Dynamic model construction and correlation analysis of college students’ academic performance

Abstract

With the increase of college students year by year, their academic performance has become one of the key directions of the majority of scholars. To find out college students’ academic performance, we take the student academic performance caused by the mutual influence between the dynamic change of research direction, building mathematical model, through the study of college students in all academic examination, was conditioned in college students, and must be rebuilt effect between college students, to determine the existence of the equilibrium point, This paper also introduces and analyses the stability of the variables similar to the basic regeneration number in mathematical epidemiology, and gives the corollary of the existence of the latter branch. Considering the random influence of campus atmosphere on students and the uncertainty and lag of the influence mechanism among students, a time-delay stochastic model is constructed based on the mathematical model. Then, the quantitative analysis of these influences was carried out, and the relationship between them was found, and due efforts were made for the in-depth study of the academic performance of college students.

Keywords

College students academic performance mathematical model time-delay stochastic model quantitative analysis

1. Introduction

College students’ academic performance is intrinsically related to their willingness to drop out [1]. Many scholars have pointed out that, among other factors, students’ intention to drop out is usually caused by poor exam results [2, 3]. Once students fail to pass the exam, they will be in a very negative mood and need to take a make-up exam. Therefore, make-up exam is a major indicator of their intention to drop out. According to The psycho-social theory of Albert Banshera, college students’ learning basically takes place in society [4]. It can be seen that the academic performance of students will be affected by their peers, which will lead to the occurrence of make-up examinations, retakes and other behaviors. To find out through, make-up examination, rebuild social influence between groups, we will from the stability theory of ordinary differential equations, the consequent branch theory and relevant theory, to lay a good foundation for the dynamic model and the corresponding analysis, and then realize the construction of college students’ academic performance dynamics model and the related analysis, in order to make clear college students through, the influence of the make-up exam, rebuild between people.

2. A related theoretical analysis

2.1 Stability theory of ordinary differential equations

In the stability theory of ordinary differential equations [5], $n$ dimensional equations are given as Eq. (1).

$\displaystyle\frac{dx}{dt}=Ax$ (1)

In Eq. (1), ${A}=({{a}_{{ij}}})_{{n}\times{n}}$ is the matrix of the system, ${x}=({{x}_{1},{x}_{2},\cdots,{x}_{n}})^{T}\in{R}^{n}$ . If all the eigenvalues of the matrices in the equations have real parts, the result (zero solution) will be unstable, otherwise, the result will be locally stable. According to Routh-Hurwitz stability criterion [6, 7, 8], the ${n}$ order algebraic equation with given constant coefficients is shown in Eq. (2).

$\displaystyle{a}_{0}{\lambda}^{n}+{a}_{1}{\lambda}^{{n}-1}+{a}_{2}{\lambda}^{{% n}-2}\cdots+{a}_{{n}-1}{\lambda}+{a}_{n}=0$ (2)

When ${a}_{0}>0$ , consider the following equation:

$\displaystyle\Delta_{1}=a_{1},\Delta_{2}\left|\begin{array}[]{cc}a_{1}&a_{0}\\ a_{3}&a_{2}\\ \end{array}\right|,\Delta_{3}=\left|\begin{array}[]{ccc}a_{1}&a_{0}&0\\ a_{3}&a_{2}&a_{1}\\ a_{5}&a_{4}&a_{3}\\ \end{array}\right|,\ldots$ (3) $\displaystyle\Delta_{n}=\left|\begin{array}[]{cccccc}a_{1}&a_{0}&0&0&\ldots&0% \\ a_{3}&a_{2}&a_{1}&a_{0}&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{2n-1}&a_{2n-2}&a_{2n-3}&a_{2n-4}&\ldots&a_{n}\\ \end{array}\right|=a_{n}\Delta_{n-1}$

When ${a}_{i}=0$ , ${i}>n$ , the necessary and sufficient conditions for all roots of Eq. (2) to be negative real parts are as follows:

$\displaystyle a_{1}>0,\Delta_{2}>0,\Delta_{3}>0,\cdots,\Delta_{n-1}>0,a_{n}>0$ (4)

2.2 Backward branching theory

In the latter branch theory, consider an ordinary differential equation with parameter $\phi$ , as shown in Eq. (5).

$\displaystyle\frac{{dx}}{{dt}}={f}({{x},\phi}),{f}:{R}^{n}\times{R}\to{R}^{n},% {f}\in{C}^{2}({{R}^{n}\times{R}})$ (5)

Without loss of generality, let 0 be the equilibrium of Eq. (5). At this point, let ${f}({0,\phi})=0$ , two conclusions can be drawn. First, at $\phi=0$ , the linear matrix at equilibrium point 0 is represented by $={D}_{x}{f}({0,0})=\left({\frac{\partial{f}_{i}}{\partial{x}_{j}}({0,0})}\right)$ , where 0 is a simple eigenvalue of the matrix, and the real parts of all other eigenvalues are negative. Second, when the eigenvalue of matrix ${A}$ is 0, there are two non-negative eigenvectors, one is the right eigenvector ${w}$ , the other is the left eigenvector ${v}$ . Let the ${f}$ component of kth be ${f}_{k}$ , then we have the following equation:

$\displaystyle{a}=\mathop{\sum}\limits_{{k},{i},{j}=1}^{n}{v}_{k}{w}_{i}{w}_{j}% \frac{\partial^{2}{f}_{k}}{\partial{x}_{i}{x}_{j}}({0,0})$ (6) $\displaystyle{b}=\mathop{\sum}\limits_{{k},{i},{j}=1}^{n}{v}_{k}{w}_{i}\frac{% \partial^{2}{f}_{k}}{\partial{x}_{i}\partial\emptyset}({0,0})$ (7)

Therefore, as Eq. (4) approaches 0, its local dynamics are determined by ${a}$ and ${b}$ .

2.3 Random stability analysis

When conducting random stability analysis [9], ${n}$ dimensional stochastic differential Eq. (8) should be considered.

$\displaystyle{dx}={f}({{x}_{t},{t}}){dt}+{g}({{x}_{t},{t}}){dB}({t})$ (8)

In Eq. (8), the initial condition is ${x}({{t}_{0}})={x}_{0}\epsilon{C}^{+}({[{-{h},0}],{R}^{n}})$ , the continuous function space goes from $[{-h,0}]$ to $R^{n}$ , and the norm is $||\psi||=\textit{sup}_{\theta\epsilon[{-h,0}]}|{\psi(\theta)}|$ . Let the Eq. (8) contain trivial solutions, and all non-negative functions $C^{2,1}({R^{n}\times[{t_{0},\infty}],R^{+}})$ defined on $R^{n}\times[{t_{0},\infty}]$ interval are continuously differentiated when ${V}({{x},{t}})$ , then the obtained sense differential operator ${L}$ can be written as:

$\displaystyle{L}=\frac{\partial}{\partial{t}}+\mathop{\sum}\limits_{{i}=1}^{n}% {f}_{i}({{x}({{t}-{\xi}}),{t}})\frac{\partial}{\partial{x}_{i}}+\frac{1}{2}% \mathop{\sum}\limits_{{i},{j}=1}^{n}[{{g}^{T}({{x}({{t}-{\xi}}),{t}}){g}({{x}(% {{t}-{\xi}}),{t}})}]\frac{\partial^{2}}{\partial{x}_{i}\partial{x}_{j}}$ (9)

If ${\varepsilon}\epsilon({0,1})$ , ${r}>0$ is satisfied for any $\hat{\delta}=\hat{\delta}(\varepsilon,r,t_{0})$ and $|{{x}_{0}}|<\delta$ , $p\{|{{x}({{t},{t}_{0},{x}_{0}})}|<r,t\geqslant{t}_{0}\}\geqslant 1-{\varepsilon}$ , then the trivial solution of Eq. (8) is randomly or probabilistically stable, and the other way around is unstable.

3. Construction of the basic model of dynamics of college students’ academic performance

In universities, students who fail their academic exams must take make-up exams, and if they fail, they have to repeat the course, which means they have to start all over again. If you fail the test again, you will be asked to leave school. In order to determine the social influence among those who pass, resit and retake the exam, we formulated a basic dynamic model of college students’ academic performance by referring to the standard SIR epidemic model [10]. During the construction period, let ${N}({t})$ represent the number of undergraduate students, and the total number of undergraduate students will change with time. Then, students are divided into three groups: student ${P}(t)$ who has passed all courses, student ${M}(t)$ who must take make-up exams, and student ${R}(t)$ who must retake them. The schematic diagram of the model is constructed, as shown in Fig. 1.

Figure 1.

Dynamic model of college students’ academic performance.

In Fig. 1, ${\mu T}+{\mu}_{1}{R}$ is the rate of freshmen passing, and the average number of years of undergraduate study is represented by $\frac{1}{{\mu}}$ . $\frac{({{\beta}_{1}{M}+{\beta}_{2}{R}}){P}}{{T}}$ is the social impact rate among the three groups of students, and $\frac{\beta_{3}MR}{T}$ is the social influence rate between ${M}(t)$ and ${R}(t)$ . Make-up exam students through all the make-up exam to go back to the ratio of ${P}(t)$ group by ${\sigma}$ , said to reconstruct the students returned to the ratio of ${P}(t)$ crowd by ${\eta}$ said, $\mu_{1}$ to rebuild the student dropout rates, make-up exam the student to the qualified students average effective negative influence by $\beta_{1}$ , dropouts to pass the student’s average effective negative influence by $\beta_{2}$ . The average effective negative influence of retaking students on resit students is represented by $\beta_{3}$ . $\beta_{1}$ , $\beta_{2}$ and $\beta_{3}$ are all AENIRP factors.Ideally, one would expect all students to belong to ${P}(t)$ . In practice, however, such expectations are hard to meet. There are students who pass all the exams as well as those who resit, retake and drop out. Therefore, the state of students in all subgroups in our visual model is the real state. In summary, we construct the following model:

$\displaystyle{\frac{dP}{dt}=\mu T+({\mu_{1}+\eta})R+\sigma M-\mu P-\frac{\beta% _{1}PM}{T}-\frac{\beta_{2}PR}{T}}$ $\displaystyle{\frac{dM}{dt}=\frac{\beta_{1}PM}{T}+\frac{\beta_{2}PR}{T}-\frac{% \beta_{3}MR}{T}-({\sigma+\mu})M}$ (10) $\displaystyle{\frac{dR}{dt}=\frac{\beta_{3}MR}{T}-({\eta+\mu+\mu_{1}})R}$

Where, the initial conditions are set as follows: ${P}(0)=P_{0}>0$ , $M(0)=M_{0}>0$ , $R(0)=R_{0}\geqslant 0$ . If all model parameters are positive, dimensionless processing can be performed on the stated variables (the proportion in the total population is represented by corresponding lowercase letters), and the simplified model can be obtained, as shown in Eq. (3).

$\displaystyle{\frac{dp}{dt}=\mu+({\mu_{1}+\eta})r+\sigma m-\mu p-\beta_{1}pm-% \beta_{2}pr}$ $\displaystyle{\frac{dm}{dt}=\beta_{1}pm+\beta_{2}pr-\beta_{3}mr-({\sigma+\mu})m}$ (11) $\displaystyle{\frac{dr}{dt}=\beta_{3}mr-({\eta+\mu+\mu_{1}})r}$

In Eq. (3), ${p}=\frac{{P}}{{T}}$ , ${m}=\frac{{M}}{{T}}$ , ${r}=\frac{{R}}{{T}}$ .

4. Analysis of stability and backward branching

Combined with the construction and analysis of the basic model above, the equilibrium point (ideal state) of the basic model of dynamics of college students’ academic performance under the equilibrium state of “no make-up examination, no retake” is presented here:

$\displaystyle X_{0}=({p_{0},m_{0},r_{0}})=({1,0,0})$ (12)

4.1 Stability analysis

Before the stability analysis, the regeneration number $R_{01}$ of the model is defined as the number of students influenced by a make-up student in the average learning cycle, and this variable is similar to the basic regeneration number of mathematical infectious diseases. After simple calculation, the obtained matrix is shown in Eq. (13).

$\displaystyle FV^{-1}=\left({{\begin{array}[]{*{20}c}{\frac{\beta_{1}}{\sigma+% \mu}}&{\frac{\beta_{2}}{\eta+\mu+\mu_{1}}}\\ 0&0\\ \end{array}}}\right)$ (13)

The maximum eigenvalue of Eq. (13) is the basic regeneration number, as shown in Eq. (14).

$\displaystyle R_{01}=\frac{\beta_{1}}{\sigma+\mu}$ (14)

In this case, the Jacobian matrix at $X_{0}$ is shown in Eq. (15).

$\displaystyle{J}({X_{0}})=\left({{\begin{array}[]{*{20}c}{-\mu}&{\sigma-\beta_% {1}}&{\mu_{1}+\eta-\beta_{2}}\\ 0&{\beta_{1}-\sigma-\mu}&{\beta_{2}}\\ 0&0&{-\eta-\mu-\mu_{1}}\\ \end{array}}}\right)$ (15)

Its characteristic polynomial is shown in Eq. (16).

$\displaystyle({\lambda+\mu})[{\lambda-({\beta_{1}-\sigma-\mu})}]({\lambda+\eta% +\mu+\mu_{1}})=0$ (16)

The eigenvalues can be computed from Eq. (17).

$\displaystyle\lambda_{1}=-\mu<0,\lambda_{2}=\beta_{1}-\sigma-\mu,\lambda_{3}=-% ({\eta+\mu+\mu_{1}})<0$ (17)

Therefore, conclusion 1 can be obtained: when $R_{01}<1$ , the ideal state is locally asymptotically stable. In the equilibrium state (subideal state) of model “with make-up examination without retaking”, and $R_{01}>1$ , its equilibrium point is:

$\displaystyle X_{1}=({p_{1},m_{1},r_{1}})=\left({\frac{\sigma+\mu}{\beta_{1}},% 1-\frac{\sigma+\mu}{\beta_{1}},0}\right)$ (18)

At this point, the number of retaking is also defined as the number of students influenced by a make-up student in the average learning cycle, which is expressed by Eq. (19).

$\displaystyle R_{02}=\frac{({\beta_{1}-\sigma-\mu})\beta_{3}}{({\mu_{1}+\mu+% \eta})\beta_{1}}=\left({\frac{\beta_{3}}{\mu_{1}+\mu+\eta}}\right)\left({1-% \frac{1}{R_{01}}}\right)$ (19)

The derivation of Jacobi matrix, polynomials and eigenvalues is roughly the same as the previous one, so no analysis is made here. The second conclusion is as follows: when $0<{R}_{02}<1$ , the subideal state is locally asymptotically stable. The model is in the state of “make-up examination and retaking” (the realistic state), so that $X^{\ast}=({p^{\ast},m^{\ast},r^{\ast}})$ is in the realistic equilibrium state. In order to find this equilibrium point, the right side of all the equations of the model is zero, then the following formula can be obtained:

$\displaystyle p^{\ast}=\frac{({\eta+\mu+\mu_{1}})({\sigma+\mu+\beta_{3}r^{\ast% }})}{\beta_{1}({\eta+\mu+\mu_{1}})+\beta_{2}\beta_{3}r^{\ast}}$ (20) $\displaystyle A({r^{\ast}})^{2}+Br^{\ast}+C=0,A=\beta_{2}\beta_{3}^{2}\mu>0$ (21) $\displaystyle{B}={\beta}_{3}{\mu}[{({{\mu}_{1}+{\mu}+{\eta}})({{\beta}_{1}+{% \beta}_{2}+{\beta}_{3}})-{\beta}_{2}{\beta}_{3}}]$ (22) $\displaystyle{C}={\mu}({{\mu}_{1}+{\mu}+{\eta}})[{{\beta}_{1}({{\mu}_{1}+{\mu}% +})+{\beta}_{3}({{\sigma}+{\mu}-{\beta}_{1}})}]={\mu\beta}_{1}({{\mu}_{2}+{\mu% }+{\eta}})^{2}({1-{R}_{02}})$ (23)

At this point, let:

$\displaystyle R_{02}^{\ast}=1-\frac{[{({\eta+\mu+\mu_{1}})({\beta_{1}+\beta_{2% }+\beta_{3}})-\beta_{2}\beta_{3}}]^{2}}{4\beta_{1}\beta_{2}({\eta+\mu+\mu_{1}}% )^{2}}$ (24)

Conclusion 3: When $R_{02}>1$ , the model has only one equilibrium point: $I^{\ast}=({p^{\ast},m^{\ast},r^{\ast}})$ , where $r^{\ast}=\frac{-B+\sqrt{B^{2}-4AC}}{2A}$ ; When $R_{02}=1,\frac{\beta_{2}+\beta_{2}+\beta_{3}}{\beta_{2}\beta_{3}}<\frac{1}{\mu% _{1}+\mu+\eta}$ , the model has only one equilibrium point: $I^{\ast}=({p^{\ast},m^{\ast},r^{\ast}})$ , where, $r^{\ast}=-\frac{B}{A}$ ; When $R_{02}=R_{02}^{\ast},\frac{\beta_{1}+\beta_{2}+\beta_{3}}{\beta_{2}\beta_{3}}<% \frac{1}{\mu_{1}+\mu+\eta}$ , the model has only one equilibrium point: $I^{\ast}=({p^{\ast},m^{\ast},r^{\ast}})$ , where $r^{\ast}=-\frac{B}{2A}$ ; When $R_{02}^{\ast}<R_{02}<1,\frac{\beta_{1}+\beta_{2}+\beta_{3}}{\beta_{2}\beta_{3}% }<\frac{1}{\mu_{1}+\mu+\eta}$ , the model has two equilibrium points: $I_{1}^{\ast}=({p_{1}^{\ast},m_{1}^{\ast},r_{1}^{\ast}})$ , $I_{2}=({p_{2}^{\ast},m_{2}^{\ast},r_{2}^{\ast}})$ , where $r_{1,2}^{\ast}=\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}$ ; For $R_{02}\leqslant 1,\frac{\beta_{1}+\beta_{2}+\beta_{3}}{\beta_{2}\beta_{3}}>% \frac{1}{\mu_{1}+\mu+\eta}$ or $R_{02}<R_{02}^{\ast}$ , there is no equilibrium.

4.2 Backward branch

When analyzing the backward branch, first let ${p}={x}_{1}$ , ${m}={x}_{2}$ , ${r}={x}_{3}$ , then the simplified model, namely Eq. (3), can be transformed into a vector form, as shown in Eq. (25).

$\displaystyle\frac{dX}{dt}=H(t)$ (25)

When ${X}=({{x}_{1},{x}_{2},{x}_{3}})^{T}$ , ${H}=({{h}_{1},{h}_{2},{h}_{3}})^{T}$ , then 10 becomes 26.

$\displaystyle{\frac{dx_{1}}{dt}=\mu+({\mu_{1}+\eta})x_{3}+\sigma x_{2}-\mu x_{% 1}-\beta_{1}x_{1}x_{2}-\beta_{2}x_{1}x_{3}:=h_{1}}$ $\displaystyle{\frac{dx_{2}}{dt}=\beta_{1}x_{1}x_{2}+\beta_{2}x_{1}x_{3}-\beta_% {3}x_{2}x_{3}-({\sigma+\mu})x_{2}:=h_{2}}$ (26) $\displaystyle{\frac{dx_{3}}{dt}=\beta_{3}x_{2}x_{3}-({\eta+\mu+\mu_{1}})x_{3}:% =h_{3}}$

By solving the Jacobian matrix, calculating the eigenvectors and related parameters, we get the fourth conclusion: if $R_{02}>1$ , $X^{\ast}$ under realistic equilibrium belongs to local asymptotic stability. However, there are also special cases, such as $R_{02}=1$ , and there is a backward branch of the model after any of the following equivalent conditions are satisfied. Equivalent condition 1, ABNIRP factor is greater than a critical value, then $\beta_{1}>\beta_{2}^{\ast}=\frac{\beta_{1}({\beta_{1}+\mu_{2}+\eta-\sigma})}{% \sigma+\mu}$ ; Equivalent condition 2, ABNIRP factor belongs to a certain critical value, then: $\beta_{1}\in\left({0,\frac{({\mu_{1}+\eta-\sigma})+\sqrt{({\mu_{1}+\eta-\sigma% })^{2}+4\beta_{2}({\sigma+\mu})}}{2}}\right)$ ; Equivalent condition 3, when the dropout rate is lower than a certain value, then $\mu_{1}<({\sigma+\mu})\frac{\beta_{2}}{\beta_{1}}+\sigma-\beta_{1}-\eta$ .

5. Numerical simulation and discussion

Here, we carry out data simulation and discuss the influence of relevant parameters on model dynamics and threshold quantity. Specifically, the sensitivity of $R_{01}$ and $R_{02}$ to model parameters was firstly analyzed, as shown in Fig. 2A and B.

Figure 2.

Sensitivity analysis of $R_{01}$ and $R_{02}$ to model parameters.

As can be seen from Fig. 2a, $\beta_{1}=1$ means that the negative impact of students who have to take make-up exams on students who pass all courses will increase or decrease by 10%, while the number of students who have to retake will increase by 10%. Both ${\eta}$ and ${\mu}$ are positive numbers, indicating that the pass rate of students who are required to take make-up exams will lead to an increase in the number of students who pass all courses. As can be seen from Fig. 2b, $R_{02}$ is sensitive not only to $\beta_{3}$ , but also to ${\sigma}$ , ${\eta}$ and $\beta_{1}$ . Then, the backward branch of the model is explored, and the results obtained are shown in Fig. 3.

Figure 3.

Model backward branch test.

In Fig. 3, the black curve and line segment are in stable equilibrium state, while the red curve and black dashed line segment are in unstable equilibrium state. It can be seen that under the condition of (0.273253012, 1), $R_{02}$ has the backward branching problem. In order to avoid this problem, the negative influence of the students who have to retake the test on the students who have to take the make-up test can be set below a certain critical value, so as to solve the backward peak problem, but to a certain extent, the number of students who have to retake the test will increase.

6. Stochastic model of general infection function

In fact, the interaction between students is not a short process. In the short term, there are three situations: first, the negative impact of students with poor academic performance on students who have completed all academic exams does not appear immediately; second, the negative impact mechanism of poor students on average students (passing all exams) is also not obvious. immature; third, students’ academic performance will also be disturbed by the campus atmosphere. Therefore, on the basis of the simplified model (Eq. (3)), we consider the following three factors: First, the negative impact of bad students on ordinary students can be expressed by a general function. Second, the negative effects of bad students on ordinary students are time-lag. Third, random influences in the school environment. Based on the consideration of these three factors, the definition of ${P}(t)$ is explained thus. In infectious diseases, ${P}(t)$ represents the possibility of an individual staying in the latent stage t time without mortality. If ${P}$ also meets the following two conditions, the time delay model can be determined.

Case 1: ${P}:[0,\infty)\to[{0,1}]$ has the characteristics of non-incremented segment continuity and is accompanied by more finite jumps. So case two $({0+})=1,\textit{lim}_{t\to\infty}P(t)=0$ and $0<\int_{0}^{\infty}{P}({u}){du}<\infty$ theta. If ${P}(t)$ is the distribution function, then:

$\displaystyle{P}(t)=\left\{{{\begin{array}[]{l}{1,t\in[{0,\tau}]}\\ {0,t>\tau}\\ \end{array}}}\right.$ (27)

The obtained delay model is as follows:

$\displaystyle{\frac{dp}{dt}=n(p)-f({p(t),m(t),r(t)})}$ $\displaystyle{\frac{dm}{dt}=\int_{0}^{h}Q(\xi)e^{-\mu\xi}f({p({t-\xi}),m({t-% \xi}),r({t-\xi})})d\xi-\beta_{3}mr-({\sigma+\mu})m}$ (28) $\displaystyle{\frac{dr}{dt}=\beta_{3}mr-({\eta+\mu+\mu_{1}})r}$

In the model, the increase rate of middle school students in ${p}$ warehouse is represented by ${n}(p)$ , the pass rate of make-up examination under the action of self-control force is represented by ${\sigma}$ , the time of undergraduate students in school is represented by $\frac{1}{\mu}$ , the repeat pass rate and dropout rate are represented by ${\eta}$ and $\mu_{1}$ respectively, and the longest incubation period of negative effects is represented by ${h}$ . ${Q}=-{P}^{\prime}$ implies that ${P}$ is piecewise continuous, so six hypotheses are proposed as follows: Hypothesis 1, ${n}$ on $R^{+}$ is a continuous function, there is $P_{0}>0$ , such that ${n}({P_{0}})=0$ , if $0\leqslant{P}\neq{P}_{0}$ , then $({n(P)-n({P_{0}})})({P-P_{0}})<0$ . Suppose two, ${f}$ is a locally Lipsitz continuous function on $R^{+}\times R^{+}\times R^{+}$ , then ${f}({{p},0,{r}})={f}({{p},{m},0})={f}({0,{m},{r}})=0$ . Hypothesis three, if ${m}$ and ${p}$ are constant, $f_{k}$ is an increasing function of ${p}$ and ${m}$ . Hypothesis four, $({p,r})=\frac{\left({p,\frac{\eta+\mu+\mu_{1}}{\beta_{3}},r}\right)-\frac{({% \sigma+\mu})\frac{\eta+\mu+\mu_{1}}{\beta_{3}}}{\mathop{\int}\nolimits_{0}^{h}% Q(\xi)e^{-\mu\xi}d\xi}}{r}$ is a bounded and strictly decreasing function of r. Hypothesis five, ${k}({p})={\lim}_{{r}\to 0^{+}}\phi({{p},{r}})$ is a continuously increasing function of ${p}$ . If necessary, hypothesis one can be converted to hypothesis six, that ${n}$ is $R^{+}$ strictly decreasing continuous function, $\exists{p}_{0}>0$ , on the interval ${n}({p_{0}})=0$ , such that R, then ${\lim}_{p\to\infty}n(p)=-\infty$ .

6.1 The existence of equilibrium point

Through verification, it is found that the time-delay model (Eq. (6)) has a unique subideal equilibrium point, which is $X_{1}=\left({P_{1},\frac{\eta+\mu+\mu_{1}}{\beta_{3}},0}\right)$ . However, the actual equilibrium state needs to be considered, so the following basic regeneration number needs to be defined:

$\displaystyle R_{0}=\frac{\left({\mathop{\int}\nolimits_{0}^{h}Q(\xi)e^{-\mu% \xi}d\xi}\right)k({p_{1}})}{\eta+\mu+\mu_{1}}$ (29)

To sum up, it can be concluded that 5, when $R_{0}>1$ , the delay model (Eq. (6)) has a realistic equilibrium state, $X^{\ast}=({p^{\ast},m^{\ast},r^{\ast}})$ . In order to prove the existence and uniqueness of the non-repeating equilibrium point $X^{\ast}$ , corresponding derivation is carried out based on the assumptions above, and the results verify the existence and uniqueness of the realistic equilibrium point $X^{\ast}$ .

6.2 Random stability

Considering that campus atmosphere will also have a random influence on students’ study, if Eq. (6) is disturbed by white noise and this interference is proportional to the difference of equilibrium point ${p}(t),m(t),r(t)$ corresponding to $\hat{p},\hat{m},\hat{r}$ , then Eq. (6) can be rewritten into a random model, as shown in Eq. (6.2).

$\displaystyle\frac{{dp}}{{dt}}={n}({p})-{f}({{p}({t}),{m}({t}),{r}({t})})+{% \sigma}_{1}({{p}-{\hat{p}}})\frac{{dB}_{1}}{{dt}}$ $\displaystyle\frac{{dm}}{{dt}}=\int_{0}^{h}{Q}({\xi}){e}^{-{\mu\xi}}{f}({{p}({% {t}-{\xi}}),{m}({{t}-{\xi}}),{r}({{t}-{\xi}})}){d\xi}-{\beta}_{3}{mr}-({{% \sigma}+{\mu}}){m}{}+{\sigma}_{2}({{m}-{\hat{m}}})\frac{{dB}_{2}}{{dt}}$ (30) $\displaystyle\frac{{dr}}{{dt}}={\beta}_{3}{mr}-({{\eta}+{\mu}+{\mu}_{1}}){r}+{% \sigma}_{3}({{r}-{\hat{r}}})\frac{{dB}_{3}}{{dt}}$

Then, the random stability of $\hat{X}=(\hat{p},\hat{m},\hat{r})$ can be studied. In order to facilitate the study, the substitution of relevant variables, such as ${x}={p}-\hat{p}$ , $y=m-\hat{m}$ , $z=r-\hat{r}$ , is carried out. Then, to simplify the calculation, let:

$\displaystyle\bar{Q}=\int_{0}^{h}Q(\xi)d\xi,\bar{Q}_{\mu}=\int_{0}^{h}Q(\xi)e^% {-\mu\xi}d\xi$ $\displaystyle f_{\hat{p}}=\frac{\partial f}{\partial\hat{p}}(\hat{p},\hat{m},% \hat{r}),f_{\hat{m}}=\frac{\partial f}{\partial\hat{m}}(\hat{p},\hat{m},\hat{r% }),f_{\hat{r}}=\frac{\partial f}{\partial\hat{r}}(\hat{p},\hat{m},\hat{r})$ (31)

Finally, in order to obtain the random stability condition of the zero solution, the random model (Eq. (6.2)) is linearized, and the results are as follows:

$\displaystyle\frac{dx}{dt}=n(\hat{p})x-f_{\hat{p}}x-f_{\hat{m}}y-f_{\hat{r}}z+% \sigma_{1}x\frac{dB_{1}}{dt}$ $\displaystyle\frac{dy}{dt}=\int_{0}^{h}Q(\xi)e^{-\mu\xi}(f_{\hat{p}}x(t-\xi),f% _{\hat{p}}y(t-\xi),f_{\hat{p}}z(t-\xi))d\xi-\beta_{3}(y\hat{r}+z\hat{m})-(% \sigma+\mu)y{}+\sigma_{2}y\frac{dB_{2}}{dt}$ (32) $\displaystyle\frac{dz}{dt}=\beta_{3}(y\hat{r}+z\hat{m})-(\eta+\mu+\mu_{1})z+% \sigma_{3}z\frac{dB_{3}}{dt}$

After the test, when the following three groups of conditions are met, $\hat{X}$ is stochastic asymptotically stable. Condition one, $\sigma_{1}^{2}<f_{\hat{p}}-f_{\hat{m}}-f_{\hat{r}}-2n^{\prime}(\hat{p})$ , condition two, $\sigma_{3}^{2}<\beta_{3}\hat{m}+2(\mu+\sigma)-2f_{\hat{m}}-k_{1}-k_{2}-k_{3}$ , condition three, $\sigma_{3}^{2}<-3\beta_{3}\hat{m}-\beta_{3}\hat{r}+2(\eta+\mu+\mu_{1})-2f_{r}$ .

In summary, through random analysis of the influence among students, it is found that in addition to the influence of classmates, students will also be interfered by other factors, such as school environment, academic ability, physical characteristics, etc. all of which may be It has become an inducement factor for students to retake, make up exams, and drop out of school, even for students with excellent grades. Worthy of our focus on the problem, although research has certain limitation, but this article is with the aid of mathematical tools and models based on risk of dropping out some ideas to build and analysis, thus have a certain scientific nature and rationality, follow-up study these limitations to enrich model can be solved by building work, make it more perfect.

7. Summary

In general, this paper firstly conducts popular science on the stability theory of ordinary calculus equations, backward branch theory and other related theoretical knowledge, which lays the foundation for the construction and analysis of the dynamic model of academic performance of college students. Secondly, based on the analysis of theoretical knowledge and the idea of the basic reproduction number of mathematical infectious diseases, the construction of the dynamic model of college students’ academic performance is realized, and the stability and backward branches are analyzed, and the corresponding conclusions are obtained, and found that there is a backward branch in the model. In order to solve this problem, a plan is proposed to help retake students’ enthusiasm for learning through incentive measures, so that they can better complete the academic assessment, and the feasibility of the mathematical results is confirmed through numerical simulation experiments. Finally, based on the model (Eq. (3)), a time-delay model is constructed, and the balance point mining and stochastic stability analysis are carried out, and it is concluded that students will be affected by other factors besides classmates. Such as family dynamics, behavioral characteristics, disciplinary policies, etc.

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