Virulence-transmission trade-off (Frank, 1996)
The classical virulence-transmission trade-off function, following the notation of (Frank 1996), is:
\[ R_0 = \frac{\beta(\nu)N}{\delta + \nu + c(\nu)}, \]
where:
\(R_0\) The basic reproduction number. The expected number of secondary infections produced by a single infected host introduced into a fully susceptible population of size \(N\). The transmission intensity multiplied by mean infectious duration.
\(\nu\) Virulence. In this formulation, \(\nu\) is the disease-induced host mortality intensity. This represents the per-capita hazard at which infected hosts die because of the pathogen.
\(\beta(\nu)\) Transmission coefficient (often called the transmission intensity). It gives the per-capita rate at which an infected host generates new infections per susceptible host, and it is assumed to depend on virulence. The biological trade-off is encoded here: increasing \(\nu\) may increase within-host replication and hence transmission, so \(\beta'(\nu) > 0\) over some range.
\(N\) Total density (or number) of susceptible hosts in the population at invasion. Because \(R_0\) is defined at the start of an epidemic, essentially all hosts are susceptible, so the force of infection scales with \(N\).
\(\delta\) Background host mortality intensity (natural death intensity unrelated to infection). This removes infected hosts independently of the pathogen’s virulence.
\(c(\nu)\) Additional loss intensity associated with infection that is not direct mortality. Examples include recovery with immunity, behavioral isolation, treatment, or reduced contact rate induced by symptoms. It may depend on virulence if, for example, more virulent infections provoke faster immune clearance or stronger behavioral avoidance.
All intensities are rates, but not all rates are intensities. A rate is a general term indicating the change of one quantity relative to another, expresed as a ratio (“rate” and “ratio” have the same linguistic root). An “intensity” is a rate, but one some characteristics is the instantaneous hazard parameter (= rate) of a stochastic process. In classical ODE epidemic models (e.g., SIR), “rate” and “intensity” are often used interchangeably because constant hazards produce exponential waiting times and yield linear per-capita ODE terms.
Note that:
\[ R_0 = \text{transmission intensity} \times \text{mean infection duration}. \]
- The numerator, \(\beta(\nu)N\), is the transmission intensity, i.e. the per-capita intensity (or rate) of generating new infections in a population of size \(N\).
- The denominator \(\delta + \nu + c(\nu)\) gives the total removal intensity of infected hosts: the rate of removal of infected hosts from the population. Thus, \(\frac{1}{\delta + \nu + c(\nu)}\) gives the mean infectious duration.
As \(R_0\) measures the net increase of infected host over time, expressed as a ratio of infected hosts gain (the numerator) to infected hosts lost (the denominator), it serves as an index of the epidemiological state of an infectious disease. - If the numerator is greater than the denominator, there will be more infections generated than removed from the population, leading to a net increase in infections: the disease in an epidemic phase. - If the numerator is less than the denominator, there infections will be removed from the population faster than new infections will occur: the disease will “die out” without becoming an epidemic.
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