Question

Hi, I'd really appreciate some help!

For an infectious disease X, its infection rate and recovery rate were estimated as 70% and 25%.

What is the basic reproductive number of this disease?

What is the minimum proportion of a population to vaccinate so as to avoid the outbreak?

Answer 1

Page No: Given - infectious disease X Infection Rate () = 70% = 70 = 0.7 Recovery Rate (0) = = 25% = 25 = 0.25 Reproduction No. = ? 100 too Answer: - Reproduction No. (Ro) can be calculated as ratio of Infection or Transmission Rate (8) divided by Recovery Rate (1) B = Bette and Y- Gamma 0.7 from defination & Ro= Y 0.25 Ro = 2.8 B ** Minimum proportion of population to vaccinate so as to avoid outbreak. is calculated by egn. of re from Herd immunity, v=1-16 = 0.64 (where seis Herd immunity threshold). * So 64% population need to be vaccinated in order for disease to become stable. if you like Auswer pls thumbs up.

Explantation and concept from theory:-

• Susceptible (S) (not infected), Infectious (I), and Recovered (R) (that is, vaccinated or recovered with immunity). R Figure 7: The SIR model of the spread of disease. We will make three assumptions of how these categories of the population relate to each other: 1 The number of infected people increases at a rate proportional to both the number of infected and the number of susceptible people. The number of susceptible people decreases at this same rate. The ratio involved is the transmission rate B (beta), the same as in the SI model. 2 The number of covered people increases at a rate proportional to the number of infected people. The ratio involved is called the recovery rate y (gamma). 3 A susceptible person who catches the disease becomes infectious immediately. We model each of S, I and R as functions of a time variable t, which is measured in days. Our variables then become S = S(t) = number of susceptible individuals at timet I = I(t) = number of infected individuals at time R = R(t) = number of recovered individuals at time t. Just as in the SI model, it makes more sense to consider the proportions of each type of individual rather than their actual number. If the total population again has size N, the new variables become: s(t) = S(t) N = the proportion of susceptible individuals at time t I(0 i(t) = = the proportion of infected individuals at timet N R(1) = the proportion of susceptible individuals at time t. N Then we can write the equation s(t) + i(t) + r(t) = 1.

As the proportion of recovered people increases, the proportion of susceptible people decreases, as shown respectively by the yellow and blue lines. The red line represents the trend of infected individuals, and how their numbers change over time. In this example, both the blue and red lines decrease until they hit zero. This means that the entire population has become infected with the disease and moved into the recovered phase. When the blue line does NOT reach zero, this means the disease has died out before everyone in the population has contracted it. In the SIR model, the basic reproduction number R, can be calculated using the ratio of the transmission rate to the recovery rate: Ro B Y

Herd Immunity Herd immunity occurs when a critical proportion of a susceptible population is immunised against a contagious disease, giving overall protection to the remainder of the 'unprotected community (herd). This minimises the chance of an outbreak occurring and allows those who are not eligible for vaccines, such as infants and pregnant women, to also receive some protection from the disease. Herd immunity works because it is more difficult for diseases to spread between individuals if large numbers are already immune, as this breaks the chain of infection. A graphical representation of this is illustrated in Figure 11 below. In Figure 11 we see that when Ro = 3, one infectious person generates 27 new cases of infection after just three generations of the disease. But if two-thirds of the population are immunised then only 1 of those 27 people gets infected after three generations. This shows that far more people in the population are protected from the disease than just those receiving the vaccination. The herd immunity threshold (v) is the proportion of the population that needs to be immune in order for an infection to be contained. This is reached when each new case leads to just a single new case, leading to the infection becoming stable within the population; that is, R = 1. (Recall that when R> 1 we have an epidemic.)

R-3 V = 2/3 Figure 11: The effects of immunity on a susceptible population. Here a solid line means the disease is transmitted, and a dotted line means the disease is not transmitted. The grey individuals are those with immunity. Initially everyone in the population is either vaccinated or they are susceptible, so S + V = 1. Rearranging this to get s = 1 - v and substituting into Equation 11 gives us Ro(1 – v) = 1. Dividing both sides by Ro gives 1 1-V= Ro and a final rearrangement gives us a formula for v in terms of Ro: V=1- Ro

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