Consider the SIR model with births and deaths, written with standard incidence: S′ = Λ − βSI/N − μS, I′ = βSI/N − (γ + μ)I and R′ = γI − μR, where N = S + I + R. The explicit Euler method with step h multiplies S by the factor 1 − hβI/N − hμ before adding the recruitment term, and multiplies I by a factor containing −h(γ + μ). Positivity is guaranteed only if these factors are nonnegative, for which h(β + μ) ≤ 1 and h(γ + μ) ≤ 1 suffice. In models with fast processes, such as rapid progression through a latent class or high contact rates in a subgroup, the bound forces steps far smaller than accuracy alone would require.

Higher order does not remove the restriction. Bolley and Crouzeix (1978) showed that for a broad class of standard one-step and multistep methods, unconditional positivity is incompatible with order greater than one, and the theory of strong stability preserving methods expresses the admissible step of a Runge-Kutta scheme as a multiple of the Euler bound (Gottlieb et al., 2001). The implicit Euler method is unconditionally positive for problems of this type in exact arithmetic, but each step requires the solution of a nonlinear system, and the iteration used to solve it may itself leave the positive region. Discrete-time epidemic models built directly as difference equations show how badly a careless discretisation can behave, including period doubling and chaos that the differential equations exclude (Allen, 1994).

Mickens (1994) proposed building difference schemes by rules that depart from textbook practice in two main ways. The step h in the denominator of the discrete derivative is replaced by a function φ(h) that behaves like h for small steps, and nonlinear terms are evaluated at more than one time level, for example a product SI as S at the new level times I at the old one. The rules are reviewed in Mickens (2002), and Anguelov and Lubuma (2001) put them on a rigorous footing, defining a scheme as nonstandard when it uses either device and proving convergence and elementary stability results. The guiding idea is dynamic consistency: the discrete system should share the qualitative properties of the continuous one, such as positivity, invariants, equilibria and their stability, for all step sizes. Anguelov et al. (2014) applied the idea systematically to epidemiological models.

A related device comes from computational fluid dynamics. Patankar (1980) treated sink terms by multiplying them by the ratio of the new value of the variable to the old one, which makes the update a division by a positive number and keeps the variable positive. Burchard et al. (2003) observed that the original trick destroys conservation, repaired it by weighting each production term in the same way as the destruction term it mirrors, and extended the construction to second order. For epidemic models the two traditions lead to closely related formulas, and the production and destruction language gives a convenient bookkeeping for the first steps of the procedure below.

Before any discretisation the modeller writes down what the scheme must reproduce: the invariant region, the equation satisfied by the total population, the equilibria, and the basic reproduction number, computed if necessary by the next generation method of van den Driessche and Watmough (2002). For the illustration, the nonnegative orthant is invariant, the total satisfies N′ = Λ − μN, there is a disease-free equilibrium with S = Λ/μ, and the reproduction number is β/(γ + μ).

Each right-hand side is written as a sum of nonnegative production terms minus a sum of destruction terms, and every destruction term is checked to contain the variable of its own equation as a factor. This is the structural reason why solutions cannot cross zero, and it holds for compartment models by construction, since an outflow from a class is proportional to the number in it. Each transfer between compartments appears twice, once as a loss and once as a gain, and the pairs are noted.

In each equation the variable being depleted is taken at the new time level in every destruction term, and all other factors at the old level or at a level already computed. The gain that mirrors a loss is given exactly the same discrete form. Writing k for the time index, the susceptible equation becomes (Sk+1 − Sk)/φ = Λ − βSk+1Ik/Nk − μSk+1, the infective equation uses the same incidence term as a gain and (γ + μ)Ik+1 as its loss, and the removed equation has gain γIk+1 and loss μRk+1.

Adding the discrete equations gives a scheme for the total population, and the denominator function is chosen so that this scheme is exact. In the illustration the sum is (Nk+1 − Nk)/φ = Λ − μNk+1, and the choice φ(h) = (eμh − 1)/μ makes Nk coincide with the exact solution of the linear balance law at every grid point. When there is no demographic turnover the total is constant and φ(h) = h may be kept. For models in which the total does not satisfy a closed equation, Mickens (2007) gives a general recipe for constructing φ from the linear loss rates of the system.

Because each unknown appears linearly in its own equation, every update is solved by one division. The compartments are processed in the order in which infection moves through them, so that a value needed at the new level is already available. Here Sk+1 is the quotient of Sk + φΛ by 1 + φ(βIk/Nk + μ); Ik+1 is the quotient of Ik(1 + φβSk+1/Nk) by 1 + φ(γ + μ); and Rk+1 is the quotient of Rk + φγIk+1 by 1 + φμ. Numerators and denominators are sums of nonnegative quantities, so positivity holds for every step size without further argument.

The list from Step 1 is now checked against the scheme by hand or with a computer algebra system. Fixed points are found by setting new and old values equal; in the illustration they satisfy the same equations as the equilibria of the differential system. Linearising the infective update at the disease-free state gives the multiplier (1 + φβ)/(1 + φ(γ + μ)), which is below one exactly when β < γ + μ, so the discrete threshold coincides with the continuous one for every h. A scheme that fails a check is revised at Step 3, usually by moving a factor from one time level to the other.

Qualitative fidelity says nothing about the size of the error. The final step compares the scheme with a high-order adaptive method run at a tight tolerance, on a sequence of halved steps, to confirm the expected first-order convergence and to find the step at which the error becomes acceptable for the purpose in hand. Table 1 sets the resulting scheme beside the standard alternatives.