Virus spread math

The second set of dependent variables represents the fraction of the total population in each of the three categories. So, if N is the total population 7,, in our example , we have. It may seem more natural to work with population counts, but some of our calculations will be simpler if we use the fractions instead. The two sets of dependent variables are proportional to each other, so either set will give us the same information about the progress of the epidemic.

Next we make some assumptions about the rates of change of our dependent variables:. No one is added to the susceptible group, since we are ignoring births and immigration. The only way an individual leaves the susceptible group is by becoming infected.

We assume that the time-rate of change of S t , the number of susceptibles, 1 depends on the number already susceptible, the number of individuals already infected, and the amount of contact between susceptibles and infecteds. In particular, suppose that each infected individual has a fixed number b of contacts per day that are sufficient to spread the disease.

Not all these contacts are with susceptible individuals. If we assume a homogeneous mixing of the population, the fraction of these contacts that are with susceptibles is s t. Thus, on average, each infected individual generates b s t new infected individuals per day. Let's see what these assumptions tell us about derivatives of our dependent variables.

Finally, we complete our model by giving each differential equation an initial condition. For this particular virus -- Hong Kong flu in New York City in the late 's -- hardly anyone was immune at the beginning of the epidemic, so almost everyone was susceptible. We will assume that there was a trace level of infection in the population, say, 10 people. In terms of the scaled variables, these initial conditions are. Note: The sum of our starting populations is not exactly N , nor is the sum of our fractions exactly 1.

The trace level of infection is so small that this won't make any difference. Our complete model is. We don't know values for the parameters b and k yet, but we can estimate them, and then adjust them as necessary to fit the excess death data. You see the problem, right?

For 30 days the risk to others seems small, and nobody follows the CDC advice to stay home. Then suddenly, with no change in the infection rate, it explodes. By the way, that graph is generated by a simple Python script, and you can change the numbers to see what happens. Click the pencil icon to edit, and hit the Play button to rerun. So this is an exponential function. In fact, if you take the rate equation above and shrink the time interval to an infinitesimally small value i.

Solving that equation gives the following:. This says that the number of infected people N depends on the starting number N 0 and e the natural number raised to the product of a and t. That's why it's called exponential growth—the driving variable, time, is in an exponent.

In our simple model, things just get worse and worse forever. But that results from two implicit assumptions: first, that the infection rate stays constant, and second, that no one recovers and ceases to be contagious.

Fortunately neither is true, or everyone in the world would be sick in very short order. Still, this model is pretty accurate for the early stages of an epidemic. But here's the important part. What if you could reduce the infection rate by just a little bit? What if the infection rate is 0. Here's a comparison over 45 days:. That's a difference of 2, people by day With exponential growth, every little bit helps.

The moral here is that individual efforts—especially early on, when it doesn't seem to matter—really, really do matter. You, all by yourself, can be a superhero and save lives. Yes, by washing your hands and practicing safe social distancing. But what about real data? Does the number of infected actually follow an exponential function? What is the real infection rate factor? You can get all sorts of data online—I'm using coronavirus numbers from Our World in Data. Here's what that looks like:.

On the other hand, a pandemic is a global epidemic that crosses international boundaries. The Antonine Plague This pandemic is believed to have been smallpox or measles that was brought to Europe by soldiers returning from the Near East. It may have killed as many as 5 million, and during a second outbreak in the middle of the third century, it was rumored that people a day were dying in Rome. Cholera The first outbreak of cholera occurred in India in , and it became a pandemic by spreading from Bengal, across India, and to China and the Caspian Sea.

The second pandemic of cholera affected Europe and North America in the late 's, and since then, there have been five other cholera pandemics. The seventh and most recent outbreak began in Indonesia and reached Bangladesh, India, and the Soviet Union. The first complaint was registered by the cook at the Fort Riley military base on March 11, , followed by additional patients before noon that same day. In the camp of 26, men, were infected, 46 fatally. Men who carried the disease, believed now to have been transmitted from birds to humans, left Fort Riley to fight in World War I.

They spread the flu at each stop, which led to a nearly-worldwide pandemic. Although estimates vary widely, more than , Americans died from the Spanish Flu; and worldwide, the death toll was between 10 and million.

Asian Flu Pandemic Unlike the virus that caused the pandemic, the Asian Flu pandemic virus was quickly identified, due to scientific advances. Vaccine production began shortly after the outbreak occurred in China, and a small supply of vaccines was available when the flu reached the U.

Still, the virus spread quickly when school started in the fall, and nearly 70, Americans were killed; worldwide, the total reached 2 million. During the pandemic, a World Health Organization panel found that spread often followed public gatherings. They also observed that in many countries the pandemic broke out first in camps, army units and schools, suggesting that avoiding crowds may be important to controlling the spread.

Even as early as , officials knew that contact with infected persons caused the virus to spread. Consequently, they prevented large crowds from gathering in theaters, churches, and schools. The public notice to the left shows one town's attempt to limit public gatherings.

Once an influenza pandemic begins, it is often unstoppable, because the virus spreads rapidly by coughing and sneezing. In fact, infected people can pass the virus to others before symptoms even appear. Preventing people from coming in contact with one another is one way to prevent the spread of influenza. Providing vaccines to reduce the likelihood of contracting the disease, or using an antidote to treat the symptoms once they appear, are other means of control. The curve used to model this situation is called a logistic curve, an S-shaped curve that describes population growth-of both viruses and people-as well as other phenomena in economics and science.

Interesting Facts about the Logistic Curve The point of inflection is the point where the curve changes from increasing faster to increasing slower. It also marks some symmetries for the curve.