2026-367

Consider the context of a disease spreading through a global network. Which of the following statements is false?

a) If a disease has different fatality rates depending on specific factors, such as COVID-19 being more dangerous for older people, then the best vaccination strategy for minimizing the number of deaths would be to prioritize high-risk groups. However, if the objective is to eradicate the disease, the best strategy would be to prioritize nodes with the highest number of links, since these nodes contribute more to disease transmission.

b) A disease with a very high fatality rate tends to be less likely to spread globally and become a pandemic, especially if infected individuals die before transmitting the disease to many others. In network terms, death removes a node from the transmission network, reducing the number of possible future infections.

c) If a vaccine exists and 90% of the population is vaccinated at random, the remaining 10% of unvaccinated individuals will not be affected at all by the existence of the vaccine.

d) Early identification of a disease, together with measures such as quarantine and social isolation, is one of the first and most important steps to prevent a local outbreak from becoming a pandemic.

e) None of the above is false.

Original idea by: Henrique Campos Padula

2026-366

The network below shows the contacts in a small population, modelled with the SI epidemic model.

Given a transmission rate β = 0.1, what is the characteristic time τ of the epidemic?

1. τ = 4
2. τ = 5
3. τ ≈ 2.86
4. τ = 0.5
5. None of the above

Original idea by: Gustavo P. C. P. da Luz

2026-365

How do assortative degree correlations typically influence the initial spread of a pathogen compared to neutral or disassortative networks?

1. They have no noticiable impact on epidemic dynamics, since the network scale regime influences the initial spread more than degree correlations.
2. They slow down the spread because hubs are disconnected from susceptible nodes.
3. They increase the epidemic threshold, making outbreaks less likely.
4. They accelerate the spread because highly connected nodes are more likely to be linked to other highly connected nodes.
5. None of the above.

Original idea by: Giuliano Macedo.

2026-364

A zombie infection spreads according to the SI model under the homogeneous mixing assumption. Researchers observe that faster zombies encounter healthy individuals more frequently, so the infection rate is proportional to the zombie walking speed:

$$\beta =c{v}_z,$$

where $v_z$ is the average zombie speed and $c$ is a proportionality constant.

A mutation doubles their average speed. The contact network remains unchanged, with

$$⟨k⟩=8$$

and

$$⟨k^2⟩=100.$$

According to the SI model on networks, by what factor does the characteristic time change after the mutation?

A) It is reduced by half.

B) It doubles.

C) It increases by a factor of four.

D) It remains unchanged.

E) None of the above.

Original idea by: George Gigilas Junior

2026-363

Analyze the following statements about the impact of degree correlations on spreading phenomena in networks, and determine whether each statement is True (T) or False (F):

1. In the SIS model applied to scale-free networks whose second moment diverges, the epidemic threshold remains zero regardless its degree correlation (assortative, disassortative, or neutral).
2. Assortative correlation accelerates pathogen propagation because hubs, which are typically among the first nodes to become infected, preferentially connect to other hubs, facilitating rapid transmission through the most connected parts of the network.
3. In scale-free networks, a weakly infectious pathogen will always die out despite hubs existence, unlike random networks where the homogeneous connectivity allows the pathogen to spread steadily across the population.
4. In networks with finite second moment, disassortative correlations raise the epidemic threshold λ_c, making it harder for a pathogen to sustain itself in the network, while assortative correlations produce the opposite effect by lowering λ_c.

Choose the correct alternative:

A) T-T-T-F

B) F-T-F-T

C) T-T-F-T

D) T-F-T-T

E) None of the above

Original idea by: Gabriela Caspa

2026-362

Consider an epidemic spreading on a large scale-free network whose degree distribution follows

$$P(k) \sim k^{-\gamma}$$

with \(2 < \gamma < 3\). According to Network Epidemics theory field, which of the following statements is correct?

A) The epidemic spreads more slowly than in a random network because hubs act as bottlenecks.

B) In the SIS model, the epidemic threshold \(\lambda_c\) tends to zero as the network size becomes very large (\(N \to \infty\)).

C) The characteristic spreading time increases with the second moment \(\langle k^2 \rangle\).

D) The early-time behavior of the SI model is linear in time rather than exponential.

E) None of the above.

Original idea by: Carlos Trindade

2026-359

A research team is modeling the spread of a highly contagious pathogen in a densely packed, fully connected community where the homogeneous mixing assumption holds. 

Because they are modeling the disease beyond its initial early stages, they cannot use the simple exponential approximation. Instead, they use the exact SI model equation:

i(t)=i0exp(βt)1−i0+i0exp(βt)

At the start of the observation (t=0), exactly 10% of the population is infected (i0=0.1). After a certain number of days t, the infection's exponential growth factor reaches eβt=9.

What is the fraction of infected individuals i(t) in the population at this time?

a) 81.8%

b) 33.3%

c) 10.0%

d) 50.0%

e) None of the above.

Original idea by: Matheus de Oliveira Saldanha

2026-357

When modeling the spread of a pathogen or an information cascade, the choice between the SI, SIS, and SIR compartmental frameworks depends entirely on the biological or behavioral traits of the agents involved. Under the homogeneous mixing assumption, each of these three classical models leads to a fundamentally different outcome in its final regime.

Which of the following statements correctly identifies the long-term behavior (final regime) of the SIR model and explains how it differs from the SI and SIS models?

A) In the SIR model, the fraction of infected individuals eventually drops to zero because infected individuals transition into a removed state where they develop permanent immunity or die.

B) The SIR model is the only framework where the entire population eventually ends up infected at the same time, unlike the SIS model which always maintains a mix of healthy and sick individuals.

C) In the final regime of the SIR model, the system reaches a steady endemic state where a fixed, non-zero fraction of the population remains actively infected forever.

D) The SIR model differs because it completely lacks an initial exponential growth regime, making its spread linear and predictable from day one.

E) None of the above.

Original idea by: Maria Luiza Ramos da Silva

2025-308

Consider a pathogen spreading on a highly heterogeneous contact network whose degree distribution follows a power law with exponent between 2 and 3.  Based on network epidemic theory and immunization strategies, which of the statements below is correct ?

A) In networks of this type, the epidemic threshold is positive. Therefore, random immunization is sufficient to stop the epidemic if at least 30% of the population is vaccinated.

B) Strategies that immunize the acquaintances of randomly selected individuals tend to target high-degree nodes, increasing the epidemic threshold.

C) The strategy of immunizing acquaintances of random individuals does not improve epidemic control in real networks, because there is no correlation between the degree of a node and the degree of its neighbors.

D) For networks with diverging second moment in their degree distribution, SI and SIS models have the same epidemic threshold, since hubs do not influence their dynamics.

E) None of the above.

 Original idea by: Pedro Zaffalon da Silva 

2025-307

Which of the following statements correctly describes the relationship between network structure and the spread of pathogens, ideas, or computer viruses?

A) The structure of the underlying network is irrelevant to the spreading process; the only factor that matters is the inherent contagiousness of the agent (virus or idea). 

B) Biological diseases spread through contact networks, whereas digital viruses and social phenomena spread randomly without following any specific network topology. 

C) The contact network serves as the transmission medium for spreading processes, meaning that the network's topology significantly determines the speed, reach, and pattern of the diffusion. 

D) Spreading phenomena are strictly limited to biological systems (like the SARS outbreak), and network science concepts cannot be effectively applied to digital or social contexts.

E) None of the above.

Original idea by: Augusto Cruz

2025-306

Consider the propagation of a virus modeled by the SIS (Susceptible-Infected-Susceptible) system in two distinct networks:

• A Random Network (Erdős-Rényi).
• A Scale-Free Network.

Both networks have the same number of nodes and the same average degree.

Given that the epidemic threshold is defined by the critical spreading rate, below which the virus dies out exponentially, select the correct statement regarding the behavior of this threshold and immunization strategies:

1. Since both networks have the same average degree , the epidemic threshold will be identical for both, as the average connection density determines the initial spreading speed in any topology.
2. The Random Immunization strategy is equally effective in both types of networks, meaning that the same fraction of nodes have to be immunized in both networks to stop the epidemic.
3. In Scale-Free Networks, with degree exponent \(\gamma\) between 2 and 3, the hubs are less relevant, and the network behaves similarly to a random network.
4. In the Scale-Free Network, the epidemic threshold is given by \(\langle k \rangle / \langle k^2 \rangle\).
5. None of the above.

Original idea by: Giancarlo Maldonado Cárdenas

2025-305

Consider immunization strategies discussed in the context of network epidemics. Which of the following statements is incorrect?

1. Random immunization performs poorly in scale-free networks because it rarely targets highly connected hubs.
2. Targeted immunization raises the epidemic threshold by removing the hubs responsible for rapid spreading.
3. Random immunization always outperforms targeted immunization in networks where \(\langle k^2\rangle\) is finite.
4. Immunizing a randomly selected neighbor of a randomly chosen individual tends to target high-degree nodes due to the friendship paradox.
5. None of the above

Original idea by: Jhonatan Cléto

2025-304

In the context of epidemic spreading on networks, which statement correctly explains why scale-free networks allow epidemics to spread more easily than random networks?

1. Because scale-free networks have a lower average degree \(\langle k\rangle\), making it harder to contain infections.
2. Because scale-free networks follow homogeneous mixing, allowing each node the same chance to infect others.
3. Because scale-free networks contain hubs with very high degree, causing \(\langle k^2\rangle\) to diverge and making the epidemic threshold \(\lambda_c\) approach zero.
4. Because scale-free networks have stronger community structure, slowing down infection and increasing the epidemic threshold. 
5. None of the above.

Original idea by: Thiago Soares Laitz

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 are 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