Mathematical epidemiology

Mathematical epidemiology uses mathematical models and statistical inference to study the occurrence, transmission, and control of disease in populations.

Each undirected cross-group partnership is counted once from each group's contact total. Thus group size times per-person contact rate times cross-group fraction must agree. If one group occupies fraction $p$ while its cross-group per-person rate stays fixed, aggregate cross-group contact is order $p$ and the other group's cross-contact fraction is order $p$. Assigning both per-person rates independently generally breaks this balance.

A macroparasite burden model tracks the distribution of within-host parasite burdens and couples it to transmission or environmental infectious stages. Mean burden alone is insufficient when worm mating, nonlinear fecundity, immunity or host mortality depend on that distribution.

For sexually reproducing worms, an infected host need not contain both sexes. Independent random sexes split a Poisson total burden of mean $m$ into independent male and female Poisson distributions of mean $m/2$, giving the displayed probability of a fertile mixed-sex burden. For an arbitrary burden probability generating function $G$, this probability is $1-2G(1/2)+G(0)$. Its low-burden behaviour can introduce an Allee effect in transmission, so an asexual mean-burden invasion calculation is not automatically valid.

If host-specific equilibrium mean burden has a gamma distribution, conditional Poisson distributions produce a Poisson-gamma mixture, hence a negative binomial distribution. With mean $m$ and shape $k$, its zero-burden probability is $(1+m/k{)}^{-k}$. Decreasing $k$ increases aggregation: a smaller infected-host prevalence can accompany the same mean burden and more heavily infected individuals.

Independent worm acquisition at rate $\lambda$ and loss at rate $\mu j$ produce a birth-death process for within-host parasite burden. Summing its forward equations against $j$ gives $d\mathbb{E}[j]/dt=\lambda -\mu \mathbb{E}[j]$. At constant exposure, detailed balance $\lambda p_j=\mu (j+1)p_{j+1}$ yields a Poisson distribution with mean $\lambda /\mu$. Heterogeneous acquisition rates change the pooled burden distribution even when each individual host follows this model.

A stochastic epidemic model represents infection and recovery as random events. A continuous-time Markov chain can track integer counts, while a stochastic SIR model illustrates random fade-out and finite-population outbreak variability that a deterministic limit does not show.

In a continuous-time epidemic model, a constant recovery rate $\gamma$ gives an exponential distribution of infectious duration with mean $1/\gamma$. Other duration distributions require additional stages or infection-age structure.

Quarantine restricts contacts of exposed individuals whose infection status is uncertain, with monitoring to detect illness. It differs from isolation of recognized infected or ill individuals. An incubation period quantile can enter a duration calculation, but assumptions about exposure time, symptom detection and infectiousness are needed before the calculation describes transmission risk.

The incubation period is the interval from infection to symptom onset in a person who develops symptoms. Its probability distribution describes onset times, not the full timing of infectiousness. Exposure may only be known to fall in an interval, requiring a corresponding observation model when estimating this distribution.

Observed symptom-onset intensity is the convolution of past infection intensity with the incubation delay density: $\mu (t)={\int }_0^{\infty }h(t-s)f(s)ds$. Back-calculation infers the infection history from that relation. Include pre-observation infections unless their absence is known.

For equal-width time bins and infections assigned to bin-end time $t_i$, let $q_{\ell }=\mathrm{Pr}((\ell -1)\Delta \le T<\ell \Delta )$. The next onset-bin mean is ${\mu }_k={\sum }_{i<k}{h}_i{q}_{k-i}$. Assigning infections instead to bin starts shifts the index by one. Bin probabilities integrate a delay density; they are not the density itself.

Calendar time locates an event on a shared external timeline, in contrast with an individual's age or the time elapsed since an individual event.

Incidence measures the occurrence of new cases in a population during a specified period.

The infection age of an infected individual is the time elapsed since that individual's infection.

An imported infection is acquired outside the population or observation system being modelled and enters it as an external source of incidence.

An infectivity profile describes how an infected individual's transmission rate varies with infection age.

The generation interval is the time between infection of a source individual and infection of a secondary case caused by that source.

The generation-interval distribution gives the probability law of the generation interval. In discrete time, its probabilities $g_{\tau }$ form a nonnegative sequence summing to one.

The infectious disease renewal equation expresses current incidence as a convolution of past incidence with an infectivity profile, multiplied by a time-varying transmission level.

For incidence $I_t$ and discrete generation-interval probabilities $g_{\tau }$, the total infectiousness at time $t$ is ${\Lambda }_t={\sum }_{\tau \ge 1}{g}_{\tau }{I}_{t-\tau }$.

A time-varying reproduction number describes transmission at a specified calendar time while allowing transmission conditions to change during an epidemic.

The instantaneous reproduction number freezes the transmission conditions at calendar time $t$ and counts the expected secondary infections produced over a complete infectious lifetime under those conditions.

The case reproduction number is the expected number of secondary infections actually generated by a person infected at time $t$, allowing transmission conditions to change during that person's infectious lifetime. It follows an infection cohort through the subsequent conditions it experiences. The instantaneous reproduction number instead freezes conditions at the reference time. Cori and colleagues distinguish these two constructions in their framework for time-varying reproduction numbers. ^[ref].

An effective reproduction number describes transmission in the current population, accounting for available susceptibility and transmission conditions. The phrase is used with different time conventions, so a definition must specify whether it means an instantaneous reproduction number or a case reproduction number. The CDC uses a time-varying effective quantity $R_t$ to describe the direction of community transmission. ^[ref].

For the closed, homogeneous SIR model with constant $\beta ,\gamma$, the instantaneous convention givesThus the infectious compartment grows when $R_{\mathrm{eff}}>1$ and declines when $R_{\mathrm{eff}}<1$, provided $i>0$. At the SIR infection peak it equals one. This freezes the current susceptible fraction; it does not count a newly infected person's actual future secondary infections while $s$ continues to decrease. The distinction comes directly from the SIR model equations and the two time conventions above.