Maternal Immunity

Overview

This app will teach you about the concept of passive immunity through the utilization of an MSEIR type model. It will also show you how variable population size can affect disease dynamics. You can read more in-depth information about the model in the “Model tab”. Tasks that will help you understand the model and concept are present in the “What to do” tab".

The Model

Model Overview

For this app, we will use the MSEIR type compartmental model. This model covers the concept of passive immunity, where a number of newborns receive maternal antibodies from their mother, which protects them from a disease for a short amount of time. We allow for 5 different compartments:

In this model, we will allow for the population size to be dynamic. Because of that, N will represent the total population size at any given time. The dynamic model for N is shown below.

The following processes define the dynamics of the model:

Model Diagram

The following shows the general flow diagram for the presented MSEIR model:

Flow diagram for this model.

Flow diagram for this model.

Model Equations

The following set of differential equations dictates the relationship described above:

\[\dot M = mR- (p + n)M\] \[\dot S = mS + p M - b_E S I / N - nS + w R\] \[\dot E = b_E S I / N - (b_I + n)E\] \[\dot I = b_I E - (g + n)I\] \[\dot R = g I - (w + n)R\] \[N = M + S + E + I + R\] Here, N is dependent on \(m\) and \(n\). This means that population size is subject to change over time depending on the values of these parameters. The population is growing if \(m > n\), decaying if \(m < n\), and constant if \(m = n\). It is also important to note that transmission rate is scaled with the population size. This is a different feature than in other apps in this package.

What to do

In most traditional types of models (and models in this package), total population size remains constant. However, this is not always the case when the natural birth and death rates are not balanced. This also occurs when disease-related death is high. Some examples of diseases like this are the plague, measles, smallpox, malaria, and tuberculosis. This app will show the dynamic of a MSEIR model with examples shown of constant population size and dynamic population size. It is important to consider the fact that when considering an exponentially increasing/decreasing population size, parameters need to be carefully selected so that the model retains biological relevance.

The tasks below are described in a way that assumes everything is in units of days (rate parameters, therefore, have units of inverse days). If any quantity is not given in those units, you need to convert it first (e.g. if it says a week, you need to convert it to 7 days).

Task 1

First we will observe the dynamics of the MSEIR model with a constant population size.

Now run the simulation. Make sure that an outbreak occurs. What can you observe from the results? Make sure to take note of the relationship between the M population and the S population. Also make note of the final number of infecteds.

Task 2

Run the simulation with the same parameters as in Task 1, except this time run it for 1200 days to observe the long term dynamics at place. What do you notice? Does the population reach steady state? What does this say about the dynamics at place?

Task 3

Sometimes it is not appropriate to assume that a population will be constant over a period of time. Here, we will observe what kind of effect an exponentially increasing population would have on the dynamics of the MSEIR model.

Compare the effects of an increasing population to what you saw in Task 1. How does the increasing population affect the M population and the S population. Does it appear that this is an endemic disease?

Task 4

Now we will compare the effects of a decreasing population on disease dynamics with an MSEIR model.

How do these results compare to those in Task 3. What compartments are most affected by the decreasing population?

Task 5

Now we are going to observe what disease dynamics are like with a short latent period and a short recovery period. We will observe the long term dynamics of such a disease by doing the following:

Task 6

Set the simulation to run with the same parameters used in Task 5, including the \(b_E\) values that was used. Adjust the value for the waning immunity parameter, w. What do you observe? What compartments are especially affected by increasing this parameter to > 0? Hint: you will likely need to decrease the tmax parameter.

Further Information

References

Hethcote, Herbert W. 2000. “The Mathematics of Infectious Diseases.” SIAM Review 42 (4): 599–653.