99
Tianjin Fourth Teaching Center 1.00 1.00
Tianjin Fifth Teaching Center 0.92 0.92
Tianjin Sixth Teaching Center 1.00 1.00
Tanggu Teaching Center 1.00 1.02
Dagang branch 1.00 1.16
Tertiary Teaching Center 1.00 2.21
Shenzhen Teaching Center 1.00 2.33
Jinzhou Teaching Center 1.00 1.09
Huludao Teaching Center 1.00 1.08
Zhengzhou Teaching Center 0.97 0.97
Tianjin First Teaching Center 1.00 3.57

#### Input-output data sets of online education institutes in different regions

The teaching center The evaluation index Input indicators Output indicators
The average number of (X1) Number of receiving devices (X2) Number of multimedia classrooms (X3) Number of networked computers (X4) Number of students graduating in the year (Y1) Employment rate (Y2) Average Grade (Y3)
Tianjin First Teaching Center 41 5 5 150 54 32.3 78
Tianjin Second Teaching Center 71 1 7 180 247 20.5 76
Tianjin Third Teaching Center 44 1 6 40 75 60.8 72
Tianjin Fourth Teaching Center 90 2 5 150 243 33.4 79
Tianjin Fifth Teaching Center 55 1 5 248 102 46.2 78
Tianjin Sixth Teaching Center 19 1 3 80 86 50.6 75
Tanggu Teaching Center 34.5 1 9 250 270 61.8 85
Dagang branch 9.5 1 2 40 117 42 72
Tertiary Teaching Center 59 1 10 85 356 68.9 73
Shenzhen Teaching Center 39.5 1 15 300 146 76.2 82
Jinzhou Teaching Center 51 1 4 200 107 80 66
Huludao Teaching Center 33 2 11 150 301 83.6 54
Zhengzhou Teaching Center 42 1 1 60 43 75 69

#### Organizational structure, responsibilities and functions of online education in colleges and universities

Set up agencies Organizational functions and responsibilities
Administration department Responsible for the daily affairs and management of the Network College; Responsible for the personnel management of the Network College; Responsible for the coordination of various departments and rooms within the college, the external liaison work of the network college and the internal financial management of the college.
Academic affairs department Responsible for the recruitment of network college; Responsible for the development of various types of education programs, the development of teaching plans, teaching programs; Organize and hire teachers to implement network teaching, including teaching, online tutoring and answering questions, homework correction, experiment and practice teaching, graduation defense, etc., the development of network courses and various teaching course assessment, credit recognition, certificate issuance; Student course selection management.
Technical department Responsible for the construction and management of off-campus teaching stations; Network college all teaching equipment management and maintenance; Research and development of network teaching platform; Network college website construction and maintenance, update and daily management.

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