Human Health and Disease Set-1

Q1. A novel pathogen has basic reproduction number $R_0=3$. A vaccine reduces susceptibility with efficacy $e=0.80$. Using the herd-immunity requirement $v=\dfrac{1-1/R_0}{e}$, what minimum fraction $v$ of the population must be vaccinated to stop sustained transmission?

Q2. In a community with disease prevalence $P=0.02$, two diagnostic tests have characteristics: Test A: sensitivity $Se_A=0.95$, specificity $Sp_A=0.90$; Test B: $Se_B=0.85$, $Sp_B=0.98$. Using $PPV=\dfrac{Se\cdot P}{Se\cdot P+(1-Sp)(1-P)}$, which test gives the higher positive predictive value (PPV) and approximately what is that PPV?

Q3. A patient harbors a bacterial population of size $N=10^9$. The spontaneous mutation rate to resistance to each antibiotic (A or B) is $\mu=10^{-6}$ per cell division. The expected numbers of pre-existing mutants are approximately $N\mu$ (single-drug resistant) and $N\mu^2$ (resistant to both). Which treatment strategy is least likely to encounter bacteria already resistant to both drugs at the start of therapy?

Q4. In a population the annual incidence of a chronic infectious disease fell from 50 to 30 cases per 100,000, but measured prevalence rose from 200 to 300 per 100,000. Using the relation $Prevalence \approx Incidence \times Duration$ (steady state), which epidemiological change best explains the simultaneous fall in incidence and rise in prevalence?

Q5. In a low-prevalence population ($P=0.005$) two independent tests have $Se_1=0.99,\;Sp_1=0.95$ and $Se_2=0.98,\;Sp_2=0.97$. Consider two strategies: series testing (positive only if both tests positive) and parallel testing (positive if either test positive). Which strategy yields the higher PPV and approximately what is its PPV? (Hint: for series $Se_{ser}=Se_1Se_2$, $Sp_{ser}=1-(1-Sp_1)(1-Sp_2)$.)

Q6. In an outbreak, the attack rate among unvaccinated individuals was $30\%$ and among vaccinated individuals was $6\%$. Using the vaccine efficacy formula $VE=\dfrac{AR_u-AR_v}{AR_u}\times100\%$, what is the vaccine efficacy?

Q7. For a disease with basic reproduction number $R_0=5$, a vaccine has efficacy $e=80\%$ (i.e. $e=0.8$). The effective reproduction number after vaccinating fraction $p$ of the population is $R_e=R_0(1-p\,e)$. What minimum vaccination coverage $p$ is required to ensure $R_e<1$?

Q8. A 20-year-old patient has recurrent Neisseria meningitidis infections. Laboratory tests show normal serum $C3$ concentration but markedly reduced total haemolytic complement activity (CH50) and impaired complement-mediated lysis of Neisseria. Which complement component deficiency best explains this pattern?

Q9. Two diagnostic tests for a disease have these performances: Test A — sensitivity $95\%$, specificity $90\%$; Test B — sensitivity $85\%$, specificity $99\%$. If disease prevalence in the screened population is $1\%$, which test gives the higher positive predictive value (PPV)? (Use $PPV=\dfrac{\text{sensitivity}\times\text{prevalence}}{\text{sensitivity}\times\text{prevalence}+(1-\text{specificity})\times(1-\text{prevalence})}$.)

Q10. Consider a bacterial infection with population size $N=10^{10}$. The per-cell-division mutation rates conferring resistance to antibiotics A and B are $\mu_A=10^{-7}$ and $\mu_B=10^{-7}$ respectively, and mutations are independent. Approximate the probability that a bacterium resistant to both drugs exists before treatment, and choose the strategy that minimizes the chance such double-resistant bacteria are present at treatment start.