Refresh MathBio 101 with Zika

By now, you must have got introduced to Zika. May have also heard the heart-wrenching stories of children born with small head. They call it Microcephaly. It is probably connected to Zika virus infection of pregnant mothers.  The epidemic of Zika virus is causing havoc in some places in south america. WHO has recently declared it as a Public Health Emergency of International Concern (PHEIC). 

Doctors, scientists, public health workers, across the globe, are working hard to contain the disease and to develop vaccine and drugs against it. The conspiracy theorists are also working hard. Am sure you have read articles, flooding the social media, connecting Zika epidemic with greedy pharma companies. Conspiracy or not, you must have thought that how come all of a sudden this virus is causing such a havoc. It seems, as if it appeared from thin air and is spreading like an avalanche. 

But there is nothing unusual about it. Epidemics spreads like that and mathematical models of epidemics explain such avalanche. Let us check one of the simple mathematical models of epidemic to understand the Zika epidemic. This model is taught in the introductory course in Mathematical Biology. Let's refresh MathBio 101.

                                                        Image source: BBC

An infectious disease spreads through contact between an infected and an uninfected person. In some cases, the contact may not be direct. For Zika, the contact is through  mosquito. Whatever be the mode of transfer, to spread the diseases there must be some infected people in the community. Size of that infected population may be very small; but epidemic can not start from zero. 

Zika was reported, first time, in a paper published in 1952. It was the first report of isolation of this virus from rhesus monkey caged in the canopy of Zika Forest of Uganda.There were subsequent sporadic reports of  human infection by this virus across the globe. In 2015, there were reports of Zika infection in Brazil, the center of current crisis. So, there was already an infected population with potential to spread it to susceptible people. 

Let us call the fraction of the population with infection as I. The fraction of susceptible people who are still not infected be S.  The disease spreads from I to S and size of the infected population (I) increases. With time, I can decrease, as some of the infected people recover, develop immunity or (sadly) die. Let us name that fraction of population that have recovered or died as R.  

As the infection spreads, with time, sizes of these three populations change. We can write three Ordinary Differential Equations (ODE) to capture this population dynamics. 

Remember, here, S + I + R = 1 and people who have recovered (R) do not get infected again.

This is called SIR model of epidemics. It is a generic model. We have not considered any particular mechanisms of spread of infection or any particular means of disease remission. It relies on the simple idea that infection spreads by interaction between susceptible and infected people and some people either die or recovers. 

We can simulate these model by numerical integration of these three ODEs. For that we require numerical values for the constant terms a and b. In this model, a, grossly, represents how frequently one susceptible person gets infected when he/she comes in contact with an infected one. For Zika, this would depend upon mosquito, their numbers, their behavior and also on the behavior of the virus. 

The other constant b, represent how frequently infected people either die or recovers. Again this will depend upon health of individual, condition of their immune system, existence of healthcare facilities and also on the virus. 

For simulation, let us take a = 0.2 and b = 0.05. We also have to specify, values of S, I and R at the beginning (t = 0). Say those are, S = 0.999; I = 0.001 (very few people are infected) and  R = 0. 

We have simulated the SIR model with these values. The results are shown here in the figure. 

Initially, most of the people are uninfected. With time, number of infected people increases exponentially. Some of the infected people either die or recover. Therefore, size of the infected population, I, can not increase forever. It reaches a peak and then start falling. Remember, those who die or recover do not get infected again. As R increases, there is less and less people, left to be infected and the disease stop spreading further. 

Check the second ODE (eq. 2) carefully. It represents, rate of change of I with time. 

When, a = b/S, 

dI/dt = (b/S).S.I - b.I = 0. 

That means, when a = b/S, the infection will not spread. 

Suppose, some people are infected with the virus but size of the infected population is very small. Say I = 0.001 and S = 0.999. Like the previous simulation, consider b = 0.05. So, b/S = 0.05005. 

For some reason (may be weather), the constant a is very low. Say a = 0.05005. This makes a = b/S. Therefore, dI/dt will be zero and the size of the infected population will not change with time. Though there is circulation of the infection in the community, it will not become an epidemic. 

Suppose, after 100 days, something happens; like mosquito population increases enormously due to a change in weather. This will change the constant a. Now, say a = 0.3. As a becomes greater than b/S, infection will start spreading very fast and the size of the infected population I will increase exponentially. 

Now, we have a full blown epidemic. Eventually, it will recede as people will recover, get immuned or die. This dynamics of sudden appearance of epidemics is shown in the following figure. Here, till 100 days (shown by arrow), a = 0.05005. After that a is changed to 0.3. 

This is one of the simple models for epidemics. This may not correctly explain the current Zika crisis. There are many complicated models for epidemics. Some are disease specific and consider finer details. 

Even then, this simple model explains, how all of a sudden an epidemic can start like an avalanche. This also explain how common precautions, like using mosquito net or vaccination reduces chance of an epidemic. All these steps reduce the value of the constant a. As long as we keep a less or equal to b/S, we are safe.  

Update: Very recently a paper is published that has modeled the transmission dynamics of Zika virus in French Polynesia. There was an outbreak of Zika in these islands in 2013-14. 

They have used a mathematical model very similar to the SIR. Only, this model is bit elaborate. 

The model includes the dynamics of mosquito population. There is a susceptible mosquito population (Sv) that can get the virus from infectious people (IH). Once they have the virus, we call them exposed (Ev). Some of these exposed mosquitoes become infectious (Iv) and infect susceptible human (SH).  

They have also included a human population (EH) that is exposed to Zika through mosquito bite but the infection is in latent stage.

This is called susceptible-exposed-infectious-removed (SEIR) model. Just like the SIR model, ODEs are used to model the dynamics of all the populations. Here, we have seven different populations. So they have used seven ODEs. They have another additional ODE, for cumulative number of infected people. 

For details, look into the paper. It is freely available at Biorxiv. 

By fitting the model to the population data of the out-break, they have made an interesting prediction. Suppose, people, recovered from infection, get life-long natural immunity to Zika. In such case, the model predicts that  it would take at least a decade before re-invasion of Zika in this island population. Some relief for the health workers!!