In this session, we will discuss some basic concepts in infectious disease modeling. A model is simply our understanding of how things work in the real world. In our everyday lives, we rely on models to make decisions. For example, when we try to decide what kind of transportation to take to go from one place to another, we inevitably have some model in mind about the time and cost associated with each available option. The model could be based on our own personal experience or some kind of knowledge base such as Google Map. In essence, we develop models for understanding and predicting the behavior of systems that we’re interested in. To this end, models are often approximations or simplifications of real-world systems because, as the eminent evolutionary biologist and geneticist John Maynard Smith said, “Describing complex, poorly-understood reality with a complex, poorly-understood model is not progress.” For example, in drug development, animal models are often used to learn about human biological pathways, though it is obvious that observations made in animal models are not generalizable always to humans. Mathematical models use a bottom-up approach to integrate assumptions about the underlying processes and empirical data collected from the real world to describe the behavior of the system. The most well-known examples of mathematical models are those we learned in high school physics. For example, the mathematical model based on Newton’s law of motion and empirical measurements of mass and gravity has been a cornerstone for understanding and predicting the movement of objects on the planet. Another familiar but much more complex example is weather forecast, which serves as a good learning model for what infectious disease modeling is trying to do: To develop robust mathematical models that can explain and predict the behavior of systems in nature. Let us now look at a simple example of epidemic models known as the SIR model. Consider a hypothetical epidemic scenario where (i) a novel pathogen invades a population that has no immunity against it; (ii) infected individuals gain permanent immunity against reinfection after recovery; and (iii), the timescale of the epidemic is much faster than changes in population demographics so that we can ignore the effect of births and immigration. The SIR model makes the following simplifying assumptions regarding disease transmission: First, at any given time t, each person belongs to one of three groups: susceptible, infected, or removed. The susceptible group corresponds to those who are susceptible, and have not yet been infected. The infected group corresponds to those who are currently infected and spreading the disease. And the removed group corresponds to those who have either died or recovered from the disease. Let’s call the number of people in each group at time t S(t), I(t), and R(t). Note that the disease prevalence at time t is simply I(t). The second assumption is that the rate of infection for a susceptible person at any given time t is proportional to the disease prevalence. Let’s call the proportional constant beta, which can be interpreted as the rate of making contacts that conduce infection if the contact is made between a susceptible and an infected person. We will learn more about contact patterns from Professor Marc Lipsitch later in this course. The rate of infection per susceptible has a special name - force of infection. The third assumption is that infected people are infectious for D days on average after which they would recover or die and can no longer infect other people. This is the same as saying that infected people recover at rate 1/D. Let us now look at what the epidemic dynamics look like under these assumptions. Consider an arbitrary time point t and a small time-step delta_t. The time-step should be small compared to the average infectious duration. For example, if the average infectious duration is three days, delta_t should be smaller than half a day or so. In the SIR model, the number of susceptibles at time t plus delta_t is simply the number of susceptibles at time t, minus the number of infections that occur between time t and t plus delta_t. Under assumption two, the number of infections between time t and t plus delta_t is equal to S(t) which is the number of people subject to the risk of infection, times beta I(t), which is the force of infection, times delta_t, which is the length of time that we are considering. Similarly, the number of infected people at time t plus delta_t is equal to the number of infected people at time t plus the number of infections that occur between time t and t plus delta_t, minus the number of recoveries that occur during the same time interval. The number of recoveries is simply I(t), the number of infected people that may recover, times 1/D, which is the rate of recovery, times delta_t. These two simple equations together specify the epidemic dynamics in the SIR model. Starting with the initial conditions at time zero, for example 100 infected people seeded into the population of 1,000,000, we can use these equations iteratively to simulate the epidemic. The resulting epi-curve has the hallmarks of a typical acute epidemic. The epidemic grows exponentially at the beginning, then slows down and peaks, and finally fades out. The most important feature of dynamical epidemic models is that they explicitly take into account the effect of herd immunity in disease transmission. Herd immunity is one of the fundamental lessons in infectious disease epidemiology. It refers to the phenomenon that immunity of an individual not only protects the individual himself from infection, but also reduces the risk of infection for everyone in the population. Let us illustrate the mechanism of herd immunity using the following example. Suppose the transmission chain of the epidemic looks like this after the primary case has been seeded into the population. Now let’s look at how the transmission chain will change if some of the people are immune before the primary case is seeded, for example, with a vaccine that provides 100 percent protection from infection. In addition to the vaccinated individuals, three unvaccinated individuals who would have been infected in the original transmission chain are now indirectly protected from infection. Therefore, they have been protected by herd immunity. A corollary of herd immunity is that not everyone in the population has to be immunized in order to prevent an infectious disease from taking foothold and becoming widespread in the population. Let us illustrate this principle with the SIR model. Suppose we vaccinate a proportion p of the population with a vaccine that provides 100 percent protection from infection. This proportion p is called vaccine coverage. Vaccination reduces the susceptible proportion from 1 to 1 minus p, which means that vaccination reduces the reproductive number from R0 to (1-p) times R0. Because an epidemic cannot take off when the reproductive number is smaller than one, we can prevent an epidemic as long as the vaccine coverage is above 1-1/R0 This minimal coverage required to prevent an epidemic is called the critical vaccination coverage. For example, if R0 is 2, we can in theory prevent an epidemic if we vaccinate at least 50 percent of the population. To summarize, in this session, we have discussed some basic concepts in infectious disease modeling.