PRE2019 3 Group16

Group Members

Name	Student Number	Study	Email
Efe Utku	1284290	Applied Physics	e.utku@student.tue.nl
Roel den Hoet	1248170	Computer Science	r.d.hoet@student.tue.nl
Venislav Varbanov	1284401	Computer Science	v.varbanov@student.tue.nl

Problem Statement

Infectious disease outbreaks are a fundamental problem of humans. There are various settings, worldwide, that might lead to an epidemic or a pandemic. Although these outbreaks have various impacts on the society; one of them is the economical consequences of it. Here, we suggest a drone operation responsible for collecting and testing nasopharyngeal specimen from people living in preselected disease-prone regions and communities. By keeping a precise track of more people in less time compared to currently used strategies; we aim to decrease the effects of the outbreak on the community and to evaluate our results in terms of the economic impact of the strategy.

Subject

Epidemics are defined as local infectious disease outbreaks that occur in a community or region. These outbreaks have major impacts on the daily life of community members as economical, social and political issues. The economical problems are mostly due to measures taken to prevent the spreading of the disease: e.g. working, transportation and gatherings in public areas are halted. To minimize these impacts one must keep an up-to-date record of regions that are prone, people who might be infected and people who are more susceptible to infections; because in the bigger picture the main problem is to identify and track reported cases. Only this way, the spreading can be reduced and the distribution of new cases per day can be minimized.

An efficient way to do this is to detect the “local source” of an outbreak and reinvestigate the timeline of the spread. However, this approach has cost and logistic complications within, depending on the number of cases and days passed since the identification outbreak. So, we suggest an alternative way to keep an outbreak under control; which is to use an aerial drone-based operation for specimen collection and accurate case identification.

In core of the drone operations is to provide a faster logistic solution for case reporting; hence, providing a faster tactic to act and take precautions regarding the spread. This subject is going to be investigated in terms of its’ effects on different stakeholders and the its’ numerical impact on the way the disease spreads. The later, technical, part also consists of 2 components. First one is the mathematical model describing the population dynamics with and without the drone strategy, and the second one is an optimization problem to get a realistic point of view on the costs and possibilities of this strategy. Then, by combining these two technical components a feasibility study will be conducted to compare the total cost/ economical impact of the outbreak on the community and the total cost of the drone operations. The economic impact is going to be calculated based on GDP per capita per day and the progress of the epidemic without the drone operations. The cost of the drone operations is going to be calculated based on the cost of a single drone, number of drones operating, progress of the epidemic with the drone operations and other logistic costs.

Objectives

Deliverables

The list of the deliverables and their explanations are given below.

Mathematical Model and Simulation of Population Dynamics

An epidemiological compartmental model describing the population dynamics of a community. The model is given by a system of non-linear differential equations.

A MATLAB script for simulating mentioned mathematical model; which is used to investigate the impact of the drone operation on the spreading of the disease.

Feasibility Study of the Operation

Two cost analysis are performed for estimating the economical impact of an outbreak and the cost of drone operations; these studies are then combined to check the feasibility of the proposed approach.

User, Society and Enterprise

User

The users of our project are sick people, and people that are vulnerable to the disease. The users can now be diagnosed at home without having to travel to a hospital, which saves the sick people a trip to the hospital.

Society

For the society, keeping the possibly sick people inside will decrease the exposure of the healthy people to the virus, and hence this will decrease the spreading of the virus. More people in society will remain healthy, and the spreading of the virus can be stopped earlier thanks to the drones.

Enterprise

This invention will help drone manufacturing companies with new work, but it will also help the medical sector by providing more jobs on the operation bases for people to work on. There will also be an increased demand for nose swabs, as these are necessary for our product to work.

Description of the Operation

(This text can be divided to the sections below)

We operate from a drone base in a central location in the region. From this base, drones will fly towards people whose nose swabs we would like to collect. These people have been notified earlier by the use of a phone app. When the drone arrives, the person uses a nose swab to collect specimen, and gives this to the drone in a marked container. After the drone has collected all specimen on its route, it will return to the drone base. Here, all collected specimen will be tested. When the test is over, the results will be communicated to the user via the phone app.

Through the phone app, the user will also receive a recommendation based on the results of the test. This recommendation will imply if they should stay at home isolated/ should receive medical treatment. Also, at the end of specific time intervals users might also receive information regarding the isolated regions and districts close to them, which also counts as an intervention policy.

Population Dynamics

Mathematical Model

Use of mathematical methods have proven to be a successful way for estimating population dynamics. This approach dates back to mid 18th century and Bernoulli’s works on smallpox. He is also the first one to clearly define some of the most crucial epidemiological parameters, which are still used now. (Dietz, 2000) Although these methods are being improved since then, his work has also been essential to the theory of disease control. (Smolinski et al., 2003) Disease control theory refers to the applied intervention policies regarding infectious diseases and their systematic study as mathematical models. These models, consisting of various parameters, provide the framework to investigate how each intervention policy will affect the dynamics of population groups in case of an outbreak.

Based on the drone operation intervention policy mentioned in this report; here, we propose a compartmental epidemiological model to study the impact of this strategy. Compartmental models are deterministic, helpful for simplifying the problem and hold an assumption that individuals in the same compartment have the identical characteristics; thus, mean values are used. The model we used is derived from the iconic “S-E-I-R” Model (Kermack et al., 1927) and the model used in the study of Chowell et al. on early detection of Ebola virus, whose results are complementary to the core aim of our approach.

We define a model where the population is divided into 5 compartments. Namely; Susceptible (“S”), Exposed (“E”), Infected (“I”), Isolated (“J”) and Removed (“R”). The schematic of the model given below displays the transitions between the states.

(Schematic will be added, eq.s going to be edited)

And the corresponding differential equations are given as;

dS/dt=μ(N-S(t))-λ(t)S(t)+δdR(t),

dE/dt= λ(t)S(t)-(μ+α(1+ϵ))E(t),

dJ/dt=αϵ(E(t)+I(t))-(γ_r+μ)J(t),

dI/dt=αE(t)-(γ+μ+αϵ)I(t),

dR/dt=γ_r J(t)+γI(t)-(μ+δd)R(t).

Where;

λ(t)=β ((I(t+(1-r)lJ(t)))/(N-rJ(t))

is defined as the “force of infection” by Chowell et al..

μ,δ,γ,γ_r,r,l,d,β and α are consecutively the natural death rate, removal rate from recovery to be susceptible again, removal rate of infectious individuals, removal rate of isolated infectious individuals, effectiveness of isolation, relative transmissibility of isolated infectious individuals, probability of being susceptible after recovery, mean transmission rate and rate of individuals getting isolated.

In addition,

ϵ=(Number of Tests*Succes rate of the Test )/Population is the term responsible for impact of the operations on the outbreak.

The model has the boundaries: (S(0),E(0),J(0),I(0),R(0))∈{(S,E,J,I,R)∈[0,N^5 ]:S≥0,E≥0,J≥0,I≥0,R≥0,S+E+J+I+R=N}

Although a natural death rate is present in the model, it has been set equal to the birth rate so the total population, “N”, is assumed to be constant. Also, it has to be noted that this transition schematic doesn’t show the death rate from each compartment. From these boundaries, in the limit t→∞ it can be proven that a Disease-Free Equilibrium and an Endemic Equilibrium exists for different initial conditions.

Planning

Week 3: Make plan - research algorithm and model

Week 4: Research algorithm and drone - create model

Week 5: Implement algorithm - research drone

Week 6: Simulate algorithm - research drone

Week 7: Create presentation

Week 8: Give presentation

Milestones

Week 3: New subject chosen - plan made

Week 4: Research of algorithm done - model done

Week 5: Algorithm implemented and tested - drone research done

Week 6: Case example simulated - drone component list done

Week 7: Wiki finalized

Week 8: Presentation finalized

Task Division

Main

Efe Utku - Work on wiki page, Research on Mathematical Model/ Feasibility/ Drone Ops./ Population Dynamics Simulation

Roel den Hoet - Research of algorithms, implementation and testing of algorithms, work on wiki page

Venislav Varbanov - Research, implementation, testing and description of algorithms

Weekly Contribution

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02- 08/03/2020

Efe:

-Written Problem Statement, Subject

-Updated the WikiPage Template

-Research on Epidemic Modeling and Adjusting the Model

-Worked on MATLAB Simulation for Pop. Dynamics

Roel:

- Researched algorithms

- Updated the wiki page on User, Society and Enterprise

Venislav:

- Worked on simulating the spread of disease

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09- 15/03/2020

Efe:

-Written Drone Ops., Pop. Dynamics Model

-Researched on test conducting, predictive methods for region/ people selection, existing datasets on epidemics

-Worked on MATLAB Simulation for Pop. Dynamics

Roel:

Venislav:

- Worked on simulating work of the drones

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10- 16/03/2020

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Algorithm notes (Venislav)

Input:

- social network: undirected graph, vertices represent people and have coordinates and condition, edges between people who often communicate

- number of drone bases, coordinates of each base, number of drones per base

- range, flight time(entire day?, or add recharge time), speed and capacity of drones

Output/score: total number of people that got sick and/or time until no more people could get sick

Two ways we can choose which people get sick:

1. People that got sick the previous day or earlier and are not yet diagnosed have a chance x of transmitting the disease to each neighbor. A person with n sick undiagnosed neighbors has chance of getting sick min(100%,n*x). (preferred by me)

2. Use provided formula to compute x - the increase of sick people in a day, and pick x random people with a sick undiagnosed neighbor (if there are enough such people) and make them sick.

Day 1: Initially some people are sick (P). They get some of their neighbors (N) sick (S) and their connections are removed. Each of N picks a time window from 08:00 to 00:00 the same day. The drones try to cover as many of N as possible (D).

Day 2: Until 08:00 results of D are ready and connections of identified sick people are removed. People of S-D get some of their neighbors sick (S’). We consider all neighbors (N’) of N-D. Each of N-D and N’ picks a time window from 08:00 to 00:00 the same day. The drones try to cover as many of N-D and N’ as possible (D’).

Day 3: Until 08:00 results of D’ are ready and connections of identified sick people are removed. People of S-D-D’ and S’-D’ get some of their neighbors sick (S’). People of S-D-D’ self-diagnose and remove their connections. Let M be the neighbors of S-D-D’. We consider all neighbors (N’’) of M-D’. Each of M-D’ and N’’ picks a time window from 08:00 to 00:00 the same day. The drones try to cover as many of M-D’ and N’’ as possible (D’’).

…

References

Chowell, D., Safan, M., & Castillo-Chavez, C. (2016). Modeling the Case of Early Detection of Ebola Virus Disease. Mathematical and Statistical Modeling for Emerging and Re-Emerging Infectious Diseases, 57–70. doi: 10.1007/978-3-319-40413-4_5

Dietz, K., & Heesterbeek, J. A. P. (2000). Bernoulli was ahead of modern epidemiology. Nature, 408(6812), 513–514. doi: 10.1038/35046270

Katriel, G. (2013). Stochastic Discrete-Time Age-Of-Infection Epidemic Models. International Journal of Biomathematics, 06(01), 1250066. doi: 10.1142/s1793524512500660

Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(772), 700–721. doi: 10.1098/rspa.1927.0118

Smolinski, M. S., Hamburg, M. A., & Lederberg, J. (2003). Microbial threats to health: emergence, detection, and response:Washington, DC: National Academies Press.