3.1.3 Mathematical modelling

• Mathematical modelling is the process of creating a mathematical representation of a real-world problem or system
• What is the purpose of mathematical modelling?

  Represent and analyze systems
• The key in mathematical modelling is to identify the relevant variables, relationships, and constraints
• Match the type of mathematical model with its description:

  Deterministic Models ↔️ Assume no uncertainty or randomness

  Stochastic Models ↔️ Incorporate randomness or probability

  Empirical Models ↔️ Based on observed data

  Mechanistic Models ↔️ Derived from fundamental principles
• What assumption do deterministic models make about uncertainty?

  No uncertainty
• Mechanistic models are derived from fundamental physical, chemical, or biological principles.


  True
• Empirical models are based on observed data and statistical analysis rather than first principles.

  True
• Mathematical modelling involves translating real-world relationships into mathematical expressions.


  True
• In science and engineering, mathematical models are used for designing efficient transportation systems.
• What is a deterministic model?

  No randomness is assumed
• What is the primary use of deterministic models?

  Making precise predictions
• When modeling bacterial growth, assuming constant growth rate simplifies the model.


  True
• What is the limitation of analytical solutions for solving mathematical models?

  Limited to simple models
• Match the validation method with its key aspect:

  Comparison to Real-World Data ↔️ Confirms accuracy and identifies discrepancies

  Sensitivity Analysis ↔️ Evaluates the impact of input variable changes

  Reviewing Model Assumptions ↔️ Ensures assumptions are reasonable
• Mathematical modelling is a powerful tool for solving complex real-world problems.


  True
• Match the problem domain with an example of a problem that can be modeled mathematically:

  Economics and Finance ↔️ Predicting stock market trends

  Science and Engineering ↔️ Modeling the spread of diseases

  Social Sciences ↔️ Analyzing voting patterns

  Environmental Studies ↔️ Predicting natural disasters
• Steps involved in mathematical modelling

  1️⃣ Identify key variables and relationships

  2️⃣ Translate into mathematical expressions

  3️⃣ Solve the mathematical model

  4️⃣ Evaluate and refine the model
• Empirical models are based on observed data and statistical analysis
• Stochastic models incorporate randomness or probability distributions to account for uncertainty
• Match the model type with its description:

  Deterministic ↔️ Output uniquely determined by input variables

  Stochastic ↔️ Incorporates randomness

  Empirical ↔️ Based on observed data

  Mechanistic ↔️ Derived from fundamental principles
• What is the key variable in the bacterial population growth model?

  $N(t)$
• What is an example of a real-world problem in economics that can be modeled mathematically?

  Predicting stock market trends
• Empirical models are based on observed data and statistical analysis
• Mechanistic models are derived from fundamental physical, chemical, or biological principles
• Steps to develop a mathematical model based on assumptions:

  1️⃣ Define the problem

  2️⃣ State the assumptions

  3️⃣ Formulate the mathematical equations

  4️⃣ Solve the equations

  5️⃣ Validate the model
• Validating a model involves comparing its predictions to real-world data
• Validating a model includes comparing its predictions to real-world data
• By creating mathematical models, we can make informed decisions
• Mathematical modelling helps us understand and solve complex real-world problems.

  True
• The choice of model type depends on the specific problem, the available data, and the level of understanding
• Stochastic models use probability distributions to account for uncertainty in the system.


  True
• What type of mathematical model assumes no uncertainty or randomness?

  Deterministic
• Which type of mathematical model is derived from fundamental principles?

  Mechanistic
• Steps to develop a mathematical model based on assumptions:

  1️⃣ Define the problem and key variables

  2️⃣ State the assumptions to simplify the model

  3️⃣ Formulate the mathematical equations

  4️⃣ Solve the equations

  5️⃣ Validate the model with real-world data
• Match the aspect with its corresponding field:

  Purpose of mathematical modelling ↔️ Analyze real-world systems

  Purpose of pure mathematics ↔️ Develop abstract theories

  Outcome of mathematical modelling ↔️ Practical insights

  Outcome of pure mathematics ↔️ Theoretical knowledge
• Mathematical models can help test hypotheses in a controlled environment.

  True
• Stochastic models incorporate randomness to account for uncertainty.


  True
• Match the model type with its description:

  Deterministic Models ↔️ Output uniquely determined by input variables

  Stochastic Models ↔️ Incorporate randomness or probability distributions

  Empirical Models ↔️ Based on observed data and statistical analysis

  Mechanistic Models ↔️ Derived from fundamental principles
• What is the solution to the equation \frac{dN}{dt} = k \cdot N(t)</latex>?

  $N(t) =$$N_{0} \cdot e^{kt}$
• Numerical methods provide exact solutions for mathematical models.

  False