Mathematical modelling of infectious diseases

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments: susceptible, ${\displaystyle S(t)}$; infected, ${\displaystyle I(t)}$; and recovered, ${\displaystyle R(t)}$. The compartments used for this model consist of three classes:^[25]

• ${\displaystyle S(t)}$ is used to represent the individuals not yet infected with the disease at time t, or those susceptible to the disease of the population.
• ${\displaystyle I(t)}$ denotes the individuals of the population who have been infected with the disease and are capable of spreading the disease to those in the susceptible category.
• ${\displaystyle R(t)}$ is the compartment used for the individuals of the population who have been infected and then removed from the disease, either due to immunization or due to death. Those in this category are not able to be infected again or to transmit the infection to others.

Other compartmental models

There are many modifications of the SIR model, including those that include births and deaths, where upon recovery there is no immunity (SIS model), where immunity lasts only for a short period of time (SIRS), where there is a latent period of the disease where the person is not infectious (SEIS and SEIR), and where infants can be born with immunity (MSIR).^{[citation needed]}

Infectious disease dynamics

Mathematical models need to integrate the increasing volume of

• antigenic shift
• epidemiological
  networks
• evolution and spread of resistance
• immuno-epidemiology
• intra-host dynamics
• Pandemic
• pathogen population genetics
• persistence of pathogens within hosts
• phylodynamics
• role and identification of
  infection reservoirs
• role of host genetic factors
• spatial epidemiology
• statistical and mathematical tools and innovations
• Strain (biology) structure and interactions
• transmission
  , spread and control of infection
• virulence

Mathematics of mass vaccination

If the proportion of the population that is immune exceeds the

The herd immunity level will be denoted q. Recall that, for a stable state:^[citation needed]

${\displaystyle R_{0}\cdot S=1.}$

In turn,

${\displaystyle R_{0}={\frac {N}{S}}={\frac {\mu N\operatorname {E} (T_{L})}{\mu N\operatorname {E} [\min(T_{L},T_{S})]}}={\frac {\operatorname {E} (T_{L})}{\operatorname {E} [\min(T_{L},T_{S})]}},}$

which is approximately:^{[citation needed]}

${\displaystyle {\frac {\operatorname {\operatorname {E} } (T_{L})}{\operatorname {\operatorname {E} } (T_{S})}}=1+{\frac {\lambda }{\mu }}={\frac {\beta N}{v}}.}$

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:

${\displaystyle {\begin{aligned}&R_{0}\cdot (1-q)=1,\\[6pt]&1-q={\frac {1}{R_{0}}},\\[6pt]&q=1-{\frac {1}{R_{0}}}.\end{aligned}}}$

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 q_c). 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.

${\displaystyle q_{c}=1-{\frac {1}{R_{0}}}.}$

Because the fraction of the final size of the population p that is never infected can be defined as:

${\displaystyle \lim _{t\to \infty }S(t)=e^{-\int _{0}^{\infty }\lambda (t)\,dt}=1-p.}$

Hence,

${\displaystyle p=1-e^{-\int _{0}^{\infty }\beta I(t)\,dt}=1-e^{-R_{0}p}.}$

Solving for ${\displaystyle R_{0}}$, we obtain:

${\displaystyle R_{0}={\frac {-\ln(1-p)}{p}}.}$

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 q_c. Such a program will protect vaccinated individuals from disease, but may change the dynamics of transmission.^{[citation needed]}

Suppose that a proportion of the population q (where q < q_c) is immunised at birth against an infection with R₀ > 1. The vaccination programme changes R₀ to R_q where

${\displaystyle R_{q}=R_{0}(1-q)}$

This change occurs simply because there are now fewer susceptibles in the population who can be infected. R_q 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 A_q in those who have been left unvaccinated.

Recall the relation that linked R₀, A and L. Assuming that life expectancy has not changed, now:^{[citation needed]}

${\displaystyle R_{q}={\frac {L}{A_{q}}},}$

${\displaystyle A_{q}={\frac {L}{R_{q}}}={\frac {L}{R_{0}(1-q)}}.}$

But R₀ = L/A so:

${\displaystyle A_{q}={\frac {L}{(L/A)(1-q)}}={\frac {AL}{L(1-q)}}={\frac {A}{1-q}}.}$

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

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.^[citation needed]

Eradication

Elimination everywhere at the same time such that the infectious agent dies out (for example, smallpox and rinderpest).^{[citation needed]}

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

References