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MBW:Effects Of Human Decision On Infectious Diseases

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Introduction

Standard SIR Model and Assumptions

The SIR model is used to model infectious diseases where the individuals enter an immune state after infection where they can no longer be susceptible.

                                       Susceptible\rightarrow Infected\rightarrow Recovered

This model can be represented as a simple system of ordinary differential equations as seen below.

{\frac  {dS}{dt}}=-\beta SI


{\frac  {dI}{dt}}=\beta SI-\gamma I


{\frac  {dR}{dt}}=\gamma I


Where S is the susceptible population, N is the total population, \beta is the infection rate, I is the infected population, \gamma is the recovery rate, and R is the recovered population.

The recovered population can be individuals who are immune, dead, or removed because of isolation. The assumptions that go into a simple standard SIR model are as follows:

  1. Constant population i.e. duration of epidemic is short compared to lifetime to neglect birth and disease unrelated deaths
  2. Homogeneous mixing i.e. every person is equally likely to encounter an infected person
  3. Unchanging governing laws i.e. people are not trying to avoid infection

It is clear that sometimes these assumptions are not true of how populations respond to infectious diseases. The first assumption could be incorrect for diseases that affect the elderly or soldiers in battle because the death rate from causes other than the infectious disease could be non-negligible. The second assumption would not be true if infected individuals were present only in one neighborhood or if people we not allowed to travel from an infected area to others like a quarantined area. The third would not be true if people changed their behavior to avoid infection such as wearing face masks, using hand sanitizer, or being vaccinated. The third assumption is interesting to investigate further because often when there is an outbreak of a disease individuals in a population can make decisions on how to act that can strongly affect the outcome of the epidemic. Math biologists have been investigating types of human decision and how to model it to better model how infectious diseases will affect populations.

Human Decision and Information, S. Funk et al. 2010

S. Funk et al. 2010 is a review on the modeling of human behavior in SIR models. In order for accurate modeling, human decision needed to be focused in order to be systematically incorporated into mathematical models. For the purpose of modeling, human decision was focused on:

  1. self-initiated, voluntary behavior
  2. human decision purely based on information people gain from the outside world

The information from the outside world was then again categorized for the model. They broke information down into two categories:

  1. Local information: word of mouth, social interaction
  2. Global information: news, public health authorities, media

The distinction between these types of information can be very important. Local information can occur in clusters which can strongly affect disease dynamics. For example, Boulder County is a local community which has the lowest school-wide vaccination rate in Colorado and one of the highest per capita rates of whooping cough in the United States.

Figure 1: United States Whooping Cough Incidence (Center for Disease Control)

This is due in part to the belief that immunization causes unwanted side effects such as autism and a weakened immune system. The Shining Mountain Waldorf School is an even smaller subset of this community in which nearly half of the 292 students are not immunized to whooping cough and the other 21 infectious diseases that have childhood immunizations. The low vaccination rate in Boulder is beginning to affect the other local communities in Longmont and the Denver metropolitan area causing high rates of whooping cough. Figure 1 shows the rate of whooping cough or pertussis in the United States as a whole. The local concept that vaccination causes side effects such as autism and a weakened immune system is causing a nationwide increase in whopping cough. It is seen in Figure 1 that when the first whooping cough vaccination, DTP, was introduced around 1950 the number of whooping cases dropped dramatically. Around the mid-nineties when people began fearing side effects, the number of cases of whooping cough has begun climbing on a national scale demonstrating the power of local information spread.

Behavioral Changes

Human decision and the flow of information leads to behavioral changes which ultimately impact the epidemic. Behavioral changes can be of two types:

  1. Belief based: can occur at time points, detached from actual disease dynamics, highly subjective
  2. Prevalence based: objective, linked to disease dynamics

An example of a belief behavioral change is choosing not to vaccinate children against measles-mumps-rubella (MMR) because the vaccine was rumored to cause autism. This is not linked to disease dynamics of MMR nor is it scientifically accurate. An example of a prevalence based behavioral change is getting the annual flu vaccine when reports surface that many cases have been identified in your area.

Why is human behavior important in understanding infectious diseases?

Starting the in the 1960's infectious diseases began being eradicated in the developed world because of the introduction of antibiotics, vaccines, and improved sanitation. While infectious diseases are still extremely prevalent in developing countries they are also starting to make a come back in the developed world due to bacterial resistance, reduced childhood immunization, and newly discovered diseases. Diseases that are prevalent in many developing countries include: malaria, dengue, and yellow fever. Reemerging diseases in the developed world include: whooping cough and the measles. Antibiotic resistant bacteria include: tuberculosis, pneumonia, and gonorrhea. New emerging diseases recently discovered include: Lyme disease, Legionnaire's disease, toxic-shock syndrome, hepatitis C, hepatitis E, and hantavirus.

New Diseases

Figure 2: Severe acute respiratory syndrome map of infection. (WHO)

New diseases are rapidly surfacing due to the the effects of new ecosystems, global warming, environmental degradation, increased international travel, and changes in economic patterns. Outbreaks of new diseases can lead to epidemics because new diseases are generally not well understood and there is not yet a treatment plan in place. It is important that while infectious diseases do not have a cure, that individuals behave in a way that does not increase the likelihood of an epidemic. A good example of a disease spread into an epidemic by negligent human behavior is severe acute respiratory syndrome (SARS). SARS was a newly discovered disease that led to an epidemic that reached 29 countries, affected 8,098 people, and killed 774 in 2003. Figure 2 shows the widespread effects of SARS. According to the World Health Organization, there have been no reported cases of SARS worldwide since 2004. After the disease was discovered in late 2002 early 2003, scientists immediately began trying to understand it and were then successful in ending the epidemic and preventing the disease from resurfacing into another epidemic. The disease can only be spread by close human contact. The main mode of the disease spreading to many countries allowing for an epidemic, was people traveling in airplanes. Even during the outbreak and epidemic individuals continued to travel to infected area and then return home to their previously uninfected homes. After word of the severity of the disease was released to the public by the World Health Organization, people began cancelling travel plans and wearing face mask. This model could be used to study the affect of these behaviors on the disease dynamics. New diseases are not well understood and therefore it takes time to prevent spread and alter the disease dynamics to end epidemics which makes human behavior even more relevant to the dynamics. .

Reemerging Diseases

Infectious diseases are reemerging because parents are choosing not to immunize their children. As discussed above, there is a large incidence in some communities of non-immunization because parents fear possible side effects. This is allowing whooping cough to again become a problem in schools. Some parents plan on "herd immunity" and therefore believe that their children can be safe while avoiding the side effects. Herd immunity however is not an approach that works in populations where many parents stop immunizing or for diseases where the immunizations are not complete. When individuals are immunized from whooping cough they can still get the disease if they are exposed to individuals with the disease. If many individuals try to use herd immunity the mass population is no longer vaccinated and the concept of herd immunity is no longer valid. The behavior of non-immunization can be studied using this model based on the local information in communities about vaccinating children.

Antibiotic Resistance

Figure 3: Percentage of methicillin resistant S. Aureus around the world.

Drug resistant bacteria is interesting to study because a large factor in developing resistance is the human act of over prescribing or improperly using antibiotics. Antibiotics are commonly over prescribed by doctors which increases the strains of bacteria that can develop resistance. When an individual does not properly take their antibiotics more strains can survive the weaker doses or shorter doses and can therefore develop antibiotic resistance. Human behavior could be modeled to better understand how to design antibiotics or how to prescribe them to ensure that our actions are not aiding in developing drug resistance. Human decision could also be important to study because once bacteria have become drug resistant, they can spread quickly and it is important to see how human behavior and response to information about the spread can change disease dynamics. Methicillin-resistant Staphylococcus Aureus (MRSA) is a drug resistant contagious bacterial infection spread by skin contact or personal items like band aids, towels, razors etc. MRSA was also spread through contact with gym equipment at athletic facilities. Figure 3 shows the prevalence of MRSA in countries around the world. When information was spread about the dangers of MRSA, gyms began providing towels and cleaning solution and placing hand sanitizer at entrances. This behavior and the effect of the information on the dynamics and spread of MRSA is important to understand in order to combat the spread and predict the future of MRSA. Resistance maps can be generated for many bacteria and well as for the antibiotics currently on the market to better understand the severity of antibiotic resistance in the world.

For these reasons it is important to study how humans behave to local and global information to understand how to eradicate reemerging diseases, new diseases, and how to attack bacterial resistance.

Simple SIR Model with Behavior Modification

Description of the Model

In order to demonstrate the effect of human decision on infectious disease, a simple model was constructed in Polymath. SIRS models account for diseases that do not develop immunity and therefore individuals who were once infected can again become susceptible. Diseases like the common cold are SIRS diseases. The flow of population compartments is seen below:

                    Susceptible\rightarrow Infected\rightarrow Recovered\rightarrow Susceptible

The model described below will demonstrate a prevalence based behavior changes as a result of global information for an SIRS infectious disease. The population is modeled to change as a whole together and not in local clusters. The model also incorporates prevalence based assumptions because the number of individuals protecting themselves from infection will be proportional to the level of infection in the population. This model will incorporate another class of individuals P which will represent the protected population or those who are trying to prevent themselves from contracting the disease.

Model

The populations of individuals in the different compartments can been modeled by the flow diagram seen in Figure 4.

Figure 4: Population flow diagram for the compartments in the model.

The model can be described by the following system of differential equations.

{\frac  {dS}{dt}}=-IS\beta +\pi R-\theta S+\mu P

{\frac  {dI}{dt}}=IS\beta -\gamma I

{\frac  {dR}{dt}}=\gamma I-\pi R

{\frac  {dP}{dt}}=\theta S-\mu P

Where S is the susceptible population, Iis the infected population, R is the recovered population, and P is the population that is actively protecting themselves against contracting the disease. The rate constants were chosen based on certain assumptions about the disease being modeled. \beta is a representation of the number of times a susceptible person encounters and infected person which was assumed to be once every 60 days. \gamma was chosen based on the disease having a 10 day recovery period. \pi is the time a person stays in the recovered state before they are susceptible again which was assumed to be 5 days. \theta is based on the number of infected people in the population. It is assumed that for every 10 infected people, 10% of the susceptible person protects themselves. \mu is the rate constant that determines when protected individuals go back to being susceptible. It is assumed that a for every 10 susceptible (or not infected person) 10% of the protected population becomes susceptible. These assumptions lead to the following values for constants.

\beta =0.01667

\gamma =0.1

\pi =0.2

\theta =I/100

\mu =S/100

Results

The model was simulated for both a short time scale and a long time scale to demonstrate the effect of susceptible individuals protecting themselves from infection. Figure 5 shows the short term dynamics and Figure 6 shows long term dynamics. The time scale is in days.

Figure 5: Short term population modelling. X axis is time in days, and y axis is population in number of individuals.
Figure 6: Long term population modelling. X axis is time in days, and y axis is population in number of individuals.

Implications

This simple model can be used to understand diseases like the common cold where poeple tend to wash their hands more or avoid making contact with germ ridden areas during cold season. The short term dynamics show dynamics similar to a standard SIRS model because the level of infection becomes high before people start trying to avoid infection. In the long term dynamics, one can see that as the model approaches steady state, the level of infected individuals declines and the number of protected individuals increases. This resembles a situation where in a community if a lot of people got the common cold, more people would start protecting themselves against infection and so the steady state of infected individuals would drop to a lower steady state. While this model is simplified one can see how the idea of human decision and changing behavior will alter disease dynamics and add to the understanding of how to handle infectious diseases in communities.

Impact of Information Transmission on Epidemics, Kiss et al. 2009

Description of the Model

In the paper, Kiss et al. 2009 The Impact of information transmission on epidemic outbreaks, an SIRS model is considered in which the transmission of information and its effect on the individuals behavior and therefore disease state is incorporated into the model. For this study the "recovered" class was described as the "treatment" class for those individuals who were actively taking medications and recovering from the disease. They used this model to study sexually transmitted diseases and described the populations using sub classes of responsive and non responsive. Responsive individuals were described as individuals who are aware of the disease transmission, actively try to prevent transmission, and, if infected, seek early treatment. Non responsive individuals are individuals who are either not aware or do not change their lifestyle to prevent infection. The model studied the information transmission to try and study the effectiveness of increased awareness in preventing epidemics.

The model breaks individuals down into five groups:

  1. Susceptible non-responsive S_{{nr}}
  2. Susceptible responsive S_{{r}}
  3. Infectious non-responsive I_{{nr}}
  4. Infectious responsive I_{{r}}
  5. Treatment class T

Two different types of information dissemination are considered:

  1. Direct contact between individuals given by f_{{i}} and f_{{s}} which can be considered local information
  2. Population-wide dissemination of disease related information g_{{i}} and g_{{s}} which can be considered global information

The global parameters depends on the infection prevalence and the rate at which non-responsive individuals become responsive. There is a saturation effect included to account for increasing levels of infection prevalence. The parameters h_{{s}} and h_{{i}} represent the rates at which responsive individuals become non-responsive. These parameters depend on time and prevalence. One can imagine that when prevalence in the population is low, the dissemination of information will be less effective because people do not experience the disease or see others with the disease. Individuals leave the treatment population and become susceptible again at rate r. The proportion p accounts for treated individuals who do not change their ways after they are treated and therefore re-enter the non-responsive susceptible stage.

Model

A diagram of the infectious states considered for this model can be seen below:

Figure 7: Infectious states for information transmission.

Susceptible non-responsive individuals become susceptible and responsive or infected and non-responsive. Responsive susceptible individuals become infected and responsive or susceptible and non-responsive. Responsive infected individuals can become infected and non-responsive or an individual undergoing treatment. Infected non-responsive individuals can become infected and responsive or an individual undergoing treatment. A person in the treatment phase can become susceptible and responsive or susceptible and non-responsive. What creates such interesting dynamics in this model is the parameters associated with rates of change between each population compartment and how information dissemination affects them.

The population states make sense in the context of a number of diseases and therefore make this model relevant for many applications. For example, this past year the flu season in Colorado was described as being the worst in many years. If an individual responds to this newscast and begins protecting themselves from the flu outbreak by using hand sanitizer and avoiding public infectious zones, they would be part of the susceptible responsive population. If they stop doing this out of laziness or out of disbelief that the disease is spreading, they would then enter the susceptible non-responsive group.

The following equations describe the infectious states:

{\frac  {dS_{{nr}}}{dt}}=-\beta _{{nr}}(I_{{nr}}+I_{{r}}){\frac  {S_{{nr}}}{N}}-\alpha _{{s}}f_{{s}}(S_{{nr}};S_{{r}},I_{{r}},T)-\delta _{{s}}g_{{s}}(I_{{nr}},I_{{r}})S_{{nr}}+h_{{s}}(I_{{nr}},I_{{r}})S_{{r}}+prT

{\frac  {dS_{{r}}}{dt}}=-\beta _{{nr}}(I_{{nr}}+I_{{r}}){\frac  {S_{{r}}}{N}}-\alpha _{{s}}f_{{s}}(S_{{nr}};S_{{r}},I_{{r}},T)-\delta _{{s}}g_{{s}}(I_{{nr}},I_{{r}})S_{{nr}}-h_{{s}}(I_{{nr}},I_{{r}})S_{{r}}+(1-p)rT

{\frac  {dI_{{nr}}}{dt}}=\beta _{{nr}}(I_{{nr}}+I_{{r}}){\frac  {S_{{nr}}}{N}}-\alpha _{{i}}f_{{i}}(I_{{nr}};S_{{r}},I_{{r}},T)-\delta _{{i}}g_{{i}}(I_{{nr}},I_{{r}})I_{{nr}}+h_{{i}}(I_{{nr}},I_{{r}})I_{{r}}

{\frac  {dI_{{r}}}{dt}}=\beta _{{r}}(I_{{nr}}+I_{{r}}){\frac  {S_{{r}}}{N}}+\alpha _{{i}}f_{{i}}(I_{{nr}};S_{{r}},I_{{r}},T)+\delta _{{i}}g_{{i}}(I_{{nr}},I_{{r}})I_{{nr}}-\gamma _{{r}}I_{{r}}-h_{{i}}(I_{{nr}},I_{{r}})I_{{r}}

{\frac  {dT}{dt}}=\gamma _{{nr}}I_{{nr}}+\gamma _{{r}}I_{{r}}-rT

(S_{{nr}}+S_{{r}}+I_{{nr}}+I_{{r}}+T)(t)=N for all t\geq 0 and N is the population size.

Responsive susceptible individuals are less likely to become infected and are more likely to seek treatment faster leading to the following relationships:

\beta _{{nr}}>\beta _{{r}}

\gamma _{{r}}>\gamma _{{nr}}

The functions can be defined based on the situation that is being modeled. If one wanted to model contact transmitted information and population wide information transmission, the following relationships are used for modelling:

f_{{s}}(S_{{nr}};S_{{r}},I_{{r}},T=f_{{i}}(I_{{nr}};S_{{r}},I_{{r}},T)=f(X;S_{{r}},I_{{r}},T)={\frac  {X(S_{{r}}+I_{{r}}+T)}{N}}

g_{{s}}(I_{{nr}}I_{{r}})=g_{{i}}(I_{{nr}},I_{{r}})=g(I_{{nr}},I_{{r}})={\frac  {(I_{{nr}}+I_{{r}})^{{n}}}{K+(I_{{nr}}+I_{{r}})^{{n}}}}

where n\geq 1 and K is positive.

The value of information to the individual is expected to decay with time as individuals are desensitized to the media. This causes the following relationship to be used for modeling:

h_{{i,s}}(I_{{nr}},I_{{r}})={\frac  {D_{{b}}}{M_{{b}}+(I_{{nr}}+I_{{r}})}}

Results

The model described above was modeled for three different scenarios. Figure 8 shows all three model outcomes. (a) is the trivial disease-free steady state, (b) is the non-trivial disease-free steady state, and (c) is the endemic steady state.

Figure 8: Where S_{{nr}} is the continuous line, S_{{r}} is the dashed line, and I_{{nr}}+I{r} if the dotted line. The non-dimensional parameters are plotted. Parameter values can be seen in table 1.
Table 1: Parameters used for the graphs generated in Figure 8.
Parameter Graph (a) Graph (b) Graph (c)
\gamma _{{nr}} 1/26 (weeks) 1/26 (weeks) 1/26 (weeks)
\gamma _{{r}} 1/13 (weeks) 1/13 (weeks) 1/13 (weeks)
r 1/1 (week) 1/1 (week) 1/1 (week)
d=d_{{s}}=d_{{i}} 1/12 (weeks) 1/12 (weeks) 1/12 (weeks)
p 0.5 0.5 0.5
k 0.01 0.01 0.01
\beta _{{nr}} 0.5\gamma _{{nr}} 2\gamma _{{nr}} 4\gamma _{{nr}}
\beta _{{r}} 0.5\beta _{{nr}} 0.5\beta _{{nr}} 0.5\beta _{{nr}}
\alpha _{{s}}=\alpha _{{i}} 0.5d_{{s}} 4d_{{s}} 4d_{{s}}

With non-dimensionalized initial conditions: (s_{{nr}},s_{{r}},i_{{nr}},i_{{r}},\tau )=(0.5,0,0.5,0,0)

Implications

The three graphed results give insight into how the dissemination of information affects the dynamics of the diseases. Graph (a) is the case where the disease is not present in the population. When the rate at which non-responsive people become infected and the rate of information dissemination are quadrupled, the disease is present in the population but it is not endemic. The population of infected individuals decreases as non-responsive susceptible individuals join the responsive susceptible population due to the increase level of information. In graph (c) the infection rate of non-responsive susceptible individuals is double from graph (b) conditions and the disease is now an epidemic and there is a steady high population of about 25% in the population.

The observations from this study can aid in determining how to change epidemic diseases in the population. Graph (b) demonstrates that for a disease that is moderately infectious to those who are non-responsive, information dissemination is an effective way to prevent and epidemic from occurring. Graph (c) demonstrates that if the disease is more infectious to non-responsive individuals, the dissemination of information is not effective in preventing an epidemic. This can be important for public health officials to determine their plan of action. If a disease is moderately infectious then a public health campaign may be successful in preventing an epidemic but the the disease is highly infectious a more robust effort may need to be put into place in order to prevent an epidemic.

External Links/Citations

Kiss, I. Z., Cassell, J., Recker, M. & Simon, P.L. 2009 The impact of information transmission on epidemic outbreaks. Math. Biosci. 225, 1-10.

Allen, A. 2002 Bucking the Herd. The Atlantic Monthly, Boston.

Hethcote, H.W. 2000 The Mathematics of Infectious Diseases. SIAM Review. Vol. 42, No. 4, pp. 599-653.

Funk, S., Salathe, M., Jansen, V.A.A. 2010 Modelling the influence of human behavior on the spread of infectious diseases: a review. J. R. Soc. Interface. 7, 1247-1256.