1. \cdot s=1.} in turn, r 0 = n s = μ n e ⁡ ( t l ) μ n e ⁡ [ min ( t l , t s ) ] = e ⁡ ( t l ) e ⁡ [ min ( t l , t s ) ] , == (t_)} }}= (t_)} }},} which is approximately: e ⁡ ( t l ) e ⁡ ( t s ) = 1 + λ μ = β n v . } (t_)} } (t_)}}=1+=.} graph of herd immunity threshold vs basic reproduction number with selected diseases s will be (1 − q), since q is the proportion of the population that is immune and q + s must equal one (since in this simplified model, everyone is either susceptible or immune). then: r 0 ⋅ ( 1 − q ) = 1 , 1 − q = 1 r 0 , q = 1 − 1 r 0 . &r_\cdot (1-q)=1,\\&1-q=,\\&q=1-.\end remember that this is the threshold level. die out of transmission will only occur if the proportion of immune individuals exceeds this level due to a mass vaccination programme. we have just calculated the critical immunization threshold (denoted qc). it is the minimum proportion of the population that must be immunized at birth (or close to birth) in order for the infection to die out in the population. q c = 1 − 1 r 0 . =1-.} because the fraction of the final size of the population p that is never infected can be defined as: lim t → ∞ s ( t ) = e − ∫ 0 ∞ λ ( t ) d t = 1 − p . s(t)=e^^\lambda (t)\,dt}=1-p.} hence, p = 1 − e − ∫ 0 ∞ β i ( t ) d t = 1 − e − r 0 p . ^\beta i(t)\,dt}=1-e^p}.} solving for r 0 , we obtain: r 0 = − ln ⁡ ( 1 − p ) p . =.} when mass vaccination cannot exceed the herd immunity if the vaccine used is insufficiently effective or the required coverage cannot be reached, the program may fail to exceed qc. such a program will protect vaccinated individuals from disease, but may change the dynamics of transmission. suppose that a proportion of the population q (where q < qc) is immunised at birth against an infection with r0 > 1. the vaccination programme changes r0 to rq where r q = r 0 ( 1 − q ) =r_(1-q)} this change occurs simply because there are now fewer susceptibles in the population who can be infected. rq is simply r0 minus those that would normally be infected but that cannot be now since they are immune. as a consequence of this lower basic reproduction number, the average age of infection a will also change to some new value aq in those who have been left unvaccinated. recall the relation that linked r0, a and l. assuming that life expectancy has not changed, now: r q = l a q , =,} a q = l r q = l r 0 ( 1 − q ) . ==(1-q)}}.} but r0 = l/a so: a q = l ( l / a ) ( 1 − q ) = a l l ( 1 − q ) = a 1 − q . ===.} thus, the vaccination program may raise the average age of infection, and unvaccinated individuals will experience a reduced force of infection due to the presence of the vaccinated group. for a disease that leads to greater clinical severity in older populations, the unvaccinated proportion of the population may experience the disease relatively later in life than would occur in the absence of vaccine. when mass vaccination exceeds the herd immunity if a vaccination program causes the proportion of immune individuals in a population to exceed the critical threshold for a significant length of time, transmission of the infectious disease in that population will stop. if elimination occurs everywhere at the same time, then this can lead to eradication. elimination interruption of endemic transmission of an infectious disease, which occurs if each infected individual infects less than one other, is achieved by maintaining vaccination coverage to keep the proportion of immune individuals above the critical immunization threshold. eradication elimination everywhere at the same time such that the infectious agent dies out (for example, smallpox and rinderpest). reliability models have the advantage of examining multiple outcomes simultaneously, rather than making a single forecast. models have shown broad degrees of reliability in past pandemics, such as sars, sars-cov-2, swine flu, mers and ebola.