2025-294

You are a systems analyst at a major data center, and you are tasked with evaluating the robustness of your facility's network infrastructure against random node failures.

Your analysis of the network topology reveals that the average degree <k> of the nodes is 6.25. Based on your understanding of robustness, what breakdown threshold does the network at the datacenter need to have to display enhanced robustness?

a) 0.85

b) 0.84

c) 0.17

d) 0.16

e) None of the above

Original idea by: Alexandre Petrachini

2025-303

In the context of the classic SI, SIS, and SIR epidemiological models, which of the following statements is incorrect?

A. All three models predict that the epidemic ends only after every individual has been infected at least once.

B. In the Susceptible-Infected (SI) model, each individual in the population is assumed to be either healthy or infected.

C. In the Susceptible-Infected-Susceptible (SIS) model, an individual who becomes infected with the disease can recover and return to the susceptible state, allowing reinfection.

D. In the Susceptible-Infected-Recovered (SIR) model, individuals who have been infected acquire immunity or die from the disease and therefore do not become susceptible again.

E. None of the above

Original idea by: Mylena Roberta

2025-302

Consider the SI model (Susceptible-Infected) under the homogeneous mixing assumption. In this model, the spread of the virus is driven by random contacts between individuals.

You are given the following data for a specific population at time \(t\):

• Total Population (\(N\)): 10,000 individuals

• Current Susceptible individuals (\(S\)): 9,500

• Current Infected individuals (\(I\)): 500

• Average degree (contacts per individual per unit time, \(\langle k\rangle\)): 12

• Transmission probability (\(\beta\)): 0.05

At this specific moment, what is the average rate of new infections?

A) 275

B) 285

C) 570

D) 600

E) None of the above

Original idea by: Alexandre Petrachini

2025-301

Consider the following well-known pathogens and the following time evolution graph of the fraction of infected individuals using the Susceptible-Infected-Susceptible Model (SIS):

Disease	Transmission	R0
SARS	Airborne droplet	2-5
Polio	Fecal-oral route	5-7
Measles	Airborne	12-18

Which of the curves could represent the pathogens presented?

A. f1 - SARS,  f2 - Polio,  f3 - Measles

B. f1 - Measles,  f2 - SARS,  f3 - Polio

C. f1 - Polio,  f2 - SARS,  f3 - Measles

D. f1 - Measles,  f2 - Polio,  f3 - SARS

E. None of the above

Original idea by: Jhessica Silva

2025-300

In the homogeneous-mixing SI model, which statement correctly describes the analytical solution for the time required to reach a fraction \(i(t) = 0.5\) of infected invividuals?

a) It depends only on the total population size and not on the infection rate \(\beta\)

b) It always occurs at the same time, regardless of the initial condition

c) It is inversely proportional to the contact density but not to \(\beta\)

d) It increases linearly with the initial number of infected individuals

e) None of the above. 

Original idea by: Gabriel Sato

2025-299

Under the Homogenous Mixing hypothesis, why are early quarantines one of the recommendations for the slow down of infection spreading?

A) For the Susceptible-Infected-Removed Model, quarantines are a good tool to maintain \(R_0\) above 1, reaching the "Disease-free State", where epidemics die out due to the negative \(\tau\).

B) The quarantine decreases the likelihood of a disease being transmitted, which accelerates the "Disease-free State", indicating that the infection will die exponentially.

C) In the Susceptible-Infected Model, individuals area prone to be infected again after recuperation, therefore quarantines are paramount to avoid new infections on the same individual.

D) By restricting the number of connections between individuals, the characteristic time will be greater, giving researchers more time to discover new vaccines and medicines.

E) None of the above.

Original idea by: João Medrado Gondim

2025-298

In a population of size \(N=10000\), a disease spreads according to the SI model.
The average degree is \(\langle k\rangle= 20\), the transmission probability per contact is  \(\beta = 0.04\), and initially one individual is infected.

Using the SI model solution with the homogeneous mixing hypothesis, compute the expected number of infected individuals after \(t=5\) time units.

Round to the closest integer.

A) 54

B) 350

C) 850

D) 1.100

E) None of the above

Original idea by: Yan Prada