Basic reproduction number

${\displaystyle R_{0}}$ is the average number of people infected from one other person. For example, Ebola has an ${\displaystyle R_{0}}$ of two, so on average, a person who has Ebola will pass it on to two other people.

Suppose that infectious individuals make an average of ${\displaystyle \beta }$ infection-producing contacts per unit time, with a mean infectious period of ${\displaystyle \tau }$. Then the basic reproduction number is:

$${\displaystyle R_{0}=\beta \,\tau }$$

This simple formula suggests different ways of reducing ${\displaystyle R_{0}}$ and ultimately infection propagation. It is possible to decrease the number of infection-producing contacts per unit time ${\displaystyle \beta }$ by reducing the number of contacts per unit time (for example staying at home if the infection requires contact with others to propagate) or the proportion of contacts that produces infection (for example wearing some sort of protective equipment). Hence, it can also be written as[12]

${\displaystyle R_{0}={\overline {c}}\,T\,\tau ,}$

where ${\displaystyle {\overline {c}}}$ is the rate of contact between susceptible and infected individuals and ${\displaystyle T}$ is the transmissibility, i.e, the probability of infection given a contact. It is also possible to decrease the infectious period ${\displaystyle \tau }$ by finding and then isolating, treating or eliminating (as is often the case with animals) infectious individuals as soon as possible.

Latent period is the transition time between contagion event and disease manifestation. In cases of diseases with varying latent periods, the basic reproduction number can be calculated as the sum of the reproduction numbers for each transition time into the disease. An example of this is tuberculosis (TB). Blower and coauthors calculated from a simple model of TB the following reproduction number:[13]

$${\displaystyle R_{0}=R_{0}^{\text{FAST}}+R_{0}^{\text{SLOW}}}$$

In their model, it is assumed that the infected individuals can develop active TB by either direct progression (the disease develops immediately after infection) considered above as FAST tuberculosis or endogenous reactivation (the disease develops years after the infection) considered above as SLOW tuberculosis.[14]

In populations that are not homogeneous, the definition of ${\displaystyle R_{0}}$ is more subtle. The definition must account for the fact that a typical infected individual may not be an average individual. As an extreme example, consider a population in which a small portion of the individuals mix fully with one another while the remaining individuals are all isolated. A disease may be able to spread in the fully mixed portion even though a randomly selected individual would lead to fewer than one secondary case. This is because the typical infected individual is in the fully mixed portion and thus is able to successfully cause infections. In general, if the individuals infected early in an epidemic are on average either more likely or less likely to transmit the infection than individuals infected late in the epidemic, then the computation of ${\displaystyle R_{0}}$ must account for this difference. An appropriate definition for ${\displaystyle R_{0}}$ in this case is "the expected number of secondary cases produced, in a completely susceptible population, produced by a typical infected individual".[15]

The basic reproduction number can be computed as a ratio of known rates over time: if an infectious individual contacts ${\displaystyle \beta }$ other people per unit time, if all of those people are assumed to contract the disease, and if the disease has a mean infectious period of ${\displaystyle {\dfrac {1}{\gamma }}}$, then the basic reproduction number is just ${\displaystyle R_{0}={\dfrac {\beta }{\gamma }}}$. Some diseases have multiple possible latency periods, in which case the reproduction number for the disease overall is the sum of the reproduction number for each transition time into the disease.

In reality, diseases spread over networks of contact between people. Such a network can be represented mathematically with a graph and is called the contact network.[16] Every node in a contact network is a representation of an individual and each link (edge) between a pair of nodes represents the contact between them. Links in the contact networks may be used to transmit the disease between the individuals and each disease has its own dynamics on top of its contact network. For example, individuals in a population can be assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, or Recovered) and they progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for instance, SIR means each individual is originally susceptible then changes to infectious and finally gets recovered and remained recovered (immune) forever. On the other hand, public health may apply some interventions such as vaccination or contact tracing to reduce the spread of an epidemic disease. The combination of disease dynamics under the influence of interventions, if any, on a contact network may be modeled with another network, known as a transmission network. In a transmission network, all the links are responsible for transmitting the disease. If such a network is a locally tree-like network, meaning that any local neighborhood in such a network takes the form of a tree, then the basic reproduction can be written in terms of the average excess degree of the transmission network such that:

$${\displaystyle R_{0}={\frac {\langle k^{2}\rangle }{\langle k\rangle }}-1,}$$

where ${\displaystyle {\langle k\rangle }}$ is the mean-degree (average degree) of the network and ${\displaystyle {\langle k^{2}\rangle }}$ is the second moment of the transmission network degree distribution. It is, however, not always straightforward to find the transmission network out of the contact network and the disease dynamics.[17] For example, if a contact network can be approximated with an Erdős–Rényi graph with a Poissonian degree distribution, and the disease spreading parameters are as defined in the example above, such that ${\displaystyle \beta }$ is the transmission rate per person and the disease has a mean infectious period of ${\displaystyle {\dfrac {1}{\gamma }}}$, then the basic reproduction number is ${\displaystyle R_{0}={\dfrac {\beta }{\gamma }}{\langle k\rangle }}$[18][19] since ${\displaystyle {\langle k^{2}\rangle }-{\langle k\rangle }^{2}={\langle k\rangle }}$ for a Poisson distribution.

One way to calculate ${\displaystyle R_{0}}$ is to average the expected number of new infections over all possible infected types. The next-generation method is a general method of deriving ${\displaystyle R_{0}}$ when more than one class of infectives is involved. This method, originally introduced by Diekmann et al. (1990),[20] can be used for models with underlying age structure or spatial structure, among other possibilities.[21] In this picture, the spectral radius of the next-generation matrix ${\displaystyle G}$ gives the basic reproduction number, ${\displaystyle R_{0}=\rho (G).}$[22]

Consider a sexually transmitted disease. In a naive population where almost everyone is susceptible, but the infection seed, if the expected number of gender 1 is ${\displaystyle f}$ and the expected number of infected gender 2 is ${\displaystyle m}$, we can know how many would be infected in the next-generation. Such that the next-generation matrix ${\displaystyle G}$ can be written as:[12]

$${\displaystyle G={\begin{pmatrix}0&f\\m&0\end{pmatrix}},}$$

where each element ${\displaystyle g_{ij}}$ is the expected number of secondary infections of gender ${\displaystyle i}$ caused by a single infected individual of gender ${\displaystyle j}$, assuming that the population of gender ${\displaystyle i}$ is entirely susceptible. Diagonal elements are zero because people of the same gender cannot transmit the disease to each other but, for example, each ${\displaystyle f}$ can transmit the disease to ${\displaystyle m}$, on average. Meaning that each element ${\displaystyle g_{ij}}$ is a reproduction number, but one where who infects whom is accounted for. If generation ${\displaystyle a}$ is represented with ${\displaystyle \phi _{a}}$ then the next generation ${\displaystyle \phi _{a+1}}$ would be ${\displaystyle G\phi _{a}}$.

The spectral radius of the next-generation matrix is the basic reproduction number, ${\displaystyle R_{0}=\rho (G)={\sqrt {mf}}}$, that is here, the geometric mean of the expected number of each gender in the next-generation. Note that multiplication factors ${\displaystyle f}$ and ${\displaystyle m}$ alternate because, the infectious person has to ‘pass through’ a second gender before it can enter a new host of the first gender. In other words, it takes two generations to get back to the same type, and every two generations numbers are multiplied by ${\displaystyle m}$×${\displaystyle f}$. The average per generation multiplication factor is therefore ${\displaystyle {\sqrt {mf}}}$. Note that ${\displaystyle G}$ is a non-negative matrix so it has single, unique, positive, real eigenvalue which is strictly greater than all the others.

In mathematical modelling of infectious disease, the dynamics of spreading is usually described through a set of non-linear ordinary differential equations (ODE). So there is always ${\displaystyle n}$ coupled equations of form ${\displaystyle {\dot {C_{i}}}={\operatorname {d} \!C_{i} \over \operatorname {d} \!t}=f(C_{1},C_{2},...,C_{n})}$ which shows how the number of people in compartment ${\displaystyle C_{i}}$ changes over time. For example, in a SIR model, ${\displaystyle C_{1}=S}$, ${\displaystyle C_{2}=I}$, and ${\displaystyle C_{3}=R}$. Compartmental models have a disease-free equilibrium (DFE) meaning that it is possible to find an equilibrium while setting the number of infected people to zero, ${\displaystyle I=0}$. In other words, as a rule, there is an infection-free steady state. This solution, also usually ensures that the disease-free equilibrium is also an equilibrium of the system. There is another fixed point known as an Endemic Equilibrium (EE) where the disease is not totally eradicated and remains in the population. Mathematically, ${\displaystyle R_{0}}$ is a threshold for stability of a disease-free equilibrium such that:

${\displaystyle R_{0}\leq 1\Rightarrow \lim _{t\to \infty }(C_{1}(t),C_{2}(t),\cdots ,C_{n}(t))={\textrm {DFE}}}$

${\displaystyle R_{0}>1,I(0)>0\Rightarrow \lim _{t\to \infty }(C_{1}(t),C_{2}(t),\cdots ,C_{n}(t))={\textrm {EE}}.}$

To calculate ${\displaystyle R_{0}}$, the first step is to linearise around the disease-free equilibrium (DFE), but for the infected subsystem of non-linear ODEs which describe the production of new infections and changes in state among infected individuals. Epidemiologically, the linearisation reflects that ${\displaystyle R_{0}}$ characterizes the potential for initial spread of an infectious person in a naive population, assuming the change in the susceptible population is negligible during the initial spread.[23] A linear system of ODEs can always be described by a matrix. So, the next step is to construct a linear positive operator that provides the next generation of infected people when applied to the present generation. Note that this operator (matrix) is responsible for the number of infected people, not all the compartments. Iteration of this operator describes the initial progression of infection within the heterogeneous population. So comparing the spectral radius of this operator to unity determines whether the generations of infected people grow or not. ${\displaystyle R_{0}}$ can be written as a product of the infection rate near the disease-free equilibrium and average duration of infectiousness. It is used to find the peak and final size of an epidemic.

As described in the example above, so many epidemic processes can be described with a SIR‌ model. However, for many important infections, such as COVID-19, there is a significant latency period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed). Here, the formation of the next-generation matrix from the SEIR‌ model involves determining two compartments, infected and non-infected, since they are the populations that spread the infection. So we only need to model the exposed, E, and infected, I, compartments. Consider a population characterized by a death rate ${\displaystyle \mu }$ and birth rate ${\displaystyle \lambda }$ where a communicable disease is spreading. As in the previous example, we can use the transition rates between the compartments per capita such that ${\displaystyle \beta }$ be the infection rate, ${\displaystyle \gamma }$ be the recovery rate, and ${\displaystyle \kappa }$ be the rate at which a latent individual becomes infectious. Then, we can define the model dynamics using the following equations:[21][24]

$${\displaystyle {\begin{cases}{\dot {S}}=\lambda -\mu S-\beta SI,\\\\{\dot {E}}=\beta SI-(\mu +\kappa )E,\\\\{\dot {I}}=\kappa E-(\mu +\gamma )I,\\\\{\dot {R}}=\gamma I-\mu R.\end{cases}}}$$

Here we have 4 compartments and we can define vector ${\displaystyle \mathrm {x} =(S,E,I,R)}$ where ${\displaystyle \mathrm {x} _{i}}$ denotes the number or proportion of individuals in the ${\displaystyle i}$-th compartment. Let ${\displaystyle F_{i}(\mathrm {x} )}$ be the rate of appearance of new infections in compartment ${\displaystyle i}$ such that it includes only infections that are newly arising, but does not include terms which describe the transfer of infectious individuals from one infected compartment to another. Then if ${\displaystyle V_{i}^{+}}$ is the rate of transfer of individuals into compartment ${\displaystyle i}$ by all other means and ${\displaystyle V_{i}^{-}}$ is the rate of transfer of individuals out of the ${\displaystyle i}$-th compartment, then the difference ${\displaystyle F_{i}(\mathrm {x} )-V_{i}(\mathrm {x} )}$ gives the rate of change of such that ${\displaystyle V_{i}(\mathrm {x} )=V_{i}^{-}(\mathrm {x} )-V_{i}^{+}(\mathrm {x} )}$.

We can now make matrices of partial derivatives of ${\displaystyle F}$ and ${\displaystyle V}$ such that

${\displaystyle F_{ij}={\partial \!\ F_{i}(\mathrm {x} ^{*}) \over \partial \!\ \mathrm {x} _{j}}}$ and ${\displaystyle V_{ij}={\partial \!\ V_{i}(\mathrm {x} ^{*}) \over \partial \!\ \mathrm {x} _{j}}}$ , where ${\displaystyle \mathrm {x} ^{*}=(S^{*},E^{*},I^{*},R^{*})=(\lambda /\mu ,0,0,0)}$ is the disease-free equilibrium.

We now can form the next-generation matrix (operator) ${\displaystyle G=FV^{-1}}$.[15][22] Basically, ${\displaystyle F}$ is a non-negative matrix which represents the infection rates near the equilibrium, and ${\displaystyle V}$ is an M-matrix for linear transition terms making ${\displaystyle V^{-1}}$ a matrix which represents the average duration of infectiousness. Therefore, ${\displaystyle G_{ij}}$ gives the rate at which infected individuals in ${\displaystyle \mathrm {x} _{j}}$ produce new infections in ${\displaystyle \mathrm {x} _{i}}$, times the average length of time an individual spends in a single visit to compartment ${\displaystyle j.}$

Finally, for this SEIR process we can have:

${\displaystyle F={\begin{pmatrix}0&\beta S^{*}\\0&0\end{pmatrix}}}$ and ${\displaystyle V={\begin{pmatrix}\mu +\kappa &0\\-\kappa &\gamma +\mu \end{pmatrix}}}$ and so ${\displaystyle R_{0}=\rho (FV^{-1})={\frac {\kappa \beta S^{*}}{(\mu +\kappa )(\mu +\gamma )}}.}$

If an individual, after getting infected, infects exactly ${\displaystyle R_{0}}$ new individuals only after exactly a time ${\displaystyle \tau }$ (the serial interval) has passed, then the number of infectious individuals over time grows as

$${\displaystyle n_{E}(t)=n_{E}(0)\,R_{0}^{t/\tau }=n_{E}(0)\,e^{Kt}}$$

or

$${\displaystyle \ln(n_{E}(t))=\ln(n_{E}(0))+\ln(R_{0})t/\tau .}$$

The underlying matching differential equation is

$${\displaystyle {\frac {dn_{E}(t)}{dt}}=n_{E}(t){\frac {\ln(R_{0})}{\tau }}.}$$

or

$${\displaystyle {\frac {d\ln(n_{E}(t))}{dt}}={\frac {\ln(R_{0})}{\tau }}.}$$

In this case, ${\displaystyle R_{0}=e^{K\tau }}$ or ${\displaystyle K={\frac {\ln R_{0}}{\tau }}}$.

For example, with ${\displaystyle \tau =5~\mathrm {d} }$ and ${\displaystyle K=0.183~\mathrm {d} ^{-1}}$, we would find ${\displaystyle R_{0}=2.5}$.

If ${\displaystyle R_{0}}$ is time dependent

$${\displaystyle \ln(n_{E}(t))=\ln(n_{E}(0))+{\frac {1}{\tau }}\int \limits _{0}^{t}\ln(R_{0}(t))dt}$$

showing that it may be important to keep ${\displaystyle \ln(R_{0})}$ below 0, time-averaged, to avoid exponential growth.

In this model, an individual infection has the following stages:

1. Exposed: an individual is infected, but has no symptoms and does not yet infect others. The average duration of the exposed state is ${\displaystyle \tau _{E}}$.
2. Latent infectious: an individual is infected, has no symptoms, but does infect others. The average duration of the latent infectious state is ${\displaystyle \tau _{I}}$. The individual infects ${\displaystyle R_{0}}$ other individuals during this period.
3. Isolation after diagnosis: measures are taken to prevent further infections, for example by isolating the infected person.

This is a SEIR model and ${\displaystyle R_{0}}$ may be written in the following form[26]

$${\displaystyle R_{0}=1+K(\tau _{E}+\tau _{I})+K^{2}\tau _{E}\tau _{I}.}$$

This estimation method has been applied to COVID-19 and SARS. It follows from the differential equation for the number of exposed individuals ${\displaystyle n_{E}}$ and the number of latent infectious individuals ${\displaystyle n_{I}}$,

$${\displaystyle {\frac {d}{dt}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}={\begin{pmatrix}-1/\tau _{E}&R_{0}/\tau _{I}\\1/\tau _{E}&-1/\tau _{I}\end{pmatrix}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}.}$$

The largest eigenvalue of the matrix is the logarithmic growth rate ${\displaystyle K}$, which can be solved for ${\displaystyle R_{0}}$.

In the special case ${\displaystyle \tau _{I}=0}$, this model results in ${\displaystyle R_{0}=1+K\tau _{E}}$, which is different from the simple model above (${\displaystyle R_{0}=\exp(K\tau _{E})}$). For example, with the same values ${\displaystyle \tau =5~\mathrm {d} }$ and ${\displaystyle K=0.183~\mathrm {d} ^{-1}}$, we would find ${\displaystyle R_{0}=1.9}$, rather than the true value of ${\displaystyle 2.5}$. The difference is due to a subtle difference in the underlying growth model; the matrix equation above assumes that newly infected patients are currently already contributing to infections, while in fact infections only occur due to the number infected at ${\displaystyle \tau _{E}}$ ago. A more correct treatment would require the use of delay differential equations.[27]