# Why do students quit school? Implications from a dynamical modelling study

## Abstract

In 2012, more than three million students dropped out from high school. At this pace, we will have more than 30 million Americans without a high school degree by 2022 and relatively high dropout rates among Hispanic and African American students. We have developed and analysed a data-driven mathematical model that includes multiple interacting mechanisms and estimates of parameters using data from a specifically designed survey applied to a certain group of students of a high school in Chicago to understand the dynamics of dropouts. Our analysis suggests students' academic achievement is directly related to the level of parental involvement more than any other factors in our study. However, if the negative peer influence (leading to lower academic grades) increases beyond a critical value, the effect of parental involvement on the dynamics of dropouts becomes negligible.

### 1. Introduction

A dropout is usually defined as ‘a person who stops going to a school, college, etc., before finishing: a person who drops out of school.’ It is also defined as ‘a person who stops being involved in society because he or she does not believe in its rules, customs, and values’ [1]. The term dropout in the United States is used in three different ways: event dropout, status dropout and cohort dropout [2]. The event dropout rate measures the percentage of students who left school during a specific period, such as a year, without graduating. The cohort rate measures the dropout for a specific age group such as for students in ninth grade. The status dropout is the proportion of the population without diplomas who are between 16 and 24 years old and are not in school. We use the status dropout as a definition for dropouts in this study. However, there may be differences in the use of terminology around the world such as those collected in table 1. In this study, we primarily use terms common in the USA because of our focus and sampled population.

United States | some other countries |
---|---|

freshman (∼14–15 years old) | grade 9 |

sophomore (∼15–16 years old) | grade 10 |

junior (∼16–17 years old) | grade 11 |

senior (∼17–18 years old) | grade 12 |

high school | secondary school |

dropout | early school leavers |

charter schools | autonomous/free schools (receiving reduced state funding) |

According to the National Center for Education and Statistics [3], more than three million students dropped out of high school in 2010 in the USA. This means around 17 000 students drop out daily, considering a school year of 180 days. If this trend continues, there will be approximately 31 million dropouts, roughly 10% of US current population, by 2020.

Dropouts have negative effects on various aspects of our society. According to the US Census Bureau [4], the median earnings in dollars of a non-graduate from high school was $27 470, compared to $34 197 for graduate high school earner. This difference results in about $350 000 less income over the 35 years of work between a graduate and non-graduate high school student. This will imply less taxes and revenue for the government which results in lower purchasing power and a low level of productivity. In addition, non-high school graduates are less likely to be employed versus the high school graduate [5]. Relative to those who graduate, those who do not graduate will each cost the economy around $250 000 in terms of reliance on Medicare or Medicaid, higher reliance on welfare and lower tax contributions [6].

Because the high school dropout has an enormous effect on the society as a whole, many studies and reports have analysed various sets of high school data with varied success. Even politicians have spoken about this topic and many plans and strategies have been put in place in order to understand the problem. In 2010, President Obama said: ‘This is a problem we can't afford to accept or ignore. The stakes are too high—for our children, for our economy, for our country. It's time for all of us to come together—parents and students, principals and teachers, business leaders and elected officials—to end America's dropout crisis’. [7]

Researchers have identified numerous factors for dropping out from high schools. Some of the factors include socio-economic status, academic disengagement, behavioural disengagement, family dynamics, attitudes, values and beliefs about education, school structures, school resources, student body characteristics, school environment, academic policies and practices, supervision and discipline policies and practices, location and type of school, demographic characteristics and community environment [8–10] as well as challenges associated with learning certain subjects and concepts. Most of these factors are interrelated and it is hard to determine which factor causes the others. Moreover, time and again, high parental involvement and guidance has been shown to have a major impact on a student's life, reducing the likelihood of dropping out. However, parents reported having limited time to share and difficulty of involving children in household activities [11]. Many families include two breadwinners and some are single-parent families with conflicting roles and responsibilities to support their families. Existing policies and programmes for reducing dropout rates are often not designed to meet needs of low-income and working poor families. There are trade-offs between the longer working hours of parents, the high cost associated with the development of children and better environments for children [12]. This research reviews factors driving dropouts and helps to suggest a wide range of components that may be likely to make a difference. The components include evaluating patterns of potential dropout rates, identifying at-risk students and changing the factors that schools may control to reduce dropouts.

As for the proposed solutions, The National Dropout Center/Network identified fifteen different strategies to mitigate the dropout rate [13]. These strategies consist of an ongoing evaluation system of the goals and policies; community participation and support; safety for students; parental engagement; focus on early childhood education, especially in reading and writing; continuous support through tutoring and mentoring; engaging students within the community; providing after-school and summer programmes especially for at-risk students as well as an alternative school for dropout students; maintaining an ongoing professional development for teachers who are engaged with high-risk students; student engagement through technology and individualizing students' instruction.

As mentioned above, the suggested factors for high school dropouts and the remedies are numerous and interrelated. We focus on understanding the impact of some significant primary factors such as parental involvement, demographic factors and peer influences between students, some identified from our survey, on the dynamics of high school dropout prevention. This does not undermine the importance of the rest of the factors. The reasons for limiting the model to these main factors are the feasibility of analysing the mathematical model on the one hand, and the lack of studies done on the dynamics of students on and off campus on the other hand.

Despite the complexity in dropout dynamics [14,15], Mathis *et al.* [16] argued that the United States government has been implementing policies that are not research based, hence, the urgent need for the present work. There are multiple purposes for this study including testing of a mathematical model based on parental involvement and contact between students in order to understand the dropout dynamics. Dropping out of high school has unfavourable consequences on the individual as well as the society as a whole [17]. It has been linked to numerous factors that can be divided into internal (within the school) and external(outside the school). Despite the large number of suggested solutions, the number of dropouts seems to be on the rise especially within large cities. We study the impact of parental involvement and social contacts between students inside and outside of school on the dynamics of dropout. Here, we focus on school students and their activities, collectively referred to here as school environment. It represents school time during which social relationships between current students develop within a campus, and potential off campus relationships between students and non-students or their parents. Our goal is to focus on factors of the school environment and see how the number of failing students can be decreased and, in consequence, reduce the dropout rate. Our approach consists of the development and analysis of a dynamic mathematical model using an analogy with an SIR-type infectious disease model via a set of differential equations, its analysis and simulations.

In particular, using our carefully designed study, we focus on (i) identification of factors that may be critical to students' academic achievement and quantification of the impact of parental involvement on it via a survey, (ii) development of a dynamic model using factors identified through the survey specifically designed for the model, and (iii) analysis of the model to understand the role of parental involvement and peer interactions on the dynamics of dropouts.

### 2. Methods

#### (a) Data sources

We designed and carried out a survey in a group of students of one sampled school in the Chicago Public School (CPS) system. The group contained students from 9 to 12 grade levels. An approval for conducting the survey was obtained from the institutional review board at Northeastern Illinois University in 2013. The survey was distributed to 145 participants among students of the sampled school. However, five students did not respond to the survey; therefore, 140 surveys were eventually collected. Fifteen additional surveys were considered unusable because they were incomplete. With 125 usable survey responses out of 145, the response rate was 86.2%. The survey contained 16 questions and was taken online by students. Questions in the survey resulted in quantifying metrics such as the number of failing students in different core subjects (Mathematics, Science, English and Social Sciences), degree of parental involvement in their life and number of failing and dropout friends of each student during last year. Sampled students were also asked questions related to last year's performance levels and interests of their friends in education. Additional data, which cannot be obtained from our survey, was collected from the National Center of Education Statistics [2] and CPS online databases. These last two datasets were used to estimate dropout rates and the total enrolment numbers from 2005 to 2013 for the city of Chicago and for the USA [18]. We compared demographic characteristics of national/Chicago city school students with corresponding statistics of students from our sampled school (figure 1). We also collected and relied on the information from previous studies in the literature that surveyed dropout youths in the USA [14,19,20].

CPS district is the third largest school district in the USA [18]. It encompasses a total of 681 schools, out of which 472 are elementary and 106 are high schools. Moreover, it has 96 Charter schools and seven Contract schools. CPS hosts a total of 400 545 students from which 87% come from low-income families. The CPS student population is diverse, but Hispanics and African-Americans make up around 85% of the total (figure 1).

We carried out a survey at the sampled high school in Chicago which was chosen because it is one of the most representative (falling in the lowest 5% of all schools) schools in south Chicago based on socio-economic factors related to students and the neighbourhood where they live. The total student enrolment in the sampled school in 2013–2014 was 1866 [18], where about 94% of students come from low-income families and 73.3% were Hispanic (figure 1).

In order to quantify levels of parental involvement in the school, we asked students questions related to involvement of parents in their daily activities during the previous semester. The parental involvement was defined and measured using inputs of students on three primary activities of their parents: attending parent teacher conferences, attending after-school activities, and checking grades at least once a month. We defined *high parental involvement* as a student's response mentioning parents being present for all the above three activities. A *moderate parental involvement* means parents fulfilled two out of the three activities. A *low parental involvement* means parents fulfilled one of the three activities. Finally, *no parental involvement* means parents failed to participate in any of the three activities of the student (figure 2*a*).

Academic achievement of a student in the school was used to categorize them into *passing*, *vulnerable* and *failing* categories. A student is grouped in a *passing* category if he or she passed all the core subjects; in a *vulnerable* category, if s/he failed only one core subject and in a *failing* category if s/he failed two or more core courses (figure 2*b*).

Peer influences that eventually lead to a student dropping out from the school were also defined and measured. These influences were termed as *negative social influences* and were manifested through multiple facets such as but not limited to the number of failing friends, the number of dropout friends, and illegal and criminal activities in the region. One way to capture the negative social influence is to quantify the attitude of friends of an individual towards school (figure 2*c*). Hence, students in the survey were asked about their friends' performance levels in the education (‘among your close friends in the school, how many are failing two or more classes?’) as well as their friends' opinion about school, for which question the answer choices were: if school is a waste of time for your friend, if sometimes it is a waste of time and if not a waste of time? (figure 2*c*).

The data were collected primarily to estimate parameters of the dynamic model discussed in §2c. We also investigated the correlation between students' achievement and negative social influence and explored the impact of parental involvement on students' achievement in the school. A typical dropout individual leaves the modelled population at the age 24 (i.e. youths leave the community/neighbourhood of the school at the rate $\approx 1/(24-18)=\frac{1}{6}$ per year). However, the average number of years a typical student spent in the school is assumed to be around 4 years, though this number could be high for a school in a poor neighbourhood with a vulnerable (to dropout) population of students.

#### (b) Model description

We developed a model based on socio-demographic characteristics of our sampled school. The total population (*T*) in the model is divided into two subpopulations: normal (*N*) and dropout (*D*). The normal population is stratified into passing (*P*), vulnerable (*V*) and failing (*F*) groups of students (table 2). Our model structure follows an SIR-type epidemic compartmental model framework [21–25]. Similar types of models have been derived for other social problems such as for fanatic behaviours [26] and alcohol drinking [27]. The model considers multiple mechanisms including rates of incoming students (*μN*+*μ*_{1}*D*), outgoing students (*μ*_{1}*D*), parental involvement (*σV*), the school effectiveness (*αP*), social influence inside the school between individuals in *V* class and in *F* class (*β*_{1}*V* (*kF*/*T*)), social influence outside the school between failing and dropout individuals (*β*_{2}(*D*/*T*)*F*), and self-dropout owing to non-school factors (*γF*), which is ignored in this research because of lack of data for it. From here onwards, we refer to the passing and vulnerable populations as the non-risky population and call the failing and dropouts the risky population (table 3). Table 4 summarizes the definition and units of parameters and table 2 contains the definition of variables. The interpretation of the model parameters is clarified in §3, where parameter estimation is carried out using data from our survey.

variables | definition |
---|---|

T |
total population of the community |

N |
population of students in the sampled high school |

P |
number of passing students in all core classes (mathematics, sciences, English and social sciences) |

V |
number of vulnerable students, who are failing one core class |

F |
number of failing students, who are failing two or more core classes |

D |
number of dropout students |

equilibrium | definition | (v) |
(f) |
(d) |
---|---|---|---|---|

E_{0} |
completely passing state (only passing students' type exist) | ${v}_{0}={\displaystyle \frac{\alpha}{\alpha +\sigma \mu}}$ | 0 | 0 |

E_{1} |
absence of dropouts | ${v}_{1}={\displaystyle \frac{\mu}{{\beta}_{1}k}}$ | ${f}_{1}={\displaystyle \frac{\alpha}{\alpha +\mu}}\left({\displaystyle \frac{{R}_{01}-1}{{R}_{01}}}\right)$ | 0 |

${E}_{1}^{\ast}$ | dropout endemic equilibrium | ${v}_{1}^{\ast}$ | ${f}_{1}^{\ast}$ | ${d}_{1}^{\ast}$ |

${E}_{2}^{\ast}$ | dropout endemic equilibrium | ${v}_{2}^{\ast}$ | ${f}_{2}^{\ast}$ | ${d}_{2}^{\ast}$ |

parameter | description | value | reference |
---|---|---|---|

α |
per capita transfer rate from passing to vulnerable owing to the school ineffectiveness |
0.59 per year | estm. |

γ |
per capita dropout rate because of personal reasons |
— | n.a. |

p |
probability of successful passing of negative influence given a contact (dimensionless) | 0.2 | estm. |

β_{1}=c_{1}p |
transmission efficiency (i.e. average effective negative influences) of dropouts (D) on vulnerable (V) students outside the campus |
0.51 per year | estm. |

β_{2}=c_{2}p |
transmission efficiency (i.e. average effective negative influences) of dropouts (D) on failing (F) students outside the campus (part of δ(D) function in the model) groups |
0.42 per year | estm. |

k |
decrease in intensity of transmission efficiency (negative social influences) of F (as compared to D) on V (dimensionless quantity). Note, ${\beta}_{1}k={c}_{1}^{\ast}p$ is the transmission efficiency between F and V |
0.6 | estm. |

σ |
per capita students' average improvement rate because of enhancement of parental involvement |
0.41 per year | estm. |

1/μ |
average number of years a typical high school student stay in the school | 4 years | estm. |

μ_{1} |
per capita local community exit rate of dropouts |
0.17 per year | assumed |

In order to analyse the model, we assume that students will graduate within 4 years. Those who graduate will leave the community and will not have contact with the current student population in the school. Furthermore, students entering high school are classified as passing students. In addition to that, according to Allensworth & Easton [28] from the University of Chicago's Consortium on Chicago School Research [28], students who fail more than one core class during one whole year are less likely to graduate on time and are called ‘off-track’. On the other hand, students who are failing, at most one core class in a whole year, are three times more likely to graduate on time and are called ‘on-track’. The core classes are mathematics, science, English and social sciences. Our definition of vulnerable students is based on this study and are students who are failing one core class. Students who are failing two or more classes are classified as ‘failing students’. It is also assumed that a low parental involvement will lead passing students to move from passing to vulnerable class. If parental involvement continues to be low, this will lead to a longer number of unhealthy contacts with failing students which will eventually lead to dropout. Figure 3 describes the dynamic of the model and it can be expressed as a system of coupled differential equations (see electronic supplementary material, equation (4.1)).

The model captures the dynamics associated with dropout rates via a system of differential equations. The passing population builds its membership from the new incoming freshmen. Owing to the ineffectiveness of the school, a proportion of students will leave the passing population and move to the vulnerable stage. Vulnerable students may improve their grades as a result of reinforced parental influence (with a rate *σV*) or may move to failing class under the influence of failing and dropout peers. The rate of transfer from vulnerable to failing is *β*_{1}*V* ((*kF*+*D*)/*T*), where *β*_{1} measures the effective negative peer pressure associated with the failing students and *k* is the decrease in intensity of contact of failing individuals compared with the corresponding influence by dropout peers. Hence, the transfer-to-failure rate is proportional (with rate constant *β*_{1}) to the number of contacts per year between vulnerable and failing. The dropout is replenished via *δ*(*D*)=*β*_{2}(*D*/*T*)*F*, which is the number of dropouts per year.

#### (c) Relevant quantities

For *γ*=0, the model has two reproduction numbers (see [29]) relative to two distinct dropout-free states, namely *E*_{0}=(*v*_{0},0,0) and *E*_{1}=(*v*_{1},*f*_{1},0) (see calculations in electronic supplementary material). The first basic reproduction number corresponding to *E*_{0} is

*R*

_{01}represents the average number of newly generated vulnerable students influenced by a typical failing and dropout peer in a completely non-risky population. It is the product of the average number of contacts of an individual and the probability of his/her successful influence

*β*

_{1}, the infectious period of a typical

*F*individual 1/(

*μ*+

*γ*), the decrease in intensity of contact

*k*and the proportion of non-risky students

*v*

_{0}.

The second basic reproduction number, corresponding to *E*_{1}, is obtained considering *D* as the only influential compartment:

*R*

_{02}represents the average number of newly generated failing students in a completely non-risky population influenced by a typical dropout peer. This number is the product of the effective (successful negative influence) average number of contacts of an individual

*β*

_{2}, the duration of influence by a typical

*F*individual 1/

*μ*and the proportion of failing students

*f*

_{1}.

Depending on the value of *R*_{01} and *R*_{02}, we can have up to four equilibrium points. In addition, mathematical analysis and numerical simulations suggest the existence of another threshold quantity *R*_{e}=(*R*_{01}−1)/(*R*_{01}+*k*) (see the electronic supplementary material for details).

#### Remark 2.1

In case of *γ*=0:

(i) the completely passing equilibrium

*E*_{0}always exists for the system irrespective of the values of*R*_{01}and*R*_{02}. If*R*_{01}>1 and*R*_{02}<*R*_{e}<1, then the failing endemic equilibrium*E*_{1}exists. If*R*_{01}>1 and*R*_{e}<*R*_{02}<1, then four equilibria ${E}_{0},{E}_{1},{E}_{1}^{\ast},{E}_{2}^{\ast}$ exist. Finally, if*R*_{01}>1 and*R*_{02}>1, then*E*_{0},*E*_{1}and ${E}_{2}^{\ast}$ exist (figure 4*a*).(ii) If

*R*_{01}<1, then*E*_{0}is locally asymptotically stable (see electronic supplementary material). If*R*_{01}>1, then*E*_{0}becomes unstable and numerical simulations suggest that stability of non-trivial equilibria (*E*_{1}, ${E}_{1}^{\ast}$ and ${E}_{2}^{\ast}$) will depend on*R*_{02}value (figure 4*b*).

#### Remark 2.2

In case of *γ*≠0, again the completely passing equilibrium *E*_{0} always exists for the system irrespective of the value of *R*_{01}=(*β*_{1}*kv*_{0})/(*μ*+*γ*)+ (*β*_{1}*γv*_{0})/*μ*(*μ*+*γ*)=*β*_{1}*k*(1/(*μ*+*γ*))*v*_{0}+*β*_{1}(*γ*/(*μ*+*γ*))(1/*μ*)*v*_{0}. If *R*_{01}<1, then *E*_{0} is locally asymptotically stable (see electronic supplementary material). If *R*_{01}>1, then one endemic equilibrium exists and numerical simulations suggest that this endemic equilibrium is locally asymptotically stable. Because we did not have data to estimate *γ* and because mathematical complexity (by taking *γ*≠0) provided immense challenge to obtaining analytical results, in this study, we ignored the interpretations of results for the case *γ*>0.

### 3. Results

#### (a) Survey interpretation

Table 5 shows the number of passing, vulnerable, and failing students for different levels of parental involvement, where high parental involvement means parents fulfil the three categories (attending parent teacher conference, attend after-school activities and check grades at least once a month). Moderate parental involvement means completing two out of the three categories. Low parental involvement means parents completing one task out of three. No parental involvement means parents fail to execute any of the three tasks. Using data from table 5, we conducted a Chi-squared test to study the relationship between parental involvement and academic achievement of students. The null hypothesis was academic achievement is independent of parental involvement. The test suggested that we do not have enough evidence to reject the null (*α*=5%, ${\chi}_{6}^{2}=13.66$, *p*=0.0336), that is, academic achievement may depend on parental involvement.

degree of parental involvement | passing | vulnerable | failing | total=n_{i} |
weight=w_{i} |
---|---|---|---|---|---|

no parental involvement | 8 | 4 | 7 | 19 | 0/3 |

low parental involvement | 50 | 7 | 14 | 71 | 1/3 |

moderate parental involvement | 21 | 2 | 1 | 24 | 2/3 |

high parental involvement | 7 | 3 | 1 | 11 | 3/3 |

total | 86 | 16 | 23 | N=125 |

Based on this survey, 48.8% of students have contact with peers who always think school is a waste of time, 27% of them have peers around them who sometimes think school is a waste of time (figure 5*a*), and 57% of dropouts do not live with their parents (figure 5*b*). The data suggest that as the degree of parental involvement increases in the vulnerable (failing) student's life, the number of their failing/dropout friends decreases (figure 6); however, this trend gets changed for failing students with increases in parental participation in their life (their dropout friends first decreases and then increases; figure 6). This may occur if students become more rebellious to a sudden increase in parental involvement beyond a certain level while being discouraged from their academic performance in education.

In summary, survey results suggest that parental involvement in a student's life is strongly related to the student's academic achievement, a large number of students are in frequent contact with individuals who think that attending school is a waste of time, and dropout rates are higher among students not living with their parents. Moreover, the type of friends of a student is positively affected by parental involvement only if the student is performing better than failing in education.

#### (b) Parameter estimates

The parameter estimates are obtained using data from our survey and literature review. Methods similar to those discussed in Romero *et al.* [30], Mubayi *et al.* [31] and Kribs-Zaleta [32] were used to derive estimates. However, there are some limitations in carrying out our estimates. Some of the parameters such as *σ*, *β*_{1}, *k* and *β*_{2} are to be estimated using temporal data on contacts. Because we have only cross-sectional data (i.e. data from one time point), we considered an ad hoc simple method to show estimation process and provide a crude estimate for these parameters.

*Estimate of σ*: in our survey samples, 17 students were freshmen, 30 were sophomores, 50 were juniors and 28 were seniors. We found that 86 students were passing all core classes, 16 were vulnerable (failing only one core class) and 23 were failing (failing two or more core classes). The parental involvement metric is a complex component and, for simplicity, we restricted the definition of parental involvement in our study using three categories: parent–teacher conference attendance, attendance at after-school activities and checking grades at least once a month. The weighted average value for parental involvement in the cohort was based on the last two columns of data collected in table 5, where weights (

*w*

_{i}) were computed using the levels (i.e. number) of activities of involvement.

*The*Hence,

*σ*estimates were obtained under the assumption that the rate of transition of students from vulnerable (*V*) to passing (*D*) state because of parental involvement is directly proportional to the degree of parental involvement in students' life.*σ*was estimated as

*N*is the sample size.

*Estimate of α*: the data collected from our sampled school included questions on number of failing core classes at two time points: at the time of the survey and five weeks (35 days) ago. Seven students reported that they were passing all core courses five weeks ago and now failing at least one course. The parameter

*α*represents the

*per capita*transition rate of students from passing to vulnerable state. We obtained the estimates of

*α*by averaging unit changes that occur from

*P*to

*V*over all students as below:

*N*=125.

*Estimate of μ and μ_{1}*: a typical student enters high school at 14 years old and graduates when around 18 years old. Therefore, the average number of years spent in high school is around four, making

*μ*=1/4 (=0.25) per year. For

*μ*

_{1}, we assumed that a typical dropout leaves the modelled population at age 24 (i.e. students leave the population at the rate ≈1/(24−18)=0.17 per year). This is because dropout students are not part of the school population of students (i.e.

*P*+

*V*+

*F*) but

*D*are in the same community.

*Estimate of β's and k*: for vulnerable students, based on the degree of parental involvement, we calculated the number of friends who are failing and the number of friends who are dropouts (table 6). Similarly for failing students, we calculated the number of failing and dropout friends for students (table 7).

*We assumed that each friend can contribute to a contact/influence. We also assume that failing and dropout contact can lead to negative social influence.*From tables 6 and 7, we can see that as parental involvement increases, the average contact with failing and dropout friends almost decreases for both vulnerable and failing students.

‘friends’ |
|||||
---|---|---|---|---|---|

vulnerable students | failing |
dropout |
|||

type of PI | count of V (C_{V}) |
count (C_{Ff}) |
average (=C_{V}/C_{Ff}) |
count (C_{Df}) |
average (=C_{V}/C_{Df}) |

no parental involvement | 4 | 13 | 3.25 | 5 | 1.25 |

low parental involvement | 7 | 18 | 2.57 | 9 | 1.29 |

moderate parental involvement | 2 | 4 | 2 | 0 | 0 |

high parental involvement | 3 | 6 | 2 | 2 | 0.67 |

total | 16 | 41 | 16 |

‘friends’ |
|||||
---|---|---|---|---|---|

failing students | failing |
dropout |
|||

type of PI | count of F (C_{F}) |
count (C_{Ff}) |
average (=C_{F}/C_{Ff}) |
count (C_{Df}) |
average (=C_{F}/C_{Df}) |

no parental involvement | 7 | 18 | 2.57 | 9 | 1.29 |

low parental involvement | 14 | 32 | 2.29 | 15 | 1.07 |

moderate parental involvement | 1 | 1 | 1 | 0 | 0 |

high parental involvement | 1 | 4 | 4 | 0 | 0 |

total | 23 | 55 | 24 |

As shown in table 6, the 16 vulnerable students have 41 failing friends and 16 dropout friends. That is, on average, every vulnerable student has 2.56 (≈41/16) failing friends and 1.0 (≈16/16) dropout friend. Table 7 shows that the 23 failing students have 55 failing friends and 24 dropout friends. On average, every failing student has 2.4 (≈55/23) failing friends and 1.05 (≈24/23) dropout friends.

There are three negative social influence parameters, namely, *β*_{1}, *β*_{2} and *β*_{1}*k*, where *β*_{1} represents the potential effective influence between a vulnerable and a dropout individual, *β*_{2} represents the potential effective influence between a failing and a dropout student and *β*_{1}*k* represents the potential effective influence between a vulnerable and a failing student. Contacts are assumed to vary over a year, so that units of these parameters are taken as per year. In order to have precise units, temporal data needs to be collected and used to estimate these parameters. Each of these three parameters is equal to the product of the average number of contacts with failing and dropout students and the probability of a successful negative social influence given a contact (table 4). Let *c*_{1} and ${c}_{1}^{\ast}$ be the average numbers of negative contacts that a vulnerable student has with a failing and dropout friend, respectively. Because of the difficulty of calculating the number of negative contacts inside and outside the school, we use the average number of failing friends as an average number of negative contacts. We use data in table 6 to estimate these parameters *c*_{1}=41/16=2.6 and ${c}_{1}^{\ast}=16/16=1.0$. However, because of limited data our definition of computing average number of contacts is a simple one and could be made more sophisticated if longitudinal data can be collected or available.

The student population in the city of Chicago faces multiple challenges such as low income, low parental involvement and high crime rate [33]. Students spend on average 8 h (480 min) per day in school considering walking a few minutes to and from school. To estimate the number of minutes a student spends on average with a failing friend, we add the passing periods (7*4=28 min), a lunch period (50 min) and few minutes before (9 min) and after school (9 min) which add to 96 min equivalent to 20% of the whole day in school. Therefore, we assume the probability of a successful negative influence between vulnerable students and failing to be about 20% or higher. Because parameter *β*_{1} is a product of average number of contacts and effectiveness of an influence given a contact, this leads to *β*_{1}=*c*_{1}**p*=2.56*0.2=0.51 per year and ${\beta}_{1}\cdot k={c}_{1}^{\ast}\ast p=1\ast 0.2=0.2$ per year.

Using the formula *β*_{1}⋅*k*=0.2, we can compute *k*=0.2. However, the parameter *k*, for our purpose, was estimated by using the following criteria. Students spend on average 8 h out of 13 h a day in a school. It is assumed that on average 13 h a day, a typical student interact with others in the similar age group. Even if students are at home, now-a-days, students are in contact with their friends non-stop through social media. However, we assume negligible social media influence, because we focus on schools that are in low socio-economic regions and students in these regions are assumed to have less social media access. Hence, the proportion of influences of a *F* (they are in school for 8 h) as compared to that of a *D* (not in school whole day) student is 0.6(=8/13). Thus, we took *k*≈0.6.

*β*_{2} represents the potential negative social influences between failing students and dropouts and is equal to the product of the average number of negative contacts with dropout for failing students and the probability of a successful impact. Let *c*_{2} be the average of number of contacts that a failing student has with a dropout. We use data in table 7 to estimate it as *c*_{2}=24/23=1.1. Students spend on average 8 h per day in a school out of a 13 h day. To estimate the number of minutes a failing student spends on average with a dropout friend, we subtract 8 h from 13 h, giving an estimate of 5 h (300 min). This 300 min is equivalent to 38.5% of the whole day. Therefore, we assume the probability of a successful negative influence between failing students and dropouts to be about 38.5% or higher. Similar to *β*_{1}, we have *β*_{2}=*c*_{2}**p*=1.1*0.385=0.42 per year.

These parameter point estimates result in *R*_{01}=1.04>1, *R*_{02}=0.05 and *R*_{e}=0.03. Because *R*_{01}>1 and *R*_{02}>*R*_{e}, we have a dropout endemic equilibrium in the system.

#### (c) Simulations

Numerical simulations were carried out to study the impact of parameters on the threshold quantities and dynamics of the system. Figure 7*a* shows that as social influences increase, *R*_{01} increases linearly though the rate of its increase decreases with increases in *σ*. That is, in the presence of low parental involvement, high efforts will be needed to reduce negative social influences within school in order to reduce the failing student population significantly. The threshold quantities *R*_{02} and *R*_{e} (persistence conditions for dropouts in the population) decrease exponentially past a critical value with decrease in influences occurring on vulnerable students (captured by *β*_{1} in the model; figure 7*b*). However, *R*_{01} (persistence conditions for failing students in the population) decreases only linearly with decreases in *β*_{1}. Figure 7*c* confirms the asymptotical behaviours of the system analytically computed in §2c; the long timescale reflects changes in the overall dropout levels in the school system across generations of students.

Figure 8 shows the impact of social influences *within the school* on the proportion of dropouts in the community. We fixed the level of the social influence occurring outside the school, *β*_{2}, and carried out simulations based on low and high levels of parental guidance for a low/high level of social influence inside the school. First, we set a high level of social influence and high level of parental guidance. As the literature indicates, the number of dropouts rises at first then stabilizes at a low level. Second, for the same high level of social influence inside the school, we set a low level of parental guidance. This led to a high proportion of dropouts. Third, we lowered the level of social influence. Regardless of parental involvement, the number of dropouts was very low. We found similar results for the *F* compartment. Regardless of whether parental involvement is high or low, as long as the level of social influence is low represented by low number of contacts between vulnerable and failing students, the number of dropouts as well as the number of failing students will be very low. Hence, there is a need to find the critical level of social influence to see its effect on the number of dropouts as well as the number of failing students.

From figure 9, we can see that in the presence of high parental involvement, as long as the level of social influence represented in our model by *β*_{1} is below 0.8, the number of failing students will be low. From figure 9, in case of low parental involvement, the critical level to insure low number of failing students is 0.4 for *β*_{1}. Regardless of the level of parental involvement, if *β*_{1} is more than 0.8, the number of failing students will rise (figure 9). We found similar results for the *D* compartment. That is, regardless of the parental involvement, increasing peer influence, within the school, beyond a critical value results in an increasing number of failing students and dropouts. The critical value of peer influence is a function of parental involvement and is smaller for higher values of parental involvement.

### 4. Discussion

Dropping out from high school has been a major problem in the USA, especially in low socio-economic communities. There are many factors that may be reasons for student dropout; however, two factors, parents' guidance and peer influences, can drastically shape one's decisions to continue or discontinue school. In this study, we study dynamics of dropouts as a function of parental and social influences occurring both within and outside of school. We developed a mathematical model that captures patterns of students and dropouts with analogy to the susceptible–exposed–infected–recovered disease model [34,35]. We studied this model analytically and then illustrated our model using estimated parameters from our survey. Studies have shown a strong correlation between dropout and socio-economic status, academic disengagement, family dynamics, school culture and numerous other factors [36–38]. Our aim here is to focus on some factors that can be managed within the school. In particular, we asked, how can we lower the number of failing and dropout students using school resources, regardless of the outside factors? Primarily, we look at the role of factors related to social influences.

Data analysis suggests that as the degree of parental involvement increases in the vulnerable student's life, the number of his/her dropout friends decreases; however, for failing students, although increases in parental participation in their life result in an initial decrease in their dropout friends, eventually the number of such friends increases. Moreover, a large number of students are contacting friends who think that attending school is a waste of time. We also found that dropout rates are higher among students not living with their parents and, as expected, academic achievement is strongly dependent on parental involvement.

We model individuals of a community in an age group similar to high school students. Our model population consists of four different compartments: passing (students passing all core subjects), vulnerable (students failing one core subject), failing (students failing two or more core subjects) and dropout (those who dropped out from school and living in the school community). These individuals can transition sequentially from one compartment to another under different social mechanisms. The long-term system behaviour is governed by three threshold quantities (*R*_{01}, *R*_{02} and *R*_{e}) defined below.

The parameter *R*_{01} represents the average number of newly generated vulnerable students influenced by a typical failing or dropout peers in a population of primarily passing students. This number is equal to the sum of contributions by individuals in the *F* and *D* classes. The quantity *R*_{02} represents the average number of newly generated failing students in a vulnerable population influenced by a typical dropout peer. This number is equal to the effective (successful negative influence) average number of social contacts of an individual, the duration of influence by a typical failing student, and the proportion of failing students. *R*_{e} is a critical value of *R*_{02} below which there are no dropouts.

Our mathematical analysis suggests the existence of four different states of the system, namely: the completely passing state (the state with all students passing all core subjects; this state depends on the proportion of students who move from passing to vulnerable, the degree of parental involvement and the time spent in school before graduation), the state with no dropouts (healthy community with all students attending school; it depends on the efficiency of influences of vulnerable and failing students, the degree of parental involvement and time spent in the community) and two dropout equilibria where dropouts persist in a community irrespective of parental guidance. The completely passing state is asymptotically stable only when the average number of newly generated vulnerable students (*R*_{01}) is less than one. The state with no dropouts exists and is locally stable when *R*_{01}>1 and *R*_{02}<*R*_{e}<1. The two dropout equilibria exist when *R*_{01}>1 and *R*_{e}<*R*_{02}, but only one seems to be locally stable whenever they exist. We estimated model parameters from data that we collected from one public school in Chicago. Our results suggest that the negative social influences within the school leading to dropout and the average degree of parental guidance in a student's life are two major factors that drive the dynamics of dropout. Furthermore, parental guidance is found to be a significant factor only under a low level of negative social influence within the school. However, if the negative social influence increases beyond a critical value, then the impact of parental guidance becomes negligible.

There can be multiple factors (such as parental involvement, socio-economic conditions, family status, etc.) for creating high rates of dropouts in a community; however, owing to the magnitude of the problem and limited resources, it is challenging to precisely tease out driving mechanisms for the current state of dropouts. The results here provide preliminary understanding of mechanisms and suggest that we can lower or limit the number of dropouts and failing students by managing negative social influences occurring within a school. If schools with high dropout rates can identify early their vulnerable and failing students and can limit their homogeneous mixing (interacting only with other V/F students rather than with students passing all subjects) while fostering parental involvement in students' activities, then they can achieve sustained reduction in the number of dropouts.

Despite the interesting results that we have obtained from this data-driven study, we recognize some limitations including use of approximate values for some parameters in the absence of precise data to estimate them. The school in this research was selected because it has high dropout rates as well as being in the region of the city that is underdeveloped and crime-affected. This is a preliminary study and an extension of this work (future research) using a different school with similar student demographics can be used to validate model results. The survey was conducted at this school as authors had such access to it. A more realistic approach will be to conduct surveys in a randomly selected group of public schools in Chicago and to estimate parameters by taking averages over all the schools and students. In this study, we assumed influences (on peers) occurring from the dropout youths are primarily negative. However, with analogy to published models of illicit drug abuse, the dropout population may also deter students from dropping out (as a result of observed consequences in their life from dropping out); hence, their influences can also have a positive impact on one's life. In this study, our main aim was to develop and present a theoretical framework where the problem of high dropout rates and a large number of failing students in certain communities can be addressed. We identified some critical trends and it is our hope that this work will inspire further work in developing a quantitative analysis related to failing and dropout epidemics in certain regions.

### Ethics

An approval for conducting the survey was obtained from the institutional review board (IRB) at Northeastern Illinois University-Chicago in 2013.

### Data accessibility

The details of the data are presented within the study. Additional materials are collected in electronic supplementary material.

### Authors' contributions

A.B. and A.M. designed the survey and collected data. All four authors contributed to thematic development of the study, model development, estimation of parameters, mathematical analysis and writing of the manuscript. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Funding

This project was partially supported by A.M.'s grants from the National Science Foundation (NSF-grant no. DMPS-0838705, and -grant no. ACI 1525012).

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