How Mathematical Models Are Saving Lives: The Role of Equations in Disease Outbreaks

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Mathematical models, particularly equation based models, are powerful tools for managing outbreaks of disease i.e. predicting disease transmission and changes related to intervention; estimating interventions in real time; and assisting public health agencies in planning(Tuite et al., 2020). Mathematical models are useful to conceptualize the transmission of disease, hypothesize paths of impact to control measures in practice, and being able to make well founded decisions in order to mitigate the harm (May, 1991). The role of mathematics cannot be neglected is certainly not trivial when employed in any field of study. More specifically, mathematical equations and expressions are not mere manipulations of variables; they are more than that, and ultimately allows us to represent real-world challenges or problems, in a mathematically simplified yet ultimately accurate and, again, reproducible and learned

Mathematical models serve as vital tools for managing disease outbreaks because they help anticipate transmission, evaluate actions, and guide public health decisions. Models such as the SIR (Susceptible, Infected, Recovered) use equations to simulate transmission dynamics to assess the impact of actions like as vaccine, quarantine, and social distance. They encourage effective resource allocation and policymaking, as evidenced by the COVID-19 (Tuite et al., 2020) and Ebola outbreaks. In addition to being abstract, these models convert challenging health issues into real, life-saving measurements.

What Are Mathematical Models in Health?

How Mathematical Models Work in Healthcare

Mathematical models are simplified representations of real-world phenomena built with equations based on sound assumptions. In public health, these models simulate disease transmission using real data or case studies to forecast future cases and evaluate interventions before they are applied in the real world.

Mathematical models in health are virtual experiments that utilize mathematics and computer simulations to show how diseases spread or how health treatments affect people. These models aid public health workers in comprehending disease dynamics, assessing the efficacy of various interventions, and forecasting future outbreaks or trends. They can help policymakers make better judgments, optimize health-care programs, and increase population health.

Mathematical models in health use mathematical equations and computing technologies to depict real-world health situations. They entail identifying variables (such as illness prevalence, infection rates, or population demographics) and the relationships between them (via equations) to simulate how a disease may spread or how treatments may affect a population. 

Mathematical Models: Types and Applications in Health

There are many types of models, but the most common in epidemiology include(Cintrón-Arias et al., 2009):

• SIR Models (Susceptible, Infected, Recovered)
• SEIR Models (adds “Exposed” to represent the incubation period)

Such mathematical models can anticipate the spread of infectious diseases, evaluate the effectiveness of vaccinations or other therapies, and guide public health responses to epidemics. These models can be used to better understand the evolution of chronic diseases, compare the efficacy of various therapies, and measure the effects of lifestyle changes. They can also be used to assess the cost-effectiveness of various healthcare interventions, improve resource allocation, and inform the formulation of public health policy (Weiss, 2013).

Mathematical models can help identify potential health issues early on, enabling for timely and successful treatment. They can also be used to assess the risk of getting specific diseases depending on a variety of circumstances, allowing individuals and healthcare practitioners to make more informed decisions. Models offer evidence-based insights that can help shape public health policies and practices.

Limitations of Mathematical Models

These mathematical models have some limitations as well. They can help optimize resource utilization by identifying the most effective and efficient treatments. Mathematical models can assist predict future health issues and prepare for probable outbreaks or emergencies.  

Models rely on assumptions and simplifications of reality, and their correctness is determined by how well these assumptions mirror the real world. Models require precise data to function, and data availability and quality might be limiting factors.

Also Read: Real life Applications of Complex Numbers

How Mathematical Models Predict and Prevent Outbreaks

When a disease begins to spread, time is critical. Governments and health organizations turn to epidemic modeling to forecast potential scenarios. Models are used to predict and prevent disease outbreaks by analyzing historical and real-time data to identify patterns and trends that signal potential outbreaks. These models, including epidemiological and machine learning models, help public health officials understand how diseases spread, enabling them to implement timely interventions and strategies to mitigate the impact of outbreaks. 

These models replicate disease propagation within a population, taking into account transmission rates, infection periods, and demography. SIR (Susceptible-Infected-Recovered) models are a simple example, but more complex models can include variables such as age structure, regional distribution, and immunity.  Epidemiological models can assist anticipate the magnitude and length of an outbreak, assess the efficacy of various measures (such as vaccination or social distancing), and identify high-risk locations. A solid public health infrastructure, including well-trained workers and comprehensive surveillance systems, is essential for efficient epidemic prevention and response. Public awareness efforts and educational initiatives emphasizing preventative habits (such as hand washing and immunization) are also critical.
Global cooperation and information exchange are critical in combating outbreaks that can quickly spread across boundaries.

Also Read: Infniity Minus Infinity: Understand the Mystery.

SIR model and COVID-19 Simulation

In the early days of COVID-19, researchers used mathematical models to estimate how contagious the virus was (basic reproduction number R₀), project the number of hospital beds needed, and evaluate the impact of lockdowns and masks(Kotola et al., 2023). The SIR model(Smith & Moore, 2004) is a fundamental mathematical model in epidemiology. It divides a population into three compartments:

•  Number of Susceptible individuals
•  Number of Infected individuals
•  Number of Recovered individuals

The model assumes a fixed population with no births or deaths and perfect immunity after recovery. It uses the following differential equations:

Source: The assumption of model

$$ \frac{dS}{dt} = -\beta SI $$
$$ \frac{dI}{dt} = \beta SI – \gamma I $$
$$ \frac{dR}{dt} = \gamma I $$

Where:

• β (beta) is the transmission rate
• γ (gamma) is the recovery rate
• The basic reproduction number, R₀ = β / γ(calculated by next generation matrix), indicates whether an infection will spread (R₀ > 1) or die out (R₀ < 1).

The following simulation represents the spread of a COVID-19-like infection in a population of 1,000,000 people, with an initial infected count of 1 and recovery rate γ = 0.1 (average recovery in 10 days). The infection rate is β = 0.3. The graph shows the dynamics of Susceptible, Infected, and Recovered individuals over time.

Source: Model simulated using MATLAB Software

Source: Model simulated using MATLAB Software

Conclusion: Mathematical Models in Health and Science

Mathematical models translate data into action. They have helped foresee, prepare for, and avoid millions of fatalities by influencing public health decisions during disease epidemics. While not perfect, their contribution to world health is undeniable. Mathematical is less time-consuming when compared to experiments, research, and studies. It also considers ethical, such as the spread of HIV/ADIS through sexual transmission. The model requires simulation rather than experiments on humans. As technology and data collecting improve, future models will become even more precise, preparing us to meet the next health catastrophe with information rather than panic.

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FAQs

Q1: What is the SIR mathematical model used for?

Answer: SIR mathematical model is generally used to simulate how a disease spreads through a population over time.

Q2: Can math really stop a pandemic?

Answer: Math or mathematical models alone cannot stop a pandemic situation, but they play a key role in planning and response.

Q3: How accurate are disease models?

Answer: Accuracy of disease models depends on data quality, but even rough predictions guide vital decisions.

References:

1. Cintrón-Arias, A., Castillo-Chávez, C., Bettencourt, L. M. A., Lloyd, A. L., & Banks, H. T. (2009). The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences & Engineering, 6(2), 261–282. https://doi.org/10.3934/mbe.2009.6.261
2. Kotola, B. S., Teklu, S. W., & Abebaw, Y. F. (2023). Bifurcation and optimal control analysis of HIV/AIDS and COVID-19 co-infection model with numerical simulation. In PLoS ONE (Vol. 18, Issue 5 MAY). https://doi.org/10.1371/journal.pone.0284759
3. Tuite, A. R., Fisman, D. N., & Greer, A. L. (2020). Mathematical modelling of COVID-19 transmission and mitigation strategies in the population of Ontario, Canada. Canadian Medical Association Journal, 192(19), E497–E505. https://doi.org/10.1503/cmaj.200476