Disease

• Project name: Disease
• Class: 4IT495 Simulation of systems (WS 2012/2013)
• Author: Štěpán Pilař, xpils00@vse.cz
• Model type: Agent-based model
• Software used: NetLogo

Problem definition

This simulation analyzes effect of vaccination and isolation/quarantine on the dynamics of an infection with a person-to-person transmission. SIRS model of a disease (susceptible, infected, resistant, susceptible again) is used.

Method

This simulation is an expansion of basic Virus model from NetLogo library adding possible countermeasures. It's benefit is simple and easy to use comparison of effects of those countermeasures on infections with variable duration of each phase.

Admittedly, it's not very well suited for modelling specific diseases because of its generalization and lack of relevant data. Most infection models use either system dynamics approach or very large-scale agent-based models in environments like EpiSims. System dynamics models use different set of parameters which cannot be easily translated, most notably basic reproduction number representing transmissivity of the disease as number of people who are infected by one patient. A chance of transmission would be much better suited for agent-based model. EpiSims studies use transmissivity rates obtained by calibration against historical data, however those rates are calculated for contact hour. Risk of transmission is then calculated by multiplying this base transmissivity rate by time spent in company of infected people, which is based on models of daily routines in real environment and accounting for different environments (home, commuting, work, leisure). Resistance period is present in most diseases but isn't usually expressed in time, rather in chance of reinfection during some time period.

Model

The model uses only one type of homogenous agents. Agents, people, are situated on 2D world in which they move randomly at low speed coming into contact with other agents in closed environment. Number of agents is constant during one simulation run (simulating short time span). One step is one day.

Many aspects of reality are abstracted away in favour of simplicity, most notably individual susceptibility due to overall physical state, simplification of the disease course to three stages (omission of latent period) and constant transmissivity and susceptibility rates.

Parameters

Population density

Population density is represented by people-density slider ranging from 10% to 300%. Density of 100% means that if agents were evenly distributed in the environment no two agents would be in contact, density 200% means that agents are in contact in pairs given an even spatial distribution. Population density can be effectively set only before running the simulation.

Initial portion of infected

This parameter is represented by initial-infected slider and determines the initial number of randomly distributed infected agents in terms of part of the whole population. When vaccination is taken into account, the model chooses non-vaccinated agents to be infected preferably (only healthy people are vaccinated). If there are more agents to be initially infected than there are non-vaccinated agents, it infects even vaccinated agents (no vaccine has 100% effectiveness). Values range from 1% to 100%.

Percentage of vaccinated people

If agent is vaccinated, he's vaccinated before the beggining of the simulation run, therefore this parameter affects model only in the setup phase. Relative number of vaccinated agents is represented by the slider vaccination-ratio ranging from 0% to 100%.

Vaccination effectiveness

As mentioned above, no vaccine is 100% effective. The slider vaccination-effectiveness ranges from 1% to 100% and represents this fact. Effect of this value is described below. Change of this and all following parameters will affect the simulation run, but such changes were not focused on.

Efficacy of isolation

Ill agents can be isolated from the rest of the population in order to prevent the infection from spreading. Even patients who are not in isolation room, usually have less contact with other people. 0% value represents ignoration of the disease (common for common cold), higher values can represent bed rest at home with contact with family members, very high values can represent isolation in a hospital.

Incubation period, disease and resistance period duration

These three parameters (sliders incubation-period-duration, disease-duration and resistance-duration respectively) represent the length of each phase of the disease, they range from 2 to 100 steps. When transitioning to a new stage of the disease, length of this stage is taken as a random number from normal distribution with mean value given by the control and standard deviation as a fraction of the duration (1/4, 1/6 and 1/2 respectively).

Transmissivity

Transmissivity is represented by transmissivity-rate slider ranging from 1% to 100%. The detailed mechanism of transmission is discussed below, but intuitively higher value means easier transmission from person to person.

States of agents

Each person can be in one of four consecutive states - healthy (green), infected (yellow), ill (red) or resistant (blue) -, can be vaccinated and can be isolated. When an agent changes state a countdown is started with initial value given by the respective control as described above; this doesn't affect healthy agents. Change from healthy to infected can occur only as a result of a transmission from other agent. The countdown value is decreased in every step and when it reaches zero, agent changes its state from infected to ill, from ill to resistant and from resistant to healthy.

Transmission

When a healthy person comes into proximity (the same patch in NetLogo) of an infected or ill person, he can contract the infection with probability given by transmissivity-rate. If the healthy person is vaccinated the probability is multiplied by 1 - vaccination-effectiveness, if the other person is ill and isolation-efficacy is positive then the probability is multiplied by 1 - isolation-efficacy.

Example: When a healthy vaccinated agent comes into contact with an ill isolated and transmissivity-rate = 10%, vaccination-effectiveness = 60% and isolation-efficacy = 90%, the probability of the healthy agent contracting the infection is 10% × (100% - 60%) / 100% × (100% - 90%) / 100% = 0.4%.

Each agent can contract the infection from any other agent within immediate distance - the test is applied among all healthy and infected/ill agents in the same area (patch).

Measured values

Two properties of the contagion are measured in a form of chart. First one is prevalence - the total number of agents by state (in medicine the term refers only to those infected, but the model provides information about all states) in the current step. Second is incidence, the number of people who became ill in this step.

Results

Results are based on manual analysis of the dynamics of prevalence and incidence. Initial values about incubation period, duration of the disease and vaccination effectiveness are based on clinical data for seasonal influenza (incubation period: 1-4 days, illness: 7-14 days, 70%-90% effectiveness). All charts are based on 180 steps long runs and default setup people-density = 300%, initial-infected = 10%, transmission-rate = 10%, incubation-period-duration = 2, disease-duration = 10, resistance-duration = 0, vaccinated-ratio = 0%, vaccination-effectiveness = 1%, isolation-efficacy = 0% with variation of one parameter described in their title.

Higher transmission rate (TR) leads to more pronounced fluctuation with lowering amplitude; larger portion of ill, smaller of healthy (even smaller than infected), prevalence of ill can go almost to 100%. With low TR incidence spikes and then stabilizes with oscillation, higher TR leads to smaller effect of initial spike, with TR > 15% incidence fluctuates with lowering amplitude.

Larger portion of initially infected agents leads to more unstable behaviour during first steps, but doesn't affect general trend in prevalence. In incidence there appears a drop after initial spike but trend is not affected.

Lower density of population leads to faster stabilization of both prevalence and incidence and generally more healthy and less ill agents in the population. In combination with low TR it can lead to eradication of the disease faster than in denser populations.

Longer duration of resistance period has a significant effect on the dynamics of the disease. Time between local maxima in prevalence stretches, fluctuation in number of infected, resistant and healthy agents becomes more pronounced as well as the incidence. The trend of levelling of all values is still apparent. Very long resistance period can lead to near eradication of the disease after first wave, but if the disease survives the following maxima are more densely clustered. Most important factor is relative length of duration of resistance and combined incubation period and disease length. The higher the transmission rate, the shorter the resistance needs to be in order to eradicate the disease in closed population.

Isolation has positive effect on the dynamics of the disease. The higher the isolation efficacy the more stable the development of the disease is after the initial surge. Very high (hence unrealistic measures in case of influenza or any other infection with low transmission and mortality rate) value can eradicate the disease with few ill patients. Isolation efficacy is in inverse relationship with transmissivity.

Vaccination with effectiveness 70%-90% has the intended effect on the proportion of healthy and ill agents. There isn't any leverage point in which the ratio would suddenly change, the effect is rather linear. For relatively longer duration of resistance, vaccination affects dynamics of the disease in a fashion similar to isolation. High vaccination ratio in combination with high effectiveness leads to (near) eradication of the disease.

Effect of both vaccination and isolation is more pronounced in sparser populations.

Conclusion

The model illustrates the effects of selected parameters of the disease and possible countermeasures in a simple way. No surprising or non-intuitive facts were found. Due to lack of applicable data and generalization, the model cannot be used to represent any specific disease.

Citations

Camacho A, Ballesteros S, Graham AL, et al. Explaining rapid reinfections in multiple-wave influenza outbreaks: Tristan da Cunha 1971 epidemic as a case study. Proc Biol Sci 2011;278(1725):3635-3643. <http://rspb.royalsocietypublishing.org/content/early/2011/04/22/rspb.2011.0300.full.html>.

Shetty, Nandini, Tang, Julian W, Andrews Julie. Infectious disease pathogenesis, prevention, and case studies. Chichester, UK: Wiley-Blackwell, 2009. ISBN 978-144-4311-921.

Stroud, Phillip, Del Valle, Sara, Sydoriak, Stephen, Riese, Jane and Mniszewski, Susan (2007). 'Spatial Dynamics of Pandemic Influenza in a Massive Artificial Society'. Journal of Artificial Societies and Social Simulation 10(4)9 <http://jasss.soc.surrey.ac.uk/10/4/9.html>.

Influenza: Vaccine use. WORLD HEALTH ORGANIZATION. World Health Organization [online]. © 2013 [cit. 2013-01-12]. <http://www.who.int/influenza/vaccines/use/en/>.

Code

File:Xpils00-disease.nlogo