Calculus is the branch of math that studies how things change, grow, or shrink. It may sound hard and boring, but calculus has many useful applications in the real world, especially in medicine. In this blog post, we will explore some of the ways that calculus can help us understand and improve human health.
Calculus and Epidemiology
Epidemiology is the study of how diseases spread and affect people. It helps us prevent and control outbreaks, identify risk factors, and evaluate treatments. Calculus is a powerful tool for epidemiology because it allows us to make models of how diseases behave and how different actions can affect them.
One example of a calculus-based model for epidemiology is the SEIR model, which divides the people into four groups: susceptible (S), exposed (E), infected (I), and recovered ®. The model uses differential equations to describe how people move from one group to another over time, depending on how contagious the disease is, how long it takes to show symptoms, how long it takes to recover, and other factors.
Differential equations are equations that involve how things change over time or space, such as population growth, heat transfer, motion, etc. Differential equations can be classified into different types based on their complexity and structure.
The SEIR model can help us estimate important numbers such as the basic reproduction number (R0), which is the average number of new infections caused by one infected person in a group of people who can catch the disease. If R0 is greater than 1, the disease will spread quickly; if R0 is less than 1, the disease will die out eventually. The SEIR model can also help us find the herd immunity threshold (HIT), which is the fraction of the people that need to be immune (either by vaccination or natural infection) to stop an epidemic.
The CDC uses a calculus-based SEIR model to study the spread of COVID-19 and the impact of vaccination and social distancing measures. The model can help us predict the future course of the pandemic and evaluate different scenarios and policies.
[Insert an image of the SEIR model or a graph of COVID-19 projections]
According to [1], the SEIR model for COVID-19 can be expressed by the following system of differential equations:
dS/dt = -βSI/N dE/dt = βSI/N – σE dI/dt = σE – γI dR/dt = γI
where S is the number of susceptible people, E is the number of exposed people, I is the number of infected people, R is the number of recovered people, N is the total number of people, β is how contagious the disease is per contact, σ is how fast exposed people become infectious, and γ is how fast infected people recover.
By solving these equations with a computer with appropriate starting values and parameters, we can simulate how COVID-19 spreads in a group of people over time and how different actions such as vaccination or social distancing can affect its behavior.
[1] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7277729/
Calculus and Pharmacology
Pharmacology is the study of how drugs work in the body. It helps us design and test new drugs, determine optimal doses and frequencies, monitor drug interactions and side effects, and improve drug delivery systems. Calculus is essential for pharmacology because it allows us to measure the processes of drug absorption, distribution, metabolism, and excretion.
One example of a calculus-based concept for pharmacology is the half-life of a drug, which is the time it takes for half of the drug to leave the body. The half-life depends on various factors such as how the drug is taken, how much blood it reaches, how fast it is cleared from the body, and how fast it breaks down in the body. Knowing the half-life of a drug can help us calculate how much drug stays in the body at any given time after taking a dose.
One example of a drug that uses calculus is aspirin, which has a half-life of about 4 hours in the body. This means that after 4 hours, half of the drug has left the body. We can use integral calculus to find how much aspirin stays in the body at any given time after taking a dose. For example, if we take 500 mg of aspirin at time t = 0, then we can use this formula to find how much aspirin is left at time t:
A(t) = 500e^(-0.173t)
where A(t) is the amount of aspirin in mg and t is the time in hours.
Integral calculus is the branch of calculus that deals with finding areas under curves or volumes under surfaces. It can also be used to find other quantities such as work done by a force, center of mass of an object, length of a curve, etc. Integral calculus is based on two main concepts: indefinite integrals and definite integrals. Indefinite integrals are functions that represent all possible antiderivatives plus a constant term. Definite integrals are numbers that represent areas under curves between two limits.
[Insert a graph of A(t) vs t]
According to [2], the half-life of aspirin can be derived by using the following formula:
t1/2 = ln(2)/k
where t1/2 is the half-life and k is how fast the drug breaks down in the body.
By using experimental data, we can estimate the value of k and then calculate the half-life.
[2] https://www.sciencedirect.com/topics/pharmacology-toxicology-and-pharmaceutical-science/aspirin
Calculus and Physiology
Physiology is the study of how organs and systems function in the body. It helps us understand how our bodies work normally and respond to changes or challenges. Calculus is useful for physiology because it allows us to measure and analyze various physiological variables such as blood flow, blood pressure, heart rate, oxygen consumption, glucose level, hormone secretion, etc.
One example of a calculus-based application for physiology is cardiac output, which is the amount of blood pumped by each side of the heart per minute. Cardiac output depends on two factors: stroke volume (SV), which is the amount of blood pumped by each side of the heart per beat; and heart rate (HR), which is the number of beats per minute. We can use differential calculus to find cardiac output by multiplying stroke volume by heart rate:
CO = SV × HR
Differential calculus is the branch of calculus that deals with finding how things change over time or space, such as slopes of curves, maxima and minima of functions, rates of growth or decay, etc. Differential calculus is based on two main concepts: limits and derivatives. Limits are values that functions get closer and closer to as their inputs get closer and closer to a certain point. Derivatives are functions that measure how fast a function changes at a given point.
Cardiac output can vary depending on various factors such as physical activity, stress level, body temperature, etc. Knowing cardiac output can help us assess heart function and diagnose heart diseases.
According to [3], the stroke volume can be calculated by using the following formula:
SV = EDV – ESV
where EDV is the amount of blood in the side of the heart at the end of relaxation and ESV is the amount of blood in the side of the heart at the end of contraction.
By using ultrasound or other imaging techniques, we can measure EDV and ESV and then calculate SV.
[3] https://www.cvphysiology.com/Cardiac%20Function/CF002
Another example of a calculus-based application for physiology is nerve impulse propagation, which is the process by which electrical signals travel along nerve cells or neurons. Nerve impulses are essential for communication between different parts of the nervous system and between the nervous system and other organs. Calculus can help us model how nerve impulses are generated and transmitted by using differential equations.
Nerve impulses are based on changes in electrical charge across the cell membrane. The electrical charge depends on the distribution and movement of ions such as sodium (Na+), potassium (K+), chloride (Cl-), and calcium (Ca2+) across the membrane through specialized channels. The channels can be classified into two types: passive channels, which are always open and allow ions to move according to their concentration differences; and active channels, which are gated and open or close in response to stimuli such as voltage, chemicals, or forces.
The resting electrical charge (RMP) is the electrical charge when a neuron is not stimulated. It is usually around -70 mV, meaning that the inside of the cell is more negative than the outside. The RMP is maintained by passive channels and a sodium-potassium pump that actively moves Na+ out and K+ in against their concentration differences.
An action potential (AP) is a rapid change in electrical charge that occurs when a neuron is stimulated above a threshold level. It consists of four phases: depolarization,
