Deriving the Within-Host Viral Basic Reproductive Number

Why Count Infection Cycles?

A viral load alone does not say whether an infection can establish itself. That depends on what happens across an infection cycle: an infectious virion enters a susceptible target cell, that infected cell produces new virions, and some of those virions survive long enough to infect further cells. The within-host basic reproductive number summarizes this cycle at the start of infection, when the target-cell population is at its infection-free level.

The earlier material on target-cell-limited models of acute viral infection supplies the basic cell-and-virus framework. Here, the goal is to derive the threshold from that framework and state precisely what the threshold does—and does not—mean.

Definition: The within-host basic reproductive number, \(R_0\), is the expected number of newly infected target cells generated, through the full infection cycle, by one newly infected cell introduced into an otherwise infection-free system at its susceptible target-cell level. In the model below, it is dimensionless and describes invasion under the specified baseline conditions.

Model and Assumptions

Let \(T(t)\) be the concentration of susceptible target cells, \(I(t)\) the concentration of infected cells, and \(V(t)\) the concentration of infectious free virions. Consider the target-cell-limited model

\begin{aligned} \frac{dT}{dt} &= \Lambda_T-d_TT-\beta TV,\\ \frac{dI}{dt} &= \beta TV-\delta I,\\ \frac{dV}{dt} &= pI-cV. \end{aligned}

Here, \(\Lambda_T\) is the target-cell supply rate, \(d_T\) is the target-cell loss rate, \(\beta\) is the effective infection rate constant, \(\delta\) is the infected-cell loss rate, \(p\) is the infectious-virion production rate per infected cell, and \(c\) is the infectious-virion clearance rate. The parameters \(d_T,\delta,c\) have units of inverse time. If cell and virion concentrations are measured per volume, \(\beta\) has units of volume per virion per time, and \(p\) has units of virions per infected cell per time.

The calculation uses the following assumptions:

Infection is well mixed and follows the mass-action incidence term \(\beta TV\).
An infected cell produces infectious virions at a constant rate \(p\) and is lost at a constant per-cell rate \(\delta\).
Infectious virions are cleared at a constant per-virion rate \(c\); virions do not replicate independently.
At the infection-free state, target cells are at their equilibrium level, and the infection introduced is small enough that this level is not appreciably depleted during the invasion calculation.
The model contains no additional delay or cell state between infection and production, and no immune feedback or treatment changing these rates during the calculation.

With no infection, the target-cell equilibrium is \(T_0=\Lambda_T/d_T\), while \(I=0\) and \(V=0\). When the target-cell concentration is instead treated as approximately constant, \(T_0\) is the specified susceptible concentration at the start of invasion. These alternatives give the same calculation once \(T_0\) is defined.

Deriving the Infection-Cycle Result

Start with one newly infected cell. Under constant loss rate \(\delta\), the probability it remains infected at time \(t\) is \(e^{-\delta t}\). Its expected total infectious-virion production over its lifetime is therefore

\int_0^\infty p e^{-\delta t}\,dt=\frac{p}{\delta}.

Now consider one infectious virion in the susceptible population \(T_0\). Its infection hazard is \(\beta T_0\), and its probability of remaining uncleared at time \(t\) is \(e^{-ct}\). Its expected number of infection events before clearance is

\int_0^\infty \beta T_0 e^{-ct}\,dt=\frac{\beta T_0}{c}.

Multiplying the expected number of virions produced per infected cell by the expected number of new infected cells per virion gives

R_0=\frac{p}{\delta}\frac{\beta T_0}{c} =\frac{\beta T_0p}{\delta c}.

The product represents two successive stages, not two independent estimates. Under the stated assumptions, each produced virion has the same expected opportunity to infect, and each newly infected cell has the same expected lifetime production. Thus the expected number of newly infected cells arising from one infected cell is the product.

Worked Example: Calculating a Baseline Reproductive Number

Suppose a hypothetical system has \(T_0=1.0\times10^6\) target cells per mL, \(\beta=2.0\times10^{-5}\) mL per virion per day, \(p=100\) virions per infected cell per day, \(\delta=0.50\) per day, and \(c=10\) per day. Assume these rates are constant and the target-cell level stays near \(T_0\) during invasion.

First, the expected lifetime production per infected cell is \(p/\delta=100/0.50=200\) virions. The infection rate per virion at \(T_0\) is \(\beta T_0=(2.0\times10^{-5})(1.0\times10^6)=20\) infections per virion per day, so one virion has \(20/10=2.0\) expected infection events before clearance. Combining the stages,

R_0=\frac{(2.0\times10^{-5})(1.0\times10^6)(100)} {(0.50)(10)} =\frac{2000}{5.0}=400.

The result is dimensionless. Under this hypothetical baseline model, one infected cell generates an expectation of 400 newly infected cells through the infection cycle. This very large value reflects the chosen illustrative rates; it is not a claim about any real virus.

Next-Generation Matrix and Threshold

The same result can be obtained by tracking the two infected states, \(I\) and \(V\), near the infection-free equilibrium. At that equilibrium, target-cell fluctuations do not affect the infected-state equations to first order: the product of a target-cell deviation and \(V\) is a higher-order term. Write \(x=(I,V)^\mathsf{T}\). Separate the appearance of new infected cells from progression and loss:

\frac{dx}{dt}=F x-\mathcal{V}x,\qquad F= \begin{pmatrix} 0&\beta T_0\\ 0&0 \end{pmatrix}, \qquad \mathcal{V}= \begin{pmatrix} \delta&0\\ -p&c \end{pmatrix}.

The matrix \(F\) represents new infection of target cells by virions. The matrix \(\mathcal{V}\) represents infected-cell loss, virion production by infected cells, and virion clearance. The next-generation matrix is \(K=F\mathcal{V}^{-1}\): it maps infected-state contributions through the expected progression and residence in infected states to new infections. Since

\mathcal{V}^{-1}= \begin{pmatrix} 1/\delta&0\\ p/(\delta c)&1/c \end{pmatrix}, \qquad K= \begin{pmatrix} \beta T_0p/(\delta c)&\beta T_0/c\\ 0&0 \end{pmatrix},

the eigenvalues of \(K\) are \(\beta T_0p/(\delta c)\) and \(0\). The spectral radius—the largest magnitude of its eigenvalues—is therefore \(R_0=\beta T_0p/(\delta c)\). The infection-cycle calculation and the matrix calculation agree.

A direct local-stability check confirms the threshold. The linearized infected-state matrix is

A= \begin{pmatrix} -\delta&\beta T_0\\ p&-c \end{pmatrix}, \qquad \det(A)=\delta c-\beta T_0p=\delta c(1-R_0).

Its trace is \(-(\delta+c)\), which is negative. The eigenvalues are real because the discriminant is \((\delta-c)^2+4\beta T_0p\), a nonnegative quantity. If \(R_0<1\), the determinant is positive and both eigenvalues are negative, so small infected-state perturbations decay locally. If \(R_0>1\), the determinant is negative, so one eigenvalue is positive and the infection-free state is unstable to invasion. At \(R_0=1\), one eigenvalue is zero, marking the threshold; this linear result alone does not determine the eventual nonlinear outcome.

The threshold concerns the local potential for invasion from a small introduction at \(T_0\). It does not require every component, such as \(V(t)\), to increase at every instant. Nor does it describe the later infection trajectory after substantial target-cell depletion or other model assumptions cease to hold.

Worked Example: A Below-Threshold Case

Consider a second hypothetical system with \(T_0=5.0\times10^5\) target cells per mL, \(\beta=1.0\times10^{-7}\) mL per virion per day, \(p=20\) virions per infected cell per day, \(\delta=0.50\) per day, and \(c=5.0\) per day. The same assumptions of constant rates and a near-constant target-cell level apply.

The infection rate per virion is \(\beta T_0=(1.0\times10^{-7})(5.0\times10^5)=0.050\) infections per day. The expected lifetime production is \(p/\delta=20/0.50=40\) virions per infected cell, and an infectious virion has an expected \(0.050/5.0=0.010\) infection events before clearance. Thus

R_0=(40)(0.010)=0.400, \qquad \text{equivalently}\qquad R_0=\frac{(1.0\times10^{-7})(5.0\times10^5)(20)} {(0.50)(5.0)} =\frac{1.0}{2.5}=0.400.

Because \(R_0\) is below one, the infection-free equilibrium is locally stable to small introductions under this model. This does not imply that no infected cells or virions can ever be observed; it means that small perturbations do not sustain invasion in the specified baseline system.

Worked Example: Finding a Target-Cell Threshold

Suppose a hypothetical model has \(\beta=4.0\times10^{-7}\) mL per virion per day, \(p=20\) virions per infected cell per day, \(\delta=0.80\) per day, and \(c=4.0\) per day. To find the target-cell concentration at which \(R_0=1\), set the reproductive number equal to one and solve for \(T_0\):

1=\frac{\beta T_{\mathrm{crit}}p}{\delta c} \quad\Longrightarrow\quad T_{\mathrm{crit}}=\frac{\delta c}{\beta p} =\frac{(0.80)(4.0)}{(4.0\times10^{-7})(20)} =\frac{3.2}{8.0\times10^{-6}} =4.0\times10^5\ \text{cells/mL}.

At \(T_0=8.0\times10^5\) cells per mL, the reproductive number is \(R_0=(4.0\times10^{-7})(8.0\times10^5)(20)/[(0.80)(4.0)]=6.4/3.2=2.0\), above the threshold. At the critical concentration the model gives \(R_0=1\); below it, \(R_0<1\), provided the other rates remain unchanged.

Interpretation, Diagnostics, and Limitations

The formula makes the contributions of the infection-cycle rates explicit. Increasing \(\beta\), \(T_0\), or \(p\) increases \(R_0\); increasing infected-cell loss \(\delta\) or virion clearance \(c\) decreases it, if other quantities are held fixed. For example, doubling \(p\) doubles \(R_0\), whereas doubling \(c\) halves it. These are model-based comparisons, not proof that changing one biological process in isolation will produce that outcome in a real system.

A useful diagnostic is a units check. The product \(\beta T_0\) has units of infections per virion per time; \(p/\delta\) has units of virions per infected cell; and \(\beta T_0/c\) has units of infections per virion. Together, the infection-cycle factors give infections per infected cell, conventionally counted as a dimensionless number of new infected cells per infected cell. If a proposed formula has units of time or concentration, a rate or concentration has likely been omitted or placed incorrectly.

The cycle calculation is often the clearest way to explain the biology. The next-generation matrix is a systematic alternative, especially when a model contains several infected states; the local stability calculation independently checks the invasion threshold in this two-state infected subsystem. These methods answer the same threshold question here, but they are not interchangeable descriptions of every detail of the infection trajectory.

Several limits matter in application. \(R_0\) depends on the specified infection-free target-cell level and parameter values; it is not a fixed property of a virus independent of its host and conditions. It is not the viral concentration, the number of virions present, or a probability that infection occurs. Estimates can also be sensitive to uncertain rates, particularly when parameters are inferred indirectly. The model omits delays between cell infection and viral production, immune effects, and changes in infectivity or clearance. Such mechanisms require an expanded model and a reproductive-number calculation consistent with that model; they should not be silently absorbed into this formula unless the effective parameters are defined accordingly.

Common Pitfalls

Omitting a lifetime factor. Using \(\beta T_0p\) alone ignores the durations of infected-cell production and virion infectivity. The correct calculation divides by both \(\delta\) and \(c\).
Using total viral load as \(R_0\). A measured concentration is not an expected number of secondary infected cells. \(R_0\) combines rates, lifetimes, and the susceptible target-cell level.
Using the wrong target-cell level. The basic reproductive number is evaluated at the infection-free target-cell level, not automatically at a depleted level during established infection.
Reading the threshold as a claim about all future dynamics. \(R_0>1\) indicates local invasion potential under the assumptions; it does not specify the eventual peak or persistence of infection.

Key takeaway: In the target-cell-limited model, an infected cell produces \(p/\delta\) virions on average, and each virion produces \(\beta T_0/c\) new infected cells on average. Their product, \(R_0=\beta T_0p/(\delta c)\), determines local invasion potential at the infection-free state.

Check Your Understanding

Answer each question using the model and assumptions stated above. Solutions follow the questions.

If \(p\) is halved while all other parameters remain fixed, what happens to \(R_0\)?
A hypothetical model has \(\beta T_0=0.12\) infections per virion per day, \(p=30\) virions per infected cell per day, \(\delta=0.60\) per day, and \(c=3.0\) per day. Calculate \(R_0\) and interpret the threshold.
Why does the formula use \(T_0\) rather than the target-cell concentration measured late in infection?
What is the main modeling error in the claim, “The viral load is 500, so \(R_0=500\)”?

Solutions. (1) \(R_0\) is halved because it is directly proportional to \(p\). (2) \(R_0=(0.12)(30)/[(0.60)(3.0)]=3.6/1.8=2.0\). It is above one, so the infection-free state is locally unstable to small introductions under these assumptions. (3) \(R_0\) is defined at the infection-free state, where target cells are at their susceptible baseline level; a late-infection concentration reflects processes after infection has altered the system. (4) Viral load is a concentration, whereas \(R_0\) is an expected number of new infections per infected cell across an infection cycle. The claim confuses distinct quantities and does not apply the rate-and-lifetime formula.