=Paper=
{{Paper
|id=Vol-2970/plppaper2
|storemode=property
|title=Modelling Infectious Disease Dynamics with Probabilistic Logic Programming
|pdfUrl=https://ceur-ws.org/Vol-2970/plppaper2.pdf
|volume=Vol-2970
|authors=Felix Weitkämper,Beatrice Sarbu,Kailin Sun
|dblpUrl=https://dblp.org/rec/conf/iclp/WeitkamperSS21
}}
==Modelling Infectious Disease Dynamics with Probabilistic Logic Programming==
https://ceur-ws.org/Vol-2970/plppaper2.pdf
Modelling infectious disease dynamics with
probabilistic logic programming
Felix Weitkämpera , Beatrice Sarbua and Kailin Sunb
a
Institut für Informatik der LMU München, Oettingenstr. 67, 80538 München, Germany
b
Department Biologie I der LMU München, Menzinger Str. 67, 80638 München, Germany
Abstract
Motivated by the SARS-CoV-2 pandemic, we implemented a stochastic, network-based model of infectious
disease transmission in the probabilistic logic programming language ProbLog. In this contribution,
we show how probabilistic logic programming lends itself to very concise, transparent and adaptable
modelling of infectious disease dynamics. We illustrate how some key features can contribute to succinct
and expressive representation of epidemiological models. Our work makes full use of the compact
relational representation, the support for stratified negation and flexible probabilities evaluated at run
time, which are supported by the ProbLog 2 engine.
Keywords
ProbLog, SI-model, Disease Dynamics, Epidemiology Probabilistic programming
1. Introduction
Motivated by the SARS-CoV-2 pandemic and its societal impact, we implemented stochastic
models of infectious disease dynamics in the declarative probabilistic logic programming lan-
guage ProbLog [1]. In this contribution, we demonstrate how the features of ProbLog contribute
to a concise and transparent implementation.
To the best of our knowledge, this is the first described use of probabilistic logic programming
for epidemiological modelling. Its suitability for expressing network relationships is well-
supported, however, and indeed link prediction in biological networks was used as a motivation
in the original publication of the ProbLog language [1, 2, 3, 4, 5].
Epidemiological models fall into two broad categories: compartmental and network-based
models. The paradigmatic example of a compartmental model is Kermack and McKendrik’s
classic susceptible-infected-recovery (SIR) model [6]. This model, illustrated in Figure 1, divides
the population into three groups: susceptible (individuals who are not immune to the disease),
infected (individuals who currently carry the disease and are able to spread it), and recovered
(individuals who have recovered and are therefore immune). Differential equations model the
transition of the proportion of the population moving from one group to another during each
time step. There are certain assumptions of SIR models which may not be realistic in every
PLP 2021
Envelope-Open felix.weitkaemper@lmu.de (F. Weitkämper); B.sarbu@campus.lmu.de (B. Sarbu); sun.kailin@campus.lmu.de
(K. Sun)
Orcid 0000-0002-3895-8279 (F. Weitkämper)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
Figure 1: A typical SIR model, showing Susceptible, Infected and Recovered groups with their modes
of transition
Figure 2: An example of a weighted network
situation; for example, SIR models assume a well-mixed, homogeneous, and large population
[7]. Intermittent lockdowns, travel restrictions and changes in social behaviour over the course
of the SARS-CoV-2 pandemic mean that populations are not always well-mixed. Individual
differences in vulnerability to SARS-CoV-2 (and whether or not individuals are asymptomatic)
negate the assumption of homogeneity. Indeed, we can assume that one or more of the above
assumptions have been violated, since there is evidence that that the spread of SARS-CoV-2
is fat-tailed [8]. In addition, a model based on the assumption of a large population may not
be helpful in modelling the spread of disease on a smaller scale, for example when combatting
local outbreaks.
In response to those limitations, epidemiologists developed stochastic or network models, in
which disease is transmitted along the edges of a contact graph. It is such a stochastic approach
that we are implementing here. A network-based model incorporates the idea of the SIR groups
into a local network-based structure. Since the model is probabilistic, random effects in the
spread of disease are accounted for. Network-based modelling also removes the assumption that
there is an infinite population size. Instead, individuals are modelled as nodes in an undirected
graph, with edges between individuals that have contact. The disease then spreads by infection
along the nodes of the graph in discrete time steps. To model different levels of contact, one can
use a weighted graph, in which the infection probability depends on the weight. An example
network is shown in Figure 2. For a thorough introduction to network-based models, see [9].
We have chosen to use ProbLog as a declarative probabilistic programming language since it
grants immediate and transparent access to the parameters and underlying assumptions of the
model. This is particularly important when modelling such a new disease as COVID-19, where
medical and epidemiological knowledge changes rapidly, and there is considerable uncertainty
about the appropriate values for the various modelling parameters. In addition, probabilistic
logic programming allows for a very compact representation of the underlying processes.
Among probabilistic logic programming languages, ProbLog stands out with its well-
maintained ProbLog 2 engine [10], which also supports useful additional features such as
flexible probabilities. Additional parameters and assumptions can be appended easily, so the
model adapts to changing demands during the course of a pandemic. ProbLog 2 is also able
to generate simulations as well as to calculate exact probabilities of infection, increasing the
versatility of our approach [11].
As far as we know, current declarative approaches to modelling infectious disease dynamics
are limited to compartmental rather than network-based models and are based on dedicated
domain-specific languages. Those DSLs are themselves implemented imperatively as part of an
integrated simulation suite. A state-of-the-art example of this is the EMULSION framework
[12]. While such a DSL can be tailored to the application domain, it is more difficult to extend or
verify its ad hoc implementation machinery. In contrast, our modelling can rely on the general
purpose implementation ProbLog 2 of the ProbLog probabilistic logic programming language.
We present three examples of our modelling approach in the following sections. First, we
explain the basic approach using a simple network model based on class enrolment data, with a
certain probability of infection between classmates. The basic set-up already showcases the
modularity and brevity of the code. Then, we introduce an extension to accommodate for
resistance to the disease, which could be acquired through surviving the illness or through
a vaccination programme. This uses stratified negation as implemented in ProbLog. In the
Section 4, we discuss an alternative model based on distance metrics, which takes into account
the precise location of the individuals. This makes use of flexible probabilities as a powerful
and flexible modelling tool.
2. Epidemiological modelling based on SI-Model
In the basic model, we assume a closed population divided into two groups: susceptible in-
dividuals and infected individuals. Infected individuals contribute to the spread of the virus.
Once a susceptible individual is infected, he or she belongs to the infected individuals, and
once the infectivity period has lapsed, they move back to the susceptible group. We assume
a closed world of individuals. Since our motivation comes from an academic setting, with its
regular recurring intervals of lecturing weeks, discrete time steps are ideally suited to model
our domain.
At the beginning of the period under consideration, most people are still susceptible. Through
contact with infected individuals, they may become infected with a certain probability and may
also spread the virus during the next week. In our ProbLog implementation, the epidemiological
parameters are completely separated from the individuals and their mutual interaction. This
makes our model very agile, as it can rapidly be adapted to different models of infectious disease
transmission or different connections of individuals, and makes maintenance of the code much
easier.
The entire infection logic of our implementation is contained in the following code:
Listing 1: SI infection logic
1 % In every given week, there is a 1 percent chance of infection outside of the setting.
2 0.01::ill(N,STUDENTx) :- student(STUDENTx), week(N).
3 0.01::ill(N,STUDENTx) :- student(STUDENTx), week(N), vulnerable(STUDENTx).
4
5 % 10 percent of students are particularly vulnerable to disease.
6 0.1::vulnerable(STUDENTx) :- student(STUDENTx).
7
8 % The average period of infectivity is two weeks.
9 0.5::ill(B, STUDENTx):- ill(A,STUDENTx), week(B), B is A+1.
10
11 % A student has a 5 percent chance of infection from a diseased classmate.
12 0.05::ill(B,STUDENTx) :- ill(A,STUDENTy), classmates(STUDENTx, STUDENTy, COURSE), week(B),
13 B is A+1, vulnerable(STUDENTx).
14
15 0.05::ill(B,STUDENTx) :- ill(A,STUDENTy), classmates(STUDENTx, STUDENTy, COURSE), week(B),
16 B is A+1.
The network of relations between different students was implemented by supplying enrolment
data as a Prolog knowledge base and adding the following clauses to compute the network.
The predicate c l a s s m a t e s / 3 maps the relationship between persons and their lectures under the
following clause:
Listing 2: Network of connections
1 classmates(PERSONx ,PERSONy, COURSE) :-
2 participate(COURSE, PERSONx), participate(COURSE, PERSONy), PERSONx \== PERSONy.
If two persons are attending the same course and the first person is different from the second
person, then they are classmates. Note that this naturally supports a structure in which students
attending more than one course together will have an independent probability in each class
to infect each other. Those different probabilities of infection, together with the additional
extrinsic infection risk, are internally processed with a noisy-or combination function; that
means, they are treated as independent sources of infection, with illness occurring if any one of
the sources causes it.
The predicate v u l n e r a b l e is used to model natural variation in the likelihood of infection.
This could come from their age, their medical history or from behavioural patterns. It could
also be used to model the effects of possibly imperfect vaccination. While a perfect vaccination
can also be modelled in this way, by removing the possibility of non-vulnerable individuals to
contract the disease, it is better treated in the context of acquired resistance using the methods
of Section 3. We again exploit the noisy-or combination to set an explicit additional likelihood
of infection of vulnerable students. In Listing 1 above, vulnerability doubles effective exposure
time by adding an additional and equal chance of infection to the baseline. On the other hand,
various protective measures can affect not just an individual’s own likelihood of contracting
the illness, but also their chance of spreading it to others. This could be modelled in a similar
fashion.
The chance of infection outside of the setting can be added or left out depending on the
purpose of the model. Such a choice would have no impact on the remainder of the code.
The varying time of infectiousness is modelled by setting the chance to be infectious in a
week following a previous week of infection to 1/𝑙, where 𝑙 is the expected time of infectivity in
weeks.
The data provided consists firstly of a list of weeks matching the term length (or alternatively
the time period under investigation), and then the enrolment data of the courses. While ProbLog
provides a library that offers a simple interface to a CSV file, we found it more efficient to load
the data into a Prolog knowledge base, since reading the data from CSV seems to add significant
processing overhead. A small excerpt from such a knowledge base can be seen in Listing 3.
Listing 3: Courses and weeks
1 participate(course1,person1).
2 participate(course1,person2).
3 participate(course2,person3).
4 participate(course2,person4).
5 participate(course3,person2).
6 participate(course3,person4).
7 participate(course3,person5).
8
9
10 week(1).
11 week(2).
12 week(3).
13 week(4).
14 week(5).
ProbLog then allows for various modelling and simulation tasks. One could simply query the
likelihood of illness across different weeks in different populations. One could also manually
intervene by adding a fact of the form i l l ( 1 , p e r s o n 1 ) . and observe how quickly a single case of
infection can spread through the population. In principle, another interesting option would be
using e v i d e n c e to condition on the actual observations of infections at a given week. However,
using e v i d e n c e results in a significant increase in computation time when sampling from the
distribution. This limits its usability in larger datasets.
The effectiveness of various policies to mitigate the disease can then be determined using
straightforward extensions to the code. For instance, one could test the effectiveness of teaching
in person classes only in even-numbered weeks by adding an appropriate test for evenness to
the clause in lines 12-13 of Listing 1. If certain classes were to be moved online, they could
simply be removed from the knowledge base.
3. Incorporating resistance
To incorporate resistance into the model, we use ProbLog’s ability to work with stratified
negation. ProbLog can process negative conditions in the body of a clause as long as there are
no cycles involving such negative conditions. On the first glance, however, there do seem to be
cycles involved in a naive formulation of an SIR contingency:
Infection is prevented by resistance, but resistance also arises from a prior infection. This
logic could be naively implemented as in Listing 4.
Listing 4: Generic model of resistance
1 % Two different students are classmates if they attend the same class.
2 classmates(STUDENTx ,STUDENTy):- participate(COURSE, STUDENTx),
3 participate(COURSE, STUDENTy), STUDENTx\==STUDENTy.
4
5 student(STUDENTx) :- participate(COURSE, STUDENTx).
6
7 % In every given week, there is a 1 percent chance of infection outside the setting.
8 0.01::ill(B,STUDENTx) :- student(STUDENTx), week(B), \+ill(A, STUDENTx),
9 \+resistant(STUDENTx, A), B is A+1.
10
11 % 10 percent of students are particularly vulnerable to disease.
12 0.1::vulnerable(STUDENTx) :- student(STUDENTx).
13
14 % The average period of infectivity is two weeks.
15 0.5::ill(B, STUDENTx) :- ill(A,STUDENTx), week(B), B is A+1.
16
17 % A student has a 5 percent chance of infection from a diseased classmate.
18
19 0.05::ill(B,STUDENTx) :- ill(A,STUDENTy), classmates(STUDENTx, STUDENTy),
20 vulnerable(STUDENTx), \+resistant(STUDENTx, A), \+ill(A, STUDENTx), week(B), B is A+1.
21
22 0.05::ill(B,STUDENTx) :- ill(A,STUDENTy), classmates(STUDENTx, STUDENTy),
23 \+resistant(STUDENTx, A), \+ill(A, STUDENTx),
24 week(B), B is A+1.
25
26 % A student is resistant as soon as he has recovered from an illness.
27
28 resistant(STUDENTx, B):- ill(A, STUDENTx), \+ill(B, STUDENTx), week(B), B is A+1.
29
30 % Resistance lasts for the following weeks.
31
32 resistant(STUDENTx,B):-resistant(STUDENTx, A), week(B), B is A+1.
Indeed, the ProbLog 2 engine throws an exception, having detected negative cycles.
On closer inspection, however, the apparent negative cycle disappears when grounding out
the program with respect to the time in weeks. This is because illness causes future resistance,
while resistance in the past prevents future illness.
We therefore partially grounded out the program with respect to weeks using a variant of
the partial reducer from [13, Section 18.1]. The resulting unfolded program is passed to the
ProbLog 2 engine. This causes no further errors, and is in fact significantly faster than the
model of Section 2, since ProbLog no longer has to take into account multiple infections of the
same individual.
The program unfolded with respect to the knowledge base of Listing 3 can be found in
Appendix A.
Incorporating resistance unlocks modelling the full SIR cycle present in many infectious
diseases. The same mechanism can also be expanded to model the effects of vaccinations.
Vaccines take individuals from being susceptible direct to being resistant, essentially bypassing
the “infected” phase. This is represented by the grey arrow in Figure 1. This feature also allows
modellers to quickly adapt the model depending on the rate of vaccination in the local population.
Indeed, a constant rate of vaccination progress could be modelled by just a single additional
clause. In cases where more than one dose of vaccine is required to reach full immunity, we can
model this by reducing the vulnerability of individuals with one dose compared to unvaccinated
individuals, then counting fully vaccinated individuals as r e s i s t a n t . The exact value of this
reduction in vulnerability can be toggled easily depending on the latest data on vaccine efficacy,
and can be adapted with the publication of new data.
As every part of our program, the treatment of resistance can also be flexibly adapted. Should
it turn out, for instance, that resistance is obtained temporarily rather than permanently, this
can be expressed simply by annotating Line 32 of Listing 4 with a non-1 probability.
4. Incorporating distance
While the previous two models implemented a network approach, one can also use ProbLog to
implement a much more local model, which can explore different parameters of disease spread
as a direct function of the distance between two individuals.
A major advantage of ProbLog2 is the support of flexible probabilities, which can be used for
this implementation. This means that the probabilities are not set in the code, but are calculated
by arithmetic expressions at run time, making it possible to specify more complex models. The
probabilities of a possible infection are determined with the help of the Euclidean distance
function. We assume a two-dimensional linear world, where the distance between two points is
determined by the length of the straight line connecting them.
As the dispersal of the virus through droplets in the air is not fully understood, it is unclear
how infectiousness depends on distance [14]. In our framework, the program can be rapidly
adjusted for different models of airborne dispersal using the ProbLog 2 built-ins.
The position of the persons is determined by coordinates. An extract of a knowledge base is
shown in Listing 5.
Listing 5: knowledge base of individuals and their coordinates
1 point(person0, 2, 3).
2 point(person1, 3, 4).
3 point(person2, 6, 9).
4 point(person3, 10, 24).
5 point(person4, 12, 16).
6 point(person5, 7, 8).
The calculation of the contagion probability and the modelling of the spread of the disease
can be represented very compactly in a single clause, as shown in Listing 6. Additionally, the
function calculating the distance can easily be altered or replaced entirely to simulate different
models of spread.
Listing 6: Infection logic for the Euclidean distance model
1 P :: infects(PERSONx,PERSONy) :- point(PERSONx, X, Y),
2 point(PERSONy, A, B), PERSONx\==PERSONy,
3 D is sqrt((A-X)^2 + (B-Y)^2),
4 P is 0.1/(D^2), D < 10.
5 ill(PERSONx):-infects(PERSONx,PERSONy), is_ill(PERSONy).
6 ill(PERSONx):-is_ill(PERSONx).
Modelling disease likelihood based on relative location opens the door to further applications
where the locations of (or the distances between) individuals are known. One use case would be
as an epidemiological component of an agent-based modelling system. Agent-based modelling,
which has developed into a promising alternative to equational and network-based models of
infectious disease dynamics, simulates the behaviour of individuals according to given rules.
An agent-based epidemiological model can be divided into several layers, one of them being the
disease model itself [15]. Since agent-based models are often used to test the consequences of
epidemiological assumptions, a flexible and rapidly specifiable system would be desirable there.
Another opportunity would be to use the data from mobile tracking applications, such as the
Corona-Warn-App used in Germany. This currently implements a much more basic model of
disease spread, which has been criticised for being over-simplistic [16].
5. Experiments
To verify the feasibility of our approach, we tested the frameworks of Sections 3 and 4 for the
SIR model and distance-dependent modelling respectively. All experiments were performed on
a KVM virtual machine with 35 CPUs, 240 GB RAM and 17 GB swap running Ubuntu 21.04.
Sampling from a network-based SIR model We obtained student enrolment data from
the Faculty for Mathematics, Computer Science and Statistics of the LMU University of Munich.
This encompassed 14828 individual enrolment events of 4594 distinct students in 104 distinct
courses. We then used the ’sampling’ mode on the code of Appendix A to simulate the progress
of the pandemic with respect to a randomly sampled subset of students from the institute. More
precisely, we randomly sampled a given number of students and then included all enrolment
events involving a sampled student in the knowledge base. We used the query i l l ( _ , _ ) and
the sampling mode of ProbLog 2 to simulate the spread of illness across five weeks.
The following run times were obtained:
Number of students sampled Run time (seconds)
50 0.7
158 4.2
331 34.7
460 94
610 213
763 290
920 504
1078 670
This demonstrates that even for realistic connection graphs taken from real-life data and
with several hundred nodes, simulations can be performed in reasonable time.
Calculating the risk of infection from a distance-based model To test the feasibility of
the distance-based infection model of Listing 6, we implemented it with synthetic data of 1,000
people scattered randomly across a two-dimensional grid of 100*100 units. We also added a
background infection rate of 1% and initialised a known case by adding a fact i s _ i l l ( p e r s o n 1 ) .
We then queried the likelihood of infection for one particular individual. This exact inference
is performed in 0,76 seconds; without the cutoff distance of 10 encoded in line 4 of Listing
6, the computation is performed in 2,23 seconds. This makes our approach viable for the
implementation of more sophisticated risk assessments than is currently provided by exposure
notification applications
6. Conclusion
We implement a stochastic model for infectious disease dynamics in the probabilistic logic
programming language ProbLog. Our model differs from the classic SIR model by incorporating
network structures. This is an advantage in scenarios with a fixed, small or heterogeneous
population. Examples of these situations include students in university or school classes, patients
in a ward, or employees in offices. We show that distances between individuals can also be
accounted for, opening the door to further applications.
Our implementation also showcases how core features of ProbLog can be harnessed in the
context of epidemiological modelling. The relational representation streamlines the specification
process and leads to concise and transparent models. Meanwhile, support for stratified negation
allows modelling immunity after infection, as we can specify that the lack of prior immunity
is a prerequisite for infection. Flexible probabilities allow for the likelihood of infection to be
computed at run time; this is essential for incorporating distance as a parameter.
The core and advanced features of ProbLog are key to making the model flexible, transparent
and compact. The implementation is accessible for audiences outside of computer science and
individuals who are not familiar with programming. For example, the model can be understood
by academic researchers or administrative leadership. Modelling parameters and assumptions
can be read off directly, making the predictions traceable and trustworthy. Adaptability is
also important for rapid epidemiological modelling. In the case of a pandemic, decisions
must be made quickly, and often at a local level without access to extensive guidance. Our
implementation allows modelling with possibly incomplete knowledge, but is also flexible to
incorporate new data and new scientific background knowledge as it arises. Although our
work was originally motivated by the SARS-CoV-2 pandemic, the techniques can be used across
infectious diseases. We show that probabilistic logic programming can provide a versatile tool
for simulation and experimental evaluation of settings and parameters, potentially supporting
local decision-making in exceptionally challenging circumstances.
Typically, the greatest challenge in harnessing probabilistic logic programming for modelling
tasks such as this one is the high computational complexity of the task. To test the feasibility
of our approach, we have implemented an SIR simulation with a network taken from real-life
course enrolment data as well as a distance-based exact inference task predicting the risk of
infection from synthetic data. In both cases run times appropriate to the respective setting have
been achieved with as many as 1,000 distinct persons involved.
Acknowledgments
We would like to thank Ella Mayer and Yagiz Teker for many constructive conversations about
the topic, and François Bry for suggesting the unfolding mechanism used for partially grounding
the program. We would also like to thank Gregor Kleen and Michael Stübiger for providing
and anonymising the data used in our experiments, and Philipp Wendler for assisting us in
obtaining the requisite data protection clearance.
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A. Code sample
Listing 7: Unfolded program incorporating resistance
1
2 0.01::ill(1,STUDENTx) :- student(STUDENTx), week(1), \+ill(0, STUDENTx), \+resistant(STUDENTx, 0).
3 0.01::ill(2,STUDENTx) :- student(STUDENTx), week(2), \+ill(1, STUDENTx), \+resistant(STUDENTx, 1).
4 0.01::ill(3,STUDENTx) :- student(STUDENTx), week(3), \+ill(2, STUDENTx), \+resistant(STUDENTx, 2).
5 0.01::ill(4,STUDENTx) :- student(STUDENTx), week(4), \+ill(3, STUDENTx), \+resistant(STUDENTx, 3).
6 0.01::ill(5,STUDENTx) :- student(STUDENTx), week(5), \+ill(4, STUDENTx), \+resistant(STUDENTx, 4).
7
8 % 0.01::ill(B,STUDENTx) :- student(STUDENTx), week(B), \+ill(A, STUDENTx), \+resistant(STUDENTx, A), B is A+1.
9
10 0.1::vulnerable(STUDENTx) :- student(STUDENTx).
11
12 % The average period of infectivity is two weeks.
13 0.5::ill(1, STUDENTx):- ill(0,STUDENTx).
14 0.5::ill(2, STUDENTx):- ill(1,STUDENTx).
15 0.5::ill(3, STUDENTx):- ill(2,STUDENTx).
16 0.5::ill(4, STUDENTx):- ill(3,STUDENTx).
17 0.5::ill(5, STUDENTx):- ill(4,STUDENTx).
18
19 % 0.5::ill(B, STUDENTx) :- ill(A,STUDENTx), week(B), B is A+1.
20
21 0.05::ill(1,STUDENTx) :- ill(0,STUDENTy), classmates(STUDENTx, STUDENTy),
22 vulnerable(STUDENTx), \+resistant(STUDENTx, 0), \+ill(0, STUDENTx).
23 0.05::ill(2,STUDENTx) :- ill(1,STUDENTy), classmates(STUDENTx, STUDENTy),
24 vulnerable(STUDENTx), \+resistant(STUDENTx, 1), \+ill(1, STUDENTx).
25 0.05::ill(3,STUDENTx) :- ill(2,STUDENTy), classmates(STUDENTx, STUDENTy),
26 vulnerable(STUDENTx), \+resistant(STUDENTx, 2), \+ill(2, STUDENTx).
27 0.05::ill(4,STUDENTx) :- ill(3,STUDENTy), classmates(STUDENTx, STUDENTy),
28 vulnerable(STUDENTx), \+resistant(STUDENTx, 3), \+ill(3, STUDENTx).
29 0.05::ill(5,STUDENTx) :- ill(4,STUDENTy), classmates(STUDENTx, STUDENTy),
30 vulnerable(STUDENTx), \+resistant(STUDENTx, 4), \+ill(4, STUDENTx).
31
32 %0.05::ill(B,STUDENTx) :- ill(A,STUDENTy), classmates(STUDENTx, STUDENTy),
33 vulnerable(STUDENTx), \+resistant(STUDENTx, A), \+ill(A, STUDENTx), week(B), B is A+1.
34
35 0.05::ill(1,STUDENTx) :- ill(0,STUDENTy), classmates(STUDENTx, STUDENTy),
36 \+resistant(STUDENTx, 0), \+ill(0, STUDENTx).
37 0.05::ill(2,STUDENTx) :- ill(1,STUDENTy), classmates(STUDENTx, STUDENTy),
38 \+resistant(STUDENTx, 1), \+ill(1, STUDENTx).
39 0.05::ill(3,STUDENTx) :- ill(2,STUDENTy), classmates(STUDENTx, STUDENTy),
40 \+resistant(STUDENTx, 2), \+ill(2, STUDENTx).
41 0.05::ill(4,STUDENTx) :- ill(3,STUDENTy), classmates(STUDENTx, STUDENTy),
42 \+resistant(STUDENTx, 3), \+ill(3, STUDENTx).
43 0.05::ill(5,STUDENTx) :- ill(4,STUDENTy), classmates(STUDENTx, STUDENTy),
44 \+resistant(STUDENTx, 4), \+ill(4, STUDENTx).
45
46 %0.05::ill(B,STUDENTx) :- ill(A,STUDENTy), classmates(STUDENTx, STUDENTy),
47 %\+resistant(STUDENTx, A), \+ill(A, STUDENTx),
48 %week(B), B is A+1.
49
50 0.05::ill(1,STUDENTx) :- ill(0,STUDENTy), classmates(STUDENTx, STUDENTy),
51 vulnerable(STUDENTx), \+resistant(STUDENTx, 0), \+ill(0, STUDENTx).
52 0.05::ill(2,STUDENTx) :- ill(1,STUDENTy), classmates(STUDENTx, STUDENTy),
53 vulnerable(STUDENTx), \+resistant(STUDENTx, 1), \+ill(1, STUDENTx).
54 0.05::ill(3,STUDENTx) :- ill(2,STUDENTy), classmates(STUDENTx, STUDENTy),
55 vulnerable(STUDENTx), \+resistant(STUDENTx, 2), \+ill(2, STUDENTx).
56 0.05::ill(4,STUDENTx) :- ill(3,STUDENTy), classmates(STUDENTx, STUDENTy),
57 vulnerable(STUDENTx), \+resistant(STUDENTx, 3), \+ill(3, STUDENTx).
58 0.05::ill(5,STUDENTx) :- ill(4,STUDENTy), classmates(STUDENTx, STUDENTy),
59 vulnerable(STUDENTx), \+resistant(STUDENTx, 4), \+ill(4, STUDENTx).
60
61 %0.05::ill(B,STUDENTx) :- ill(A,STUDENTy), classmates(STUDENTx, STUDENTy),
62 vulnerable(STUDENTx), \+resistant(STUDENTx, A), \+ill(A, STUDENTx), week(B), B is A+1.
63
64 0.05::ill(1,STUDENTx) :- ill(0,STUDENTy), classmates(STUDENTx, STUDENTy),
65 \+resistant(STUDENTx, 0), \+ill(0, STUDENTx).
66 0.05::ill(2,STUDENTx) :- ill(1,STUDENTy), classmates(STUDENTx, STUDENTy),
67 \+resistant(STUDENTx, 1), \+ill(1, STUDENTx).
68 0.05::ill(3,STUDENTx) :- ill(2,STUDENTy), classmates(STUDENTx, STUDENTy),
69 \+resistant(STUDENTx, 2), \+ill(2, STUDENTx).
70 0.05::ill(4,STUDENTx) :- ill(3,STUDENTy), classmates(STUDENTx, STUDENTy),
71 \+resistant(STUDENTx, 3), \+ill(3, STUDENTx).
72 0.05::ill(5,STUDENTx) :- ill(4,STUDENTy), classmates(STUDENTx, STUDENTy),
73 \+resistant(STUDENTx, 4), \+ill(4, STUDENTx).
74
75 %0.05::ill(B,STUDENTx) :- ill(A,STUDENTy), classmates(STUDENTx, STUDENTy),
76 \+resistant(STUDENTx, A), \+ill(A, STUDENTx),
77 %week(B), B is A+1.
78
79
80 resistant(STUDENTx, 1):- ill(0, STUDENTx), \+ill(1, STUDENTx).
81 resistant(STUDENTx, 2):- ill(1, STUDENTx), \+ill(2, STUDENTx).
82 resistant(STUDENTx, 3):- ill(2, STUDENTx), \+ill(3, STUDENTx).
83 resistant(STUDENTx, 4):- ill(3, STUDENTx), \+ill(4, STUDENTx).
84
85 %resistant(STUDENTx, B):- ill(A, STUDENTx), \+ill(B, STUDENTx), week(B), B is A+1.
86
87 resistant(STUDENTx,1):-resistant(STUDENTx, 0).
88 resistant(STUDENTx,2):-resistant(STUDENTx, 1).
89 resistant(STUDENTx,3):-resistant(STUDENTx, 2).
90 resistant(STUDENTx,4):-resistant(STUDENTx, 3).
91 resistant(STUDENTx,5):-resistant(STUDENTx, 4).
92
93 %resistant(STUDENTx,B):-resistant(STUDENTx, A), week(B), B is A+1.