Let the decimal representations of numbers and be given as: , (where can be 0), and let be the count of numbers such that and ( can also be 0). (Here, is considered the same as ).

Prove:

If the sum of all positive divisors (including itself) of a positive integer is , then is called a perfect number. For example, the sum of the positive divisors of 6 is , hence 6 is a perfect number.
Prove: There does not exist a perfect number of the form , where are positive integers, and are odd primes.

Let be positive integers, and let all be -th roots of unity, satisfying:
Prove that among , each -th root of unity appears the same number of times.

The Reproduction Rate of Diseases: Understanding the Spread of Infections

In epidemiology, the reproduction rate of a disease refers to how quickly and extensively an infectious disease can spread within a population. This concept is essential in predicting outbreaks and designing effective public health responses.



1. Basic Reproduction Number ()

The basic reproduction number, denoted as , represents the average number of secondary infections produced by a single infected individual in a fully susceptible population.

If:

• : The disease will eventually die out.
• : The disease will remain stable without increasing or decreasing.
• : The disease will spread through the population.

The value of depends on several factors:

• Infectious Period (): How long an individual remains contagious.
• Contact Rate (): The average number of susceptible individuals encountered by an infected person per unit time.
• Transmission Probability (): The likelihood that a contact results in infection.

The basic reproduction number is calculated using the formula:


Alternatively, for more complex models:


where:

• is the number of susceptible individuals.
• is the total population size.

2. Effective Reproduction Number ()

As the disease spreads, the number of susceptible individuals decreases, which affects the reproduction rate. The effective reproduction number () adjusts for these changes:


where represents the proportion of the population still susceptible.

If , the disease will gradually decline. Public health measures, such as vaccinations and social distancing, aim to reduce below 1.

3. Examples of Values for Different Diseases

Here are approximate values for some well-known diseases:

• Measles:
• COVID-19 (original strain):
• Influenza:
• Ebola:

4. Herd Immunity and the Reproduction Rate

Herd immunity occurs when a sufficient portion of the population is immune, reducing the spread of the disease. The herd immunity threshold is calculated by:


where is the fraction of the population that must be immune to prevent further outbreaks.

For example, if , at least:


of the population must be immune.

5. Modeling Disease Spread Using Differential Equations

A common mathematical model for disease spread is the SIR model, which tracks the number of Susceptible (S), Infected (I), and Recovered (R) individuals over time:


where:

• is the transmission rate.
• is the recovery rate.
• .

By solving these differential equations, we can predict the course of an epidemic and the effectiveness of intervention strategies.

6. Public Health Implications of the Reproduction Rate

Understanding and controlling is crucial for managing epidemics:

• Vaccination Programs: Aim to reduce by increasing immunity, thus lowering .
• Social Distancing: Reduces the contact rate , decreasing the spread of disease.
• Quarantine and Isolation: Limits exposure of infected individuals to others.
• Contact Tracing: Identifies and isolates potential secondary cases.

7. Conclusion

The reproduction rate is a fundamental concept in epidemiology, governing how diseases spread through populations. By controlling , public health authorities can prevent and mitigate outbreaks. Mathematical models like the SIR model provide valuable insights into the dynamics of infectious diseases and the impact of intervention strategies.

References

• Anderson, R. M., & May, R. M. Infectious Diseases of Humans: Dynamics and Control (Oxford University Press, 1991).
• Keeling, M. J., & Rohani, P. Modeling Infectious Diseases in Humans and Animals (Princeton University Press, 2008).
• Kermack, W. O., & McKendrick, A. G. "A Contribution to the Mathematical Theory of Epidemics," Proceedings of the Royal Society, 1927.
• Wikipedia: Basic Reproduction Number

Let be a set that intersects all non-finite integer arithmetic progressions, be the set of prime divisors of and be the set of prime divisors of . Suppose . Prove that ,

In the triangle let be the foot of the altitude from to the side and , , be the incenter, -excenter, and -excenter, respectively. Denote by and the other intersection points of the circle with the lines and , respectively. Prove that .

Proposed Michal Jan'ik - Czech Republic

On a cyclic quadrilateral , let and denote the midpoints of and . Let be the projection of onto and let be the reflection of over the midpoint of . Assume lies in the interior of quadrilateral . Prove that .

Let be a triangle, and let be centered at , and tangent to line at , respectively. Let line intersect again at and let line intersect again at . If is the other intersection of the circumcircles of triangles and , then prove that lines , , and either concur or are all parallel.

Advaith Avadhanam

Let , where are reals. Let be a circle with diameter and let be any other point on . Line meets the x-axis again at . Prove that angle .