Human Health and Disease Set-5

Q1. For an infectious disease with basic reproduction number $R_0 = 5$, the herd immunity threshold is given by $H = 1 - \frac{1}{R_0}$. What minimum fraction of the population must be immunized to prevent sustained transmission?

Q2. A vaccine has efficacy $E = 80\%$. For a pathogen with basic reproduction number $R_0 = 2.5$, the effective reproduction number after vaccination is $R_e = R_0\big(1 - cE\big)$, where $c$ is the fraction of the population vaccinated. To ensure $R_e < 1$ the coverage must satisfy $c > \dfrac{1 - 1/R_0}{E}$. What is the minimum vaccination coverage $c$ (in percent) required?

Q3. A screening test for disease X has sensitivity $Se = 99\%$ and specificity $Sp = 98\%$. In the screened population the prevalence is $P = 1\%$. Using Bayes' theorem, the positive predictive value is $PPV = \dfrac{Se \times P}{Se \times P + (1 - Sp)\times(1-P)}$. If a person tests positive, what is the approximate probability that they actually have disease X?

Q4. In a community of 1000 people, 90% are vaccinated with a vaccine of efficacy $VE = 75\%$. During an outbreak the attack rate among unvaccinated individuals is $AR_U = 20\%$. Using $AR_V = AR_U(1 - VE)$, calculate the percentage of all cases that will occur in vaccinated individuals (round to one decimal place).

Q5. Statement I: Therapeutic monoclonal antibodies that neutralize tumour necrosis factor-alpha (TNF‑α) are effective in treating autoimmune diseases (e.g., rheumatoid arthritis) but can precipitate reactivation of latent tuberculosis. Statement II: TNF‑α is required for the formation and maintenance of granulomas that contain Mycobacterium tuberculosis; blocking TNF‑α disrupts granuloma integrity and permits latent bacilli to proliferate. Choose the correct relationship between the two statements.

Q6. A vaccine against disease X has efficacy $E=0.80$ and the disease basic reproduction number is $R_0=5$. Using the adjusted herd-immunity coverage formula $p_c=\dfrac{1-1/R_0}{E}$, the minimum fraction of the population that must be vaccinated to prevent sustained transmission is:

Q7. In a hostel of $80$ students one index case of measles returns from travel. Among the remaining students, $10$ had documented past infection (immune) and $5$ had confirmed protective vaccination. Over the next two weeks $16$ new cases occur (index case excluded). Using $SAR=\dfrac{\text{new cases among contacts}}{\text{susceptible contacts}}\times100\%$, the secondary attack rate is closest to:

Q8. A patient harbors $N=10^{9}$ Mycobacterium tuberculosis bacilli. The spontaneous mutation rates conferring resistance to drug A and drug B are $\mu_A=10^{-8}$ and $\mu_B=10^{-8}$ per bacterium respectively. Assuming independence of mutations, the expected number of bacteria already resistant to both drugs before therapy is:

Q9. A vaccine and a vaccine— (Assertion and Reason question)
Assertion (A): Antigenic drift in RNA viruses primarily results from errors introduced by viral RNA-dependent RNA polymerases that lack proofreading activity.
Reason (R): Antigenic drift leads to gradual accumulation of amino-acid substitutions in viral surface proteins, which over time can reduce vaccine effectiveness.

Q10. A 55-year-old patient treated with a 10-day course of broad-spectrum antibiotics develops Clostridioides difficile-associated diarrhea confirmed by toxin assay. Symptoms resolve with oral vancomycin but recur twice within six weeks. Which management strategy is most likely to reduce the risk of further recurrences?

Q11. A newly emergent virus has an estimated basic reproduction number $R_0 = 4.0$. A vaccine with efficacy $E = 0.80$ (80%) is available. Under homogeneous mixing, herd immunity requires $v\times E \ge 1 - 1/R_0$, where $v$ is the fraction vaccinated. What minimum percentage of the population must be vaccinated to achieve herd immunity?

Q12. A diagnostic test for Disease X has sensitivity $Se = 90\%$ and specificity $Sp = 95\%$. In a screened population the disease prevalence is $P = 2\%$. Using Bayes' theorem,
$PPV = \dfrac{Se \times P}{Se \times P + (1-Sp)\times(1-P)}$,
what is the approximate positive predictive value (probability that a person with a positive test truly has the disease)?

Q13. Two individuals are tested by RT–PCR for the same respiratory virus. Individual A has cycle threshold $\mathrm{Ct}=18$ and Individual B has $\mathrm{Ct}=28$. Assuming each PCR cycle doubles the target (fold difference $\approx 2^{\Delta\mathrm{Ct}}$), estimate the fold difference in viral RNA between A and B and indicate which is more likely to be infectious.

Q14. A bacterial population of size $N=10^{10}$ is present in a patient before treatment. Mutation rates per cell generation conferring resistance to drug A and drug B are $\mu_A=10^{-7}$ and $\mu_B=10^{-8}$ respectively. Resistance to both drugs requires independent mutations in the same cell. Estimate the expected numbers of pre-existing mutants and state which treatment strategy minimizes the chance that resistance already exists.

Q15. A vaccine has initial efficacy $E_0 = 0.95$ and protective efficacy decays exponentially: $E(t)=E_0 e^{-kt}$ with $k=\ln 2 / t_{1/2}$. If the half-life $t_{1/2}=2$ years and the disease has $R_0=5$, herd immunity requires $E(t)\ge 1 - 1/R_0$. After approximately how long (choose nearest) will herd immunity become impossible even with 100% vaccination coverage?