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 have been a fundamental threat to humanity since its history. There are various settings, worldwide, that might lead to an epidemic or a pandemic due to bacterial, viral etc. reasons. Although these outbreaks have various impacts on the society; one of them is the failure of health systems. Every region has its own medical capacity; limited by their number of beds, doctors, nurses etc. and during an outbreak, a fast spreading disease may lead to an overwhelming number of patients seeking medical attention. Here, we suggest a drone operation that can be used in viral outbreaks for collecting and testing self-collected nasopharyngeal specimen from people living in preselected disease-prone regions and communities. By decreasing the effective contact rate significantly and keeping precise track of more people in less time compared to current strategies; we aim to decrease the consequences of an outbreak on the community and to evaluate our results in terms of the economic impact of the strategy.

Subject

Communicable diseases can be defined as diseases with the possibility to be transmitted from one individual to another and they can be “classified by a variety of methods: by clinical syndrome, mode of transmission, methods of prevention…”. (Tulchinsky et al., 2014) As, Tulchinsky et al. points out despite the advances in technology and medical sciences; they are likely to remain a challenge for public health. These communicable diseases also affect community life; so, starting from their effects on individuals, in time, they spread over and might cause epidemics/ pandemics. Epidemics are defined as local infectious disease outbreaks that occur in a community or region. Thus, collective problems of each individual start to affect the society altogether. The major impacts of these outbreaks are reflected 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, borders are closed, import/ export limitations. Social problems arise from the difficulties faced by the society as economical and state related problems begin to affect masses in time. Inevitably, social tensions lead to political issues and the occasion becomes a positive feedback cycle of problems. An example of a central problem emerging in dangerous situations is the failure of health systems; faster the disease spreads more people need medical attention. In addition, medical industry, like others, has finite resources and limitations; so, hospitals can be overwhelmed, and health system might fail. “This threat may increase as infectious diseases evolve and escape current human-developed control mechanisms.” (Tulchinsky et al., 2014)

To minimize these impacts states 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 fast identification and tracking of reported cases. Only this way, measures and intervention policies can be effectively applied; so, the spreading rate can be reduced and the distribution of new cases per day can be minimized. The most efficient way to do this is to detect “local sources” of an outbreak. These are the people who are not infected by an infectious person inside the community; one can call them “first generation” of infected people. Also, here detection refers to subjecting them to a test which gives positive/ negative outcome indicating their health status. Then, by reinvestigating the timeline of the sources; the spreading can be blocked to some extent; conditional on the mode of transmission of the outbreak. However, this approach is most of the times too difficult as reinvestigation of the timeline is merely “rational guessing” and some diseases have radically high transmission rates e.g. mostly modelled by exponentials. So, it can be difficult to distinguish between first and secondary cases. Thus, instead of focusing on actual “local sources” states focus on all cases as local sources. As time progresses, this also becomes increasingly hard to control due to the number of cases and has cost and logistic complications within. Because every identification means a test and every test means a possible source going to a hospital, laboratory and contacting crowds including doctors and nurses.

So, we suggest an alternative strategy; to use aerial drones for specimen collection and case identification. The core aim of the drone operations is to provide a faster logistic solution for case reporting, to increase the number of tests that can be conducted in a day and to make tests more remote, so less people are contacted in the process of testing. Hence, more time is provided to act and take precautions regarding the spread; such as increasing the capacity of hospitals and strengthening the health system.

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 duration 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, duration 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 employees of companies that hire us to combat the disease, or regular people in case of a governmental operation. For these people, our operation can get them tested without having to go to a hospital, which saves them from being exposed to the bacteria from other sick people. They also lose less time when being tested and they can wait for the result in the comfort of their own home instead of inside the hospital or at the doctor.

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

For the enterprises that use our product, we help them by identifying the sick people in their network and isolating them from the rest of the people. This will keep more employees healthy to work in the company, which saves a lot of downtime in the company, and keeps people from feeling endangered by the virus. By having more people being able to go to work, the company loses less money in a virus outbreak, which might save them from going bankrupt.

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.

Pre-Test Phase

Selection of the Regions

After we are hired, we will select the region where the most people of interest are located. This can range from employees in case of a company, to all people in the area in case of a government operation. When we have found possible regions, we will place the operation basis in the center of the region and then select the first batch of people to be tested.

Selection of People to be Tested

For the first wave of tests, we will randomly test people that are inside the chosen area. After the first wave, we will use predictive analysis to find out people in the area that are most likely to be infected by the disease. We will test these people and use the newly gained data to make a better analysis of the situation at hand.

Test Phase

Approach and Return of Drones to Bases

On the day that the person is supposed to get tested, they get a notification from a phone application. This notification tells them that a drone will visit them today to collect specimen. A few minutes before the drone arrives, they get another notification that the drone is nearby. When the drone has arrived, the people will put their specimen in a plastic container marked with their name. The drone will wait at the door until enough specimen is given or until x minutes have passed since its arrival. The drone then leaves and flies to the next destination, either another person or the operation basis.

How Tests are Conducted

The tests are conducted in a machine in the operation basis. When the test results are known, the people are notified using their phone application about the result.

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

---

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

---

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:

-Worked on Wiki page for Description of the Operation

Venislav:

- Worked on simulating the work of the drones

---

10- 16/03/2020

Efe:

-Updated Operations Strategies, Problem Statement, Mathematical Model

-Researched on test conducting, intervention policies, history and effects of outbreaks

-Worked on MATLAB Simulation for Pop. Dynamics

---

17- 23/03/2020

Efe:

-Updated Subject, Problem Statement, Operation Strategies

-Researched on viral outbreaks, intervention policies, history and effects of outbreaks, SEIQHR Model

---

24- 30/03/2020

---

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.

Tulchinsky, T. H., & Varavikova, E. A. (2014). Communicable Diseases. The New Public Health, 149–236. doi: 10.1016/b978-0-12-415766-8.00004-5