How to Translate Real-World Problems into Precise Mathematical Models

The core challenge of mathematical modeling is not solving equations but translating reality into mathematical language. When faced with a real problem, we must "translate" the complex situation into a concise, precise mathematical form using a systematic "reality‑to‑math dictionary"—a multi‑level, multi‑dimensional mapping system that helps us move freely between the two worlds.

Below are some common conversions.

Element Level: From Phenomena to Symbols

1.1 State Variables and Variables

Observable quantities in reality are the basic bricks of mathematical modeling. Identifying and defining variables is the first step.

Conversion rule:

Time variable → Function , e.g., population over time, object position over time.

Spatial distribution → Field function , e.g., temperature distribution, population density.

Decision choice → Decision variable , e.g., production plan output, investment portfolio allocation.

1.2 Inherent Properties and Parameters

Characteristics and constants of a real system become model parameters.

Conversion rule:

Physical constant → Parameter , e.g., spring stiffness, gravitational acceleration.

Individual differences → Parameter vector , e.g., different product prices.

Initial condition → Boundary value , e.g., initial investment, initial population.

1.3 Constraints and Feasible Domain

Feasibility boundaries in reality define the range of variable values.

Conversion rule:

Physical limitation → Inequality constraint , e.g., non‑negative speed, capacity limits.

Logical relationship → Equality constraint , e.g., budget balance.

Discrete choice → Integer constraint , e.g., number of people must be an integer.

Structure Level: From Relationships to Equations

2.1 Causal Relationships and Function Mapping

Real‑world influence mechanisms become mathematical function relationships.

Conversion rule:

Proportional relationship → Linear function , e.g., demand vs. price (law of demand).

Growth/decay → Exponential function , e.g., radioactive decay, compound interest.

Saturation effect → Logistic function , e.g., population growth with limited resources.

2.2 Equilibrium and Systems of Equations

The stable state of a real system corresponds to the solution of equations.

Conversion rule:

Mass conservation → Continuity equation , e.g., fluid mass conservation.

Force balance → Newton's equation , e.g., static equilibrium.

Market equilibrium → Supply‑demand system , e.g., equilibrium of supply and demand.

2.3 Optimization Objectives and Functionals

Optimal decisions in reality become extremum problems.

Conversion rule:

Benefit maximization → Maximize objective , e.g., profit maximization.

Cost minimization → Minimize objective , e.g., transportation cost minimization.

Multi‑objective trade‑off → Weighted sum or Pareto optimization .

2.4 Network Structure and Graph Theory

Connections in reality are described using graph‑theoretic language.

Conversion rule:

Entity set → Vertex set

Relationship connections → Edge set

Connection strength → Weight matrix

Examples: social network → graph (users as vertices, friendships as edges); transportation network → weighted directed graph (road lengths or travel times as weights).

Reasoning Level: From Logic to Computation

3.1 Causal Reasoning and Differential Equations

Rates of change in reality are translated into differential equations.

Conversion rule:

Instantaneous rate → Derivative , e.g., speed is the rate of change of displacement.

Rate dependent on state → Differential equation , e.g., cooling law.

Multivariable interaction → System of equations , e.g., predator‑prey (Lotka‑Volterra) model.

3.2 Uncertainty and Probabilistic Models

Random phenomena are described with probability language.

Conversion rule:

Random event → Random variable , e.g., coin toss outcome.

Statistical regularity → Probability distribution , e.g., measurement error, waiting time.

Expected return → Mathematical expectation , e.g., investment return.

3.3 Evolutionary Processes and Dynamic Systems

Temporal development is modeled with dynamic systems.

Conversion rule:

Discrete‑time evolution → Difference equation , e.g., yearly population update.

Continuous evolution → Dynamical system

State transition → Markov chain , e.g., weather changes represented by a transition matrix.

3.4 Conditional Logic and Piecewise Functions

Context‑dependent situations are expressed with conditional expressions.

Conversion rule:

Tiered pricing → Piecewise function

Switch control → Indicator function

Comprehensive Application: Epidemic Spread Model

We demonstrate the dictionary with a complete example—the classic SIR model.

Real description: A disease spreads among a population divided into susceptible (S), infected (I), and recovered (R). Infected individuals transmit the disease to susceptibles and eventually recover.

Mathematical translation:

1. Element identification: State variables are the three population counts; parameters are transmission rate and recovery rate; constraint is total population conservation.

2. Structure building: Transmission mechanism: infection rate proportional to the product of S and I; recovery mechanism: recovery rate proportional to I.

3. Reasoning formalization: This yields the standard SIR differential equations.

Strategy Recommendations for Using the Dictionary

1. Top‑down identification: Clarify the system’s macro goals and overall structure before detailing individual elements.

2. Analogy transfer: Adapt mature models from similar problems to new contexts.

3. Layered abstraction: Build multi‑scale models, refining from simple to complex.

4. Dimensional analysis: Use dimensional homogeneity to check equation plausibility.

5. Parameter estimation: Calibrate model parameters from real data and validate model effectiveness.

The "reality‑to‑math dictionary" is not a rigid rule set but a way of thinking. Mastering this conversion framework enables rapid development of mathematical intuition for new problems, allowing the power of mathematics to serve real‑world problem solving.