Human Health and Disease Set-3

Q1. For a contagious disease with basic reproduction number $R_0=3$, what minimum fraction of the population must be immunised (assuming a perfectly effective vaccine and homogeneous mixing) to achieve herd immunity according to $h=1-\dfrac{1}{R_0}$?

Q2. A diagnostic test has sensitivity $Se=95\%$ and specificity $Sp=90\%$. In a population where disease prevalence is $1\%$, what is the approximate probability that a person who tests positive actually has the disease? (Use $PPV=\dfrac{Se\times Prev}{Se\times Prev + (1-Sp)\times(1-Prev)}$. )

Q3. A bacterial population contains $1\%$ antibiotic-resistant mutants and $99\%$ susceptible cells. An antibiotic kills $99\%$ of the susceptible bacteria but does not affect the resistant mutants. After treatment, approximately what percentage of the surviving bacteria will be resistant?

Q4. For a disease with $R_0=5$ and a vaccine efficacy $VE=80\%$ (i.e. $VE=0.80$), the minimum fraction of the population that must be vaccinated to achieve herd immunity is given by $c_{\min}=\dfrac{1-\dfrac{1}{R_0}}{VE}$. Evaluate $c_{\min}$ and state whether herd immunity is theoretically achievable.

Q5. A chronic viral infection comprises two strains: wild-type (constituting $99\%$ of the viral load) with replication rate $r_w=1.0$ (normalized) in the absence of drug, and a drug-resistant mutant (constituting $1\%$) with replication rate $r_r=0.5$ in the absence of drug (fitness cost). A drug is given that reduces wild-type replication 100-fold so its post-drug rate becomes $r_w' = 0.01$, but the drug does not affect the resistant mutant. Which pattern of total viral load over time after starting therapy is most likely?

Q6. For a communicable disease with basic reproduction number $R_0 = 4$, the herd-immunity threshold is given by $p_c = 1 - \dfrac{1}{R_0}$. What minimum percentage of the population must be immune to interrupt sustained transmission?

Q7. A vaccine against disease X has efficacy $90\%$ and $80\%$ of the population are vaccinated. If the basic reproduction number is $R_0 = 3$ and effective immune fraction is coverage $\times$ efficacy, determine whether the community reaches herd immunity (use $p_c = 1 - \dfrac{1}{R_0}$).

Q8. A serological test for disease Y has sensitivity $95\%$ and specificity $98\%$. In a population where the disease prevalence is $1\%$, what is the probability that a person who tests positive is truly infected (positive predictive value)? Use Bayes' theorem.

Q9. Assertion: Widespread use of a vaccine that reduces disease severity but does not prevent infection or onward transmission (a non-sterilizing or "leaky" vaccine) can drive the evolution of higher pathogen virulence.
Reason: Vaccinated hosts survive infections that would otherwise kill unvaccinated hosts, so highly virulent strains that rely on host survival to transmit can persist and gain a selective advantage in a vaccinated population.

Q10. A population is split into two groups: high-contact (fraction $f_H=0.2$, contact rate $c_H=5$) and low-contact (fraction $f_L=0.8$, contact rate $c_L=1$). The basic reproduction number is proportional to the mean contact rate $\bar c = f_H c_H + f_L c_L$. If vaccine supply allows immunizing $20\%$ of the total population, which allocation minimizes the new effective reproduction number $R_{eff}$?