Survey of Population Modeling of the Flu Virus and an Extension of the Models Considering an Infection Level Based Contraction Rates and Decaying Immunity

*Adam Aaron(Grad), Patrick Caveney(Undergrad), Patrick Jones(Undergrad)*

# Introduction

The influenza virus has been apart of humankind for centuries, if not a millenia, unlike other diseases in the world what makes the influenza virus so dangerous is that it is always changing and has high durability. Along with it’s high mutation rate, there are chances that the severity of the virus could increase as well as decrease from one year to the next. Some instances of epidemics of influenza in the US, seen recently was the Swine Flu, that swept across the nation in a matter of months and had very high death rates among children and older adults. A few years before Swine Flu, there was the Bird Flu that decimated bird population in specific areas and had few cases seen in people. While influenza is highly contagious spreading from person to person via touch, from objects that have been exposed to virus, and through the air when water droplets contain the virus. However just being contagious isn’t the only factor that makes the influenza survive as long as it has in human history. Unlike smallpox, which has very few cases in first world countries, but in the past smallpox was a global killer however since the vaccination of smallpox and the virus’s in-ability to adapt, it has all but died out with very few cases seen each year. The point is that the influenza virus is extremely durable and highly adaptable. Not only can it change within a host but even without a host the virus lives for an exceptionally long time before finding another host. According to the World Health Organization, the influenza virus in the US alone costs $71 to $167 billion every year, which includes hospital health care costs and lost of productivity.

The specifics of the influenza virus from the World Health Organization, that describe the influenza virus can be categorized into two main groups: A and B. Group A has two subgroups that include strains H3N2 and H1N1, group A is the more severe of the two groups and has the most deaths to date. What makes Influenza type A so severe are the protein components that are used to describe the subgroups, the “H” stands for haemagglutinin protein and the “N” stands for neuraminidase protein, these spike-like components are antigens that reside on the surface of the virus shell, that encloses the genetic material. Due to minor changes of the genetic makeup of these parts of the virus that allow the virus to evade existing antibodies and force our society to create yearly vaccines. As mentioned previously, when major changes to the genetic makeup of the virus are pandemics seen across the world and include much higher death rates.

What drives our curiosity of modeling this virus was ignited by the research article “Modelling of H1N1 Flu in Iran” by Ali Akbar Haghdoost MD, Mohammad Mehdi Gooya MD, and Mohammad Reza Baneshi PhD, these researchers conducted a series of investigations in two cities looking at patients who had influenza, from their observations they have created a model and a series of equations that went with their model that showed how the population was affected by the severity of the influenza virus, while we later adjust this model to be more dynamic and illustrative of larger populations, the work done in “Modelling of H1N1 Flu in Iran” inspired our own model and drove us to improve upon it. What we hope our model can accomplish is to help study the spread of influenza and how to counter the virus, as well as predict future epidemics by allowing hospitals or vaccinations clinics to be better prepared for sudden outbreaks.

# Model Description

Haghdoost et al. created a model specifically tailored to H1N1 in Iran. Their model expanded upon the standard SIR model by introducing varying levels of infection, a death option, and an immune option. These additions are key to the research they hoped to achieve, namely the analysis of H1N1 on the hospital system and the total number of deaths from the disease.

Figure 1 is the diagram of the original model. The benefits of this model are the limitations of varying degrees of infection, the immune category, and the death category. The severe and very severe cases help the modelers determine the number of people who will need to be admitted to the hospital and the intensive care unit respectively. The severe patients are assumed to be admitted to the hospital, and the very severe patients are assumed to be admitted into the intensive care unit. The immune category makes the model more real. People who get influenza do have a chance of becoming immune to the specific strain. That is the benefit of the vaccine. The death category helps the researchers assess the impact of H1N1. This allows the researchers to recommend strategies to combat the disease and limit the death toll. The other benefit of the researcher’s set up is the fact that each progression through the model is a percent chance with all choices adding to one hundred percent. This allows the model to be scaled for any size population and avoids the discrete errors related to small populations.

**Figure 1. **Original Model Diagram

The model presented in the paper lacked equations to fully generate the model. The equations for the immune population and death population were missing. The equations presented were dependant on these populations, and the derivation of these equations was not obvious. As a result, we developed our own model using the methods that were presented in class.

The revised model neglected the seasonal Beta (infection rate). The seasonal Beta proposed had very minimal fluctuation. An infection rate was added to each level of infection to simulate the level of care taken when interacting with each infection level. Decay was added to immunity to simulate a vaccine wearing off or the body’s natural immunity after recovery wearing off.

**Figure 2.** Updated Model Diagram

The updated model is shown in the Figure 2. A susceptible individual becomes infected by interacting with an infected individual. The average interaction chance between a susceptible and infected individual is Alpha. The chance of becoming infected when interacting with a given infected individual is determined by that group’s Beta. Then, the chance of them developing a given level of infection is determined by the Lambda of each infection level. Finally, an individual leaves the infected state after Tau days for each infection. Once they are cured from their infection, they then become susceptible again with a chance Iota of becoming immune. Additionally, on any given day there is a percent chance Delta of a very severely infected person dieing.

## Population Groups

The following were the initial conditions and how they are imagined for the H1N1 Flu Virus.

### Susceptible s[t]

The susceptible group is made up of all the individuals who are not infected, immune, or dead. They are the individuals that can contract illness from interacting with infected individuals. For a normal case, a majority of individuals will be in this case, possibly sharing a majority with the Immune group due to vaccination. Note that this group would also account for people who were unsuccessfully vaccinated.

**Infected: Mild inm[t]**

The mildly infected population are the individuals that experience little to no symptoms but are contagious. They recover quickly. The chance of contracting the illness from a mildly infected individual is relatively high because they most likely do not know they are sick, and others around them are not taking special measures to prevent contraction. An initial group of mildly infected individuals would be plausible starting condition to study infection spread.

**Infected: Severe ins[t]**

The severely infected individuals will be sick for much longer, and show symptoms. People are less likely to be infected by a severely infected person as they show symptoms. They do not recover as quickly as a mildly infected person.

**Infected: Very Severe invs[t]**

A person who is severely infected has very little chance of contracting to other individuals. These are people who would likely be in the ICU where special precautions are taken when interacting with them. They take longer to recover than a severely infected individual. There is also a chance of a severely infected person dying.

**Dead d[t]**

The dead population count keeps track of those individuals who have died.

**Immune im[t]**

The immune population are made up of individuals who have either been vaccinated or have recovered from infection. The immunized people would be from the initial condition. The current model does not consider vaccination during the simulation.

## Parameters

The parameters can be broken into multiple categories: controls, infectivity, lethality/immunity, and infection characteristics.

Alpha is the interaction rate, that is, the average chance of an individual having an interaction with an infected individual.

Beta is the chance, for a given infectivity level, for a person to contract the illness. This is higher for mildly infected individuals and lower for severely infected due to the level of precaution taken around each group.

Lambda is the chance that, once a person becomes infected that they develop into one of the three grades of illness. These sum to 1.00.

Tau for each infection is the number of days before the infection is fought off by the person’s system.

Iota is the percent chance that, when a person is cured, they achieve immunity. Tau immunity is used, they will then lose this immunity after a duration of Tau Immunity days.

Delta is the chance on any given day that a person is severely infected that they will pass away.

Tau Immunity is the time constant at which an individual loses their immunity. This is similar to a person getting a flu shot and having the effectiveness wear off after a year.

# Results

## Case 1: Based on Iranian Hospital Parameters

Our first case study is to reproduce the results of *Modeling of H1N1 in Iran* by Haghdoost et al. This model is focused on changing the infectivity of the disease and the number of initially infected people. They are focused on the effects on the very severe population and the dead population. Haghdoost does not include an immune decay. The equations do not include a death and immune equation. The seasonal beta is not included in our model but its effect was slight. The model also does not include an interaction rate between infected and susceptible people or a chance of interaction between each population of infected people with the susceptible population. Because of this these terms are set to zero. The authors ran their model for 1000 days, so we will run ours for the same amount of time. The population of Kerman is 621,374 and Tehran is 8,244,535. Thus, an initial infected population of 10, 20 and 100 corresponds to 1.609 e-5%, 3.218 e-5%, and 1.609 e-4% in Kerman, and 1.213 e-6%, 2.426 e-6%, and 1.213 e-5% in Tehran. Below are graphs representing the effects of 10 and 100 initial infections on the dead and hospitalized populations in Tehran.

The Figure 3 shows the percentage of dead and hospitalized patients with 10 initially mildly infected people. This graph is very similar to the one in the paper and the estimated number of deaths and hospitalized patients are similar. The peak of hospitalized patients hits a little over a year after introduction of the disease.

**Figure 3. **Hospitalized and Dead Populations 10 initial cases.

The Figure 4 shows the percentage of dead and hospitalized patients with 100 initially mildly infected people. This graph is very similar to the one in the paper and the estimated number of deaths and hospitalized patients are similar. In this case the peak hospitalized cases hits a few months earlier than the case with only 10 initially infected people.

**Figure 4. **Hospitalized and Dead Populations 100 initial cases.

## Case 2: Based on Reference Neil M. Ferguson, et al.

In the article “A population-dynamic model for evaluating the potential spread of

drug-resistant influenza virus infections during community-based use of antivirals” by Neil M. Ferguson, Susan Mallett, Helen Jackson, Noel Roberts and Penelope Ward, they view the transmissivity of the influenza virus and suggest that the more severe cases of influenza make it in fact harder for the host to transmit the disease to another person due to varying factors that include, their fitness of being able to move around and spread the virus, their knowledge of having the virus will make them more aware of their virus spreading pathology such as washing their hands and avoiding contact with others around them. In our model, the parameters of the transmissivity has been accounted for with the mild cases being more likely to spread the virus than the severe, who are those that are too sick to perform in normal daily activities or the very severe who are in hospitals and are unable to actively spread the virus.

To emulate the parameters that they have purposed, which was based on a wide age range and they have showed an influenza virus that was resistant to drugs. For our model the immunity decay simulates the virus’s drug resistivity and over a year the amount of immune versus the amount of people that could die are as shown:

**Figure 5.** Immune vs Deaths in one year

## Case 3: Based on Reference Laetitia Canini and Fabrice Carrat

In the influenza research article “Population Modeling of Influenza A/H1N1

Virus Kinetics and Symptom Dynamics” by Laetitia Canini and Fabrice Carrat, they focus on the interactions between virus, host, and immune response of individual in a population. In their research they give a multitude of different parameters they have taken into considerations before creating a system and modeling it. While our model does not have all of their parameters, we do in one form or another account for them due the setup of our own model. For example, to simulate the model they described our adjustments would include adjusting infection rate, transmission rate, severity of the virus and the set the initial amount of patients who have the influenza virus to the values they have deduced from observation.

For modeling a more severe case of influenza, essential a pandemic, the parameters of our model would be adjusted for higher chances of severe to very severe cases with higher interaction and higher death rates. When this is done our model shows the same figure that was shown in Canini and Carrat’s article:

**Figure 6. **Infected Population from Laetitia Canini and Fabrice Carrat

**Figure 7. **Simulated Infected Population in One Year

*Below are interesting cases that can be observed with our model. We present an extinction event, the immunity decay parameter, and the effects of a vaccine on the populations.*

**Case 4: Extinction Event**

First is a model of an extinction event. This virus has a very high death rate and no immunity. This case is similar to an ebola virus. The infection times are all increased to a month and the likelihood of severity favors the very severe infection. From Figure 8 it is clear that the virus kills 80% of the population in only 100 days. Very few people are hospitalized because they die too soon.

**Figure 8.** Hospitalized and Dead Populations, Extinction Case.

The Figure 9 shows the total living population. The virus kills very fast and then asymptotically approaches zero.

**Figure 9.** Total Living Population, Extinction Case.

Interestingly, if the mild and severe infection times are decreased (from 30 days) the death toll rises. This is because in the model shown populations are quarantined in the mild and severe cases where they can only recover, not die.

**Case 5: Immunity Decay**

Next is a case showing the effects of the immunity decay and a vaccine. For this case most all the population is immune to begin with and only 3% of the population is mildly infected. Most of the other variables are the same as the standard case. The interesting variable to change with this case is the immunity decay. In Figures 10 and 11 the decay is set at a year. It can be seen that the immunity, vaccine, delays the onset of the disease for the time the immunity lasts. The vaccine never eradicated the disease, it only delays the response. In Figures 12 and 13 the immunity decay is set at half a year, and the onset of the disease adjusts accordingly.

**Figure 10. **Total Infected Population with low immunity decay.

**Figure 11. **Total Living Population with low immunity decay.

**Figure 12. **Total Infected Population high immune decay.

**Figure 13. **Total Living Population high immunity decay.

**Case 6: Oscillating Infection**

Finally is an interesting case showing a recovering population that mostly develops a decaying immunity. This creates an infection that keeps recurring in a dampered system of sorts. The important parameters of this model are the short infection time, so people will be quickly added back to the susceptible population, the long immunity decay rate, and the high immunity rate. This system never fully eradicates the virus and instead reaches a steady state where about 1% of the population is always infected. This case is analogous to the common cold. We do not have a decaying immunity to each cold strain, but the immunity decay can be thought of as introductions of new cold strains. This case also appears to respond like a predator prey model. The virus would be the predator and the people the prey. Notice that the scales have been changed to better show the effects on the populations. Figure 14 is the graph of the total infected population. Shown is the recurring infection and the steady state infection rate.

**Figure 14. **Total Infected Population, Oscillatory Case.

Below is the graph of the total living population. Even after 1000 days only 3-4% of the population has died. Our model does not have a birth rate so the population will be in decline as long as there is some virus, and in this case the infected population reaches a steady state. Notice also that there is a drop in population that coincides with each peak in infection.

**Figure 15. **Total Living Population, Oscillatory Case.

Below is the graph of the total hospitalized population and total dead population. Notice again that the hospitalized population is periodic in accordance with the infection rates.

**Figure 16.** Hospitalized and Dead Populations, Oscillatory Case.

# Discussion

The purpose of this project was to create and expand upon a model from a research article. To begin we needed to match our model to the model created by the researchers. Difficulty arose due to our model excluding the seasonal beta and the R0 value. The versatility of our model allowed us to still create a scenario similar to the one created by Haghdoost et al. The model has similar numbers of initially infected people, similar death rates, a similar delay between time zero and the increase in infection, and a similar hospitalization rate. Thus, our model can be used to reach the same conclusions as the original one the researchers used. The ability to predict potential epidemics is invaluable and could save the lives of thousands when the right precautions are taken. Our model can be utilized to have vaccines ready for upcoming epidemics, hospitals to be prepared with adequate staff and supplies, and the public with the knowledge to take extra care and precautions to not contract the virus. Future work could incorporate the seasonal Beta from the Iranian hospital model. Additionally, we could include continual vaccination of the susceptible population, change the model to have the infected condition evolve from Mild -> Severe -> Very Severe instead of one being selected upon infection. Also since the model can be observed for long time scales (multiple years) it would be important to add birth and death rates to the general population.

# References

**Population Modeling of Influenza A/H1N1 - **J. Virol.-2011-Canini-2764-70

**A population-dynamic model for evaluating the potential spread of**

**drug-resistant influenza virus infections during community-based use**

**of antivirals - **J. Antimicrob. Chemother.-2003-Ferguson-977-90

**PopMod: a longitudinal population model with two interacting disease states** - Jeremy A Lauer, Klaus Röhrich, Harald Wirth, Claude Charette, Steve Gribble and Christopher JL Murray

**Modelling of H1N1 Flu in Iran - **Ali Akbar Haghdoost MD, Mohammad Mehdi Gooya MD, Mohammad Reza Baneshi PhD

**Analysis of a disease transmission model in a population with varying size - **Journal of Mathematical Biology - 1990 - Busenberg - 257 - 28

**Influenza**. World Health Organization, Media Centre. March 2003. http://www.who.int/mediacentre/factsheets/2003/fs211/en/

http://www.cdc.gov/flu/pandemic-resources/tools/index.htm - CDC Tools for determining influenza outbreak

CDC 2010 Influenza Death Report

http://www.flu.gov/pandemic/history/index.html