Differential equation to verify the validity of the model of the whole-person mental health education activity in Universities

We use questionnaire surveys, psychological consultations, behaviour observations and other forms to analyse students’ ideological conditions. At the same time, we classify college students in school according to their ideological status. Infector Warehouse I: Students accept wrong public opinion and be active. Students often spread out wrong opinion in public. For example, class meetings, peer exchanges, etc. Lurker E: The student is in a period of uncertainty [3]. They are overwhelmed by wrong public opinion, and public opinion affects the everyday life of the individual. Susceptible warehouse S: The students have weak ideological awareness and are easily influenced by students in warehouse I. Educator Warehouse R: Students who come out of warehouse I after ideological education. They have ‘immunity’ and spread positive energy.

Because of the absorbent spread of thought and public opinion and the mutual influence of human behaviour, we build an SEIR DCE with constant input and a bilinear incidence rate: (1) { S.=pA−βSI−μSE.=βSI−δE−μE−εEI.=δE−γI−μI−εIR.=(1−p)A+γI+εI+εE−μR \left\{ {\matrix{ {\mathop S\limits^. = pA - \beta SI - \mu S} \hfill \cr {\mathop E\limits^. = \beta SI - \delta E - \mu E - \varepsilon E} \hfill \cr {\mathop I\limits^. = \delta E - \gamma I - \mu I - \varepsilon I} \hfill \cr {\mathop R\limits^. = (1 - p)A + \gamma I + \varepsilon I + \varepsilon E - \mu R} \hfill \cr } } \right. A is a constant input (the number of new students enrolled each year). µ is the output rate of each warehouse (for example, graduation). δ is the incidence rate from warehouse E to warehouse I. γ is the recovery rate from warehouse I to warehouse R. ε is the student’s self-regulated recovery rate. The model uses bilinear incidence. β is the infection rate coefficient [4]. Where A is a constant, µ, δ, γ, ε, β > 0, 0 < p < 1.

We use the regeneration matrix to calculate the primary regeneration number of the above SEIR model as (2) R0=βδpAμ(δ+μ+ε)(γ+μ+ε) {R_0} = {{\beta \delta pA} \over {\mu (\delta + \mu + \varepsilon )(\gamma + \mu + \varepsilon )}} In the transmission model, the reproduction number R₀ represents the average number of people that a communicator I can infect when all the populations are susceptible. Furthermore, we can use the Lyapunov function to prove that R₀ can be used as a threshold for disseminating inadequate information. Therefore, the SEIR, as mentioned above, studies the dissemination of wrong public opinion among college students R₀ ≤ 1, the influence of such wrong public opinion will gradually disappear over time under the existing ideological education system. When R₀ > 1 the existing ideological education system fails, and the influence of such wrong public opinion will exist for a long time and bring challenges to the ideological development of students. It will seriously distort the students’ outlook on life and values [5]. Through the above analysis, we get that R₀ is not only a threshold for whether wrong public opinion is spread but also a standard for evaluating the ideological health of college students and the ideological education system of colleges and universities.

Through observation and analysis, the parameters that affect the primary regeneration number R₀ are µ, ε, δ, γ, β. Since µ is the proportion of output students, it is no longer within the management scope of colleges and universities, so I will not discuss it here. Below we discuss the impact of other parameters on R₀. We calculate the partial derivative of R₀ concerning the above parameters and get: (3) ∂R0∂β=pAδμ(δ+μ+ε)(γ+μ+ε)∂R0∂p=Aβδμ(δ+μ+ε)(γ+μ+ε)∂R0∂δ=pAβ(μ+ε)μ(γ+μ+ε)(δ+μ+ε)2∂R0∂γ=−pAβδμ(δ+μ+ε)(γ+μ+ε)2∂R0∂ε=−pAβδ(δ+γ+2μ+2ε)μ(δ+μ+ε)2(γ+μ+ε)2 \matrix{ {{{\partial {R_0}} \over {\partial \beta }} = {{pA\delta } \over {\mu (\delta + \mu + \varepsilon )(\gamma + \mu + \varepsilon )}}{{\partial {R_0}} \over {\partial p}} = {{A\beta \delta } \over {\mu (\delta + \mu + \varepsilon )(\gamma + \mu + \varepsilon )}}} \hfill \cr {{{\partial {R_0}} \over {\partial \delta }} = {{pA\beta (\mu + \varepsilon )} \over {\mu (\gamma + \mu + \varepsilon )(\delta + \mu + \varepsilon {)^2}}}{{\partial {R_0}} \over {\partial \gamma }} = - {{pA\beta \delta } \over {\mu (\delta + \mu + \varepsilon )(\gamma + \mu + \varepsilon {)^2}}}} \hfill \cr {{{\partial {R_0}} \over {\partial \varepsilon }} = - {{pA\beta \delta (\delta + \gamma + 2\mu + 2\varepsilon )} \over {\mu {{(\delta + \mu + \varepsilon )}^2}{{(\gamma + \mu + \varepsilon )}^2}}}} \hfill \cr } Easy to know (4) ∂R0∂β>0,∂R0∂p>0,∂R0∂δ>0,∂R0∂γ<0,∂R0∂ε<0 {{\partial {R_0}} \over {\partial \beta }} > 0,{{\partial {R_0}} \over {\partial p}} > 0,{{\partial {R_0}} \over {\partial \delta }} > 0,{{\partial {R_0}} \over {\partial \gamma }} < 0,{{\partial {R_0}} \over {\partial \varepsilon }} < 0 Based on the mathematical analysis of the SEIR mentioned above propagation model, what we have to do is to try to reduce R₀ by increasing γ and ε, and reducing β, δ and p. We make the primary reproduction number <1 to ensure that the influence of such wrong public opinion gradually disappears among college students.

There is no initial immunity (shown in Figure 1(A)). Then assume that all members are randomly immunised, and the number of people immunised is 30% of the total population (shown in Figure 1(B)), 50% (shown in Figure 1(C)) and 90% (shown in Figure 1(D)).

Fig. 1

A certain percentage of mental health education can control the spread of wrong information from the simulation results. Figure 1(D) shows that when the popularisation rate of mental health education is not high, insufficient information will start to be controlled. But over time, wrong information will still fluctuate. In this case, the number of transmissions continues to rise after some time. In general, the number of communicators gradually decreases when the popularisation rate of mental health education increases. It can be seen that the small ratio of psychological health education only temporarily controls terrible information, but it cannot eliminate lousy information fundamentally [11]. Therefore, if the lousy information is effectively controlled through psychological health education, a considerable proportion of the population needs psychological health education. Figure 2 reflects the changing law of the number of communicators when the psychological health education rate is 0.1, 0.3 and 0.9, respectively.

Fig. 2

This article proposes the concept of targeted injection. We need to purposefully carry out psychological health education for people who have come in contact with them. This is because the number of nodes in the social network has the most potent destructive effect on disseminating lousy information, which provides us with a new idea for effective prevention. We choose the number of nodes (frequent social interactions) for the target injection (the injection ratio is 0.1). Target injection can significantly reduce the number of people spreading and the effect of controlling the spread of insufficient information is more prominent.

We observe the development trend of wrong information from the change of p₃, p₅. Under the premise that the anti-bad information education intensity (p₅) is determined, the sensitivity of the number of communicators to the rate of the communicator’s receiving education decreases slightly with the increase of the rate of education received (Figure 3). From the time of the outbreak, increasing the rate of education has a noticeable impact on the growth rate of the number of communicators. The greater the education rate, the slower the growth of communicators, and eventually stabilises gradually. Therefore, we need to intervene appropriately while assisting with other control methods.

Fig. 3

The thesis uses education intensity to reflect the school’s whole-person psychological education level, taking p₅ to take 0.2, 0.5 and 0.9 to get the simulation chart 4. From the results, when the overall psychological education level of the education school is determined (that is, p₃ is always 0.6), when the education level is 0.2, the number of transmissions fluctuates periodically, and the peak value gradually decreases. As the level of education gradually improves, and the anti-bad information education increases to a certain level; it is possible to control the lousy information at first temporarily. The number of disseminators will fluctuate significantly over time. Therefore, the author believes that anti-bad information education should be sufficient enough.

Fig. 4