R₀ of an SIS model with vaccination and two groups: formula and derivation

Q: What is R₀ for an SIS model with vaccination?

For a two-sex SIS model with transmission rates β_fm (women to men) and β_mf (men to women), recovery rates α_k, exit rates d_k, vaccination rates u_k, waning rates θ_k and vaccine efficacies ε_k, the reproduction number is R₀ = √(β_fm β_mf A_f A_m / ((α_f + d_f)(α_m + d_m))), with A_k = (d_k + θ_k + (1 − ε_k) u_k) / (d_k + θ_k + u_k). Without vaccination A_k = 1. Saldaña et al. (2023) call the value with vaccination the control reproduction number R_c.

By Maria Esther Leyva Borges · Last updated 21 October 2026

In an SIS model of a sexually transmitted infection with women (f), men (m) and an imperfect vaccine (Saldaña et al., 2023), the reproduction number is R₀ = √(β_fm β_mf A_f A_m / ((α_f + d_f)(α_m + d_m))), where A_k is the fraction of group k that vaccination leaves effectively susceptible. It is the geometric mean of the numbers of cases each group causes in the other, reduced by vaccination in both groups.

R_{0} = \sqrt{\frac{β_{f m} β_{m f} (d_{f} + θ_{f} + u_{f} (1 - ϵ_{f})) (d_{m} + θ_{m} + u_{m} (1 - ϵ_{m}))}{(α_{f} + d_{f}) (α_{m} + d_{m}) (d_{f} + θ_{f} + u_{f}) (d_{m} + θ_{m} + u_{m})}}

LaTeX: R_0 = \sqrt{\frac{\beta_{fm} \beta_{mf} \left(d_{f} + \theta_{f} + u_{f} \left(1 - \epsilon_{f}\right)\right) \left(d_{m} + \theta_{m} + u_{m} \left(1 - \epsilon_{m}\right)\right)}{\left(\alpha_{f} + d_{f}\right) \left(\alpha_{m} + d_{m}\right) \left(d_{f} + \theta_{f} + u_{f}\right) \left(d_{m} + \theta_{m} + u_{m}\right)}}

Written with A_k = (d_k + θ_k + (1 − ε_k) u_k) / (d_k + θ_k + u_k), the same formula reads:

R_{0} = \sqrt{\frac{β_{f m} β_{m f} A_{f} A_{m}}{(α_{f} + d_{f}) (α_{m} + d_{m})}}

Saldaña et al. call the value with vaccination the control reproduction number R_c(u_f, u_m) and keep R₀ for the case without vaccination (u_f = u_m = 0); the expression is the same.

What the SIS model with vaccination describes

The model follows a sexually transmitted infection in a population of women and men that only infect each other. Each group k (f or m) has three compartments: S_k susceptible, V_k vaccinated and I_k infected and infectious. It is an SIS model: recovery gives no immunity, so recovered people return to S_k. The vaccine lowers the risk of infection by a factor 1 − ε_k and its protection wanes. People enter and leave the sexually active population at the same rate, so each group keeps a constant size N_k. Its parameters are:

Symbol	Meaning	Typical unit
β_fm	Transmission rate from women to men: rate at which a susceptible man is infected, per unit of prevalence I_f/N_f among women	1/year
β_mf	Transmission rate from men to women	1/year
α_k	Recovery rate of group k; 1/α_k is the mean duration of infection	1/year
d_k	Rate of leaving the sexually active population; 1/d_k is the mean sexually active lifespan	1/year
b_k	Rate of entry into the sexually active population, as unvaccinated susceptibles (b_k = d_k)	1/year
u_k	Vaccination rate of susceptibles in group k	1/year
θ_k	Waning rate of vaccine protection; 1/θ_k is its mean duration	1/year
ε_k	Vaccine efficacy, between 0 (no protection) and 1 (full protection)	none
N_k	Size of group k	people

From the SIS equations to the infection diagram

The infection diagram is read off the equations of the infected compartments, I_f and I_m. With forces of infection

λ_{f m} = \frac{β_{f m} I_{f}}{N_{f}}, λ_{m f} = \frac{β_{m f} I_{m}}{N_{m}}

the model is:

\begin{aligned} \frac{d S_{f}}{d t} & = b_{f} N_{f} - (λ_{m f} + u_{f} + d_{f}) S_{f} + α_{f} I_{f} + θ_{f} V_{f} \\ \frac{d V_{f}}{d t} & = u_{f} S_{f} - (1 - ϵ_{f}) λ_{m f} V_{f} - (d_{f} + θ_{f}) V_{f} \\ \frac{d I_{f}}{d t} & = λ_{m f} (S_{f} + (1 - ϵ_{f}) V_{f}) - (α_{f} + d_{f}) I_{f} \\ \frac{d S_{m}}{d t} & = b_{m} N_{m} - (λ_{f m} + u_{m} + d_{m}) S_{m} + α_{m} I_{m} + θ_{m} V_{m} \\ \frac{d V_{m}}{d t} & = u_{m} S_{m} - (1 - ϵ_{m}) λ_{f m} V_{m} - (d_{m} + θ_{m}) V_{m} \\ \frac{d I_{m}}{d t} & = λ_{f m} (S_{m} + (1 - ϵ_{m}) V_{m}) - (α_{m} + d_{m}) I_{m} \end{aligned}

At the disease-free equilibrium (DFE) I_f = I_m = 0. With b_k = d_k, the S and V equations of each group give

S_{k}^{*} = \frac{N_{k} (d_{k} + θ_{k})}{d_{k} + θ_{k} + u_{k}}, V_{k}^{*} = \frac{N_{k} u_{k}}{d_{k} + θ_{k} + u_{k}}

Susceptible and vaccinated people are both infected, by the same partners, so what matters for transmission is their combination

A_{k} = \frac{S_{k}^{*} + (1 - ϵ_{k}) V_{k}^{*}}{N_{k}} = \frac{d_{k} + θ_{k} + (1 - ϵ_{k}) u_{k}}{d_{k} + θ_{k} + u_{k}}

A_k is the fraction of group k that vaccination leaves effectively susceptible: it is 1 without vaccination (u_k = 0) and falls to (d_k + θ_k) / (d_k + θ_k + u_k) with a perfect vaccine (ε_k = 1). Each term of the I_f and I_m equations becomes one arrow:

Term	Where it appears	Kind	Arrow in the diagram
λ_fm S_m	+ in dI_m/dt	New infection of susceptible men	Dashed, I_f → I_m, label β_fm S_m*/N_f
(1 − ε_m) λ_fm V_m	+ in dI_m/dt	New infection of vaccinated men	Dashed, I_f → I_m, label (1 − ε_m) β_fm V_m*/N_f
λ_mf S_f	+ in dI_f/dt	New infection of susceptible women	Dashed, I_m → I_f, label β_mf S_f*/N_m
(1 − ε_f) λ_mf V_f	+ in dI_f/dt	New infection of vaccinated women	Dashed, I_m → I_f, label (1 − ε_f) β_mf V_f*/N_m
α_f I_f	− in dI_f/dt	Outflow (recovery, returns to S_f)	Solid, out of I_f, label α_f
d_f I_f	− in dI_f/dt	Outflow (leaves the population)	Solid, out of I_f, label d_f
α_m I_m	− in dI_m/dt	Outflow (recovery, returns to S_m)	Solid, out of I_m, label α_m
d_m I_m	− in dI_m/dt	Outflow (leaves the population)	Solid, out of I_m, label d_m

There are no transfer arrows: nobody moves from I_f to I_m, so the two groups are linked only through infection. The S and V equations, with vaccination, waning and the return of recovered people, are used only to find S_k* and V_k*; their infection terms are the other side of the new-infection terms, so they are not drawn twice. See how to build an infection diagram for the general rules.

The infection diagram of the SIS model with vaccination

Both infected compartments, I_f and I_m, are infectious nodes. Each direction has two dashed new-infection arrows, one for susceptible and one for vaccinated people, and each node has an outflow α_k + d_k. Vaccination adds an entry route and lowers its rate, but it does not add any infected compartment or change the cycle I_f → I_m → I_f. The diagram separates changes in the intensity of transmission from changes in its structure.

Infection diagram of the two-group SIS model with vaccination. Red: infectious nodes; dashed: new-infection arrows; solid: outflow arrows. There are no transfer arrows.

How R₀ is derived

The diagram has a single infection cycle, I_f → I_m → I_f. R0Calc follows it and multiplies, for each node, the rate coming in by the rate going out. I_m receives the sum of its two arrows, β_fm (S_m* + (1 − ε_m) V_m*)/N_f, and loses α_m + d_m; I_f receives β_mf (S_f* + (1 − ε_f) V_f*)/N_m and loses α_f + d_f. The product counts the cases a case produces in its own group after a round trip: it infects people in the other group, who in turn infect people in the first. That is two generations, because the cycle passes through two infectious nodes, so R0Calc takes the square root to get cases per generation. Substituting the DFE values gives the formula above.

The next-generation matrix method (Diekmann, Heesterbeek and Metz, 1990; van den Driessche and Watmough, 2002) gives the same result. With infected compartments (I_f, I_m):

F = (\begin{matrix} 0 & \frac{β_{m f} (S_{f}^{*} + (1 - ϵ_{f}) V_{f}^{*})}{N_{m}} \\ \frac{β_{f m} (S_{m}^{*} + (1 - ϵ_{m}) V_{m}^{*})}{N_{f}} & 0 \end{matrix}),

V = (\begin{matrix} α_{f} + d_{f} & 0 \\ 0 & α_{m} + d_{m} \end{matrix})

FV⁻¹ has a zero diagonal, so its eigenvalues are ±√(K₁₂ K₂₁), where K₁₂ and K₂₁ are its off-diagonal entries, and the spectral radius is R₀. This is the expression Saldaña et al. obtain, and R0Calc's result equals it exactly (checked symbolically with SymPy).

Worked example

Take β_fm = 1.5 and β_mf = 1.0 per year, a one-year mean infection (α_f = α_m = 1 per year), a 40-year sexually active life (d_f = d_m = 0.025 per year), and a vaccine with efficacy ε_f = ε_m = 0.8 whose protection lasts 10 years on average (θ_f = θ_m = 0.1 per year). Without vaccination, R₀ = √(1.5 × 1.0) / 1.025 ≈ 1.19: the infection can spread. Vaccinating women at u_f = 0.2 and men at u_m = 0.05 per year gives A_f = 0.165/0.325 ≈ 0.508 and A_m = 0.135/0.175 ≈ 0.771, so R₀ ≈ 1.19 × √(0.508 × 0.771) ≈ 0.75, below the threshold. Vaccinating only women at the same rate (u_m = 0) gives R₀ ≈ 0.85. These are illustrative values, not estimates for a specific infection.

Why vaccinating one group protects both: A_f and A_m multiply under the same square root, so lowering either one lowers R₀. Each group can only be infected by the other, and every chain of transmission passes through both.

Special case: no vaccination

With u_f = u_m = 0, A_f = A_m = 1 and the formula becomes R₀ = √(β_fm β_mf / ((α_f + d_f)(α_m + d_m))):

R_{0} = \sqrt{\frac{β_{f m} β_{m f}}{(α_{f} + d_{f}) (α_{m} + d_{m})}}

LaTeX: R_0 = \sqrt{\frac{\beta_{fm} \beta_{mf}}{\left(\alpha_{f} + d_{f}\right) \left(\alpha_{m} + d_{m}\right)}}

One infectious woman infects β_fm (N_m/N_f) / (α_f + d_f) men, and one infectious man infects β_mf (N_f/N_m) / (α_m + d_m) women. R₀ is the geometric mean of these two numbers; the group sizes cancel in the product.

Compute it in R0Calc

Open the calculator, paste the YAML below into the YAML panel and choose Apply the YAML to the diagram (the ← button between the YAML and the diagram).
R₀ appears under the diagram as the formula at the top of this page. Copy it as LaTeX, or open the cycle breakdown to see the cycle I_f → I_m → I_f.
Set u_f or u_m to 0 in the labels to see the effect of vaccinating only one group, or add a second infected stage to one of the groups.

Labels in R0Calc cannot contain an asterisk, so the YAML writes the DFE values in terms of the parameters: S_m* + (1 − ε_m) V_m* = N_m A_m, and the same for women. Each direction is one entry whose label is the sum of the two routes, which keeps the result short; two entries, one per route, give the same R₀ written as a sum of the two terms.

non_infectious_infested: []
infectious_infested:
  - I_f
  - I_m
new_infections_arrows:
  - from: I_f
    to: I_m
    param: \frac{\beta_{fm} N_m (d_m + \theta_m + (1-\epsilon_m) u_m)}{N_f (d_m + \theta_m + u_m)}
  - from: I_m
    to: I_f
    param: \frac{\beta_{mf} N_f (d_f + \theta_f + (1-\epsilon_f) u_f)}{N_m (d_f + \theta_f + u_f)}
transfer_arrows: []
outs_arrows:
  - from: I_f
    param: \alpha_f + d_f
  - from: I_m
    param: \alpha_m + d_m

Frequently asked questions

What is R₀ for an SIS model with vaccination?

R₀ = √(β_fm β_mf A_f A_m / ((α_f + d_f)(α_m + d_m))), with A_k = (d_k + θ_k + (1 − ε_k) u_k) / (d_k + θ_k + u_k). Without vaccination A_k = 1. Saldaña et al. (2023) call the value with vaccination the control reproduction number R_c.

Why is there a square root in R₀ for two-sex transmission?

Because the only infection cycle passes through two infectious compartments: a case in one group causes cases in the other, which in turn cause cases in the first. The product under the root counts cases after those two generations, and R₀ counts cases per generation, so it is the square root, the geometric mean of what each group causes in the other.

Does an imperfect vaccine change the infection diagram?

It adds a second new-infection arrow into each group, for vaccinated people who are still infected at a reduced rate, and it lowers the rates through the DFE values S* and V*. It adds no infected compartment and does not change the cycle I_f → I_m → I_f.

References

Saldaña F., Steindorf V., Srivastav A. K., Stollenwerk N., Aguiar M. (2023). Optimal vaccine allocation for the control of sexually transmitted infections. Journal of Mathematical Biology, 86, 75. doi:10.1007/s00285-023-01910-y
Diekmann O., Heesterbeek J. A. P., Metz J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28, 365–382.
van den Driessche P., Watmough J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180, 29–48.