Direct transmission via households informs models of disease and intervention dynamics in cholera

While several basic properties of cholera outbreaks are common to most settings—the pathophysiology of the disease, the waterborne nature of transmission, and others—recent findings suggest that transmission within households may play a larger role in cholera outbreaks than previously appreciated. Important features of cholera outbreaks have long been effectively modeled with mathematical and computational approaches, but little is known about how variation in direct transmission via households may influence epidemic dynamics. In this study, we construct a mathematical model of cholera that incorporates transmission within and between households. We observe that variation in the magnitude of household transmission changes multiple features of disease dynamics, including the severity and duration of outbreaks. Strikingly, we observe that household transmission influences the effectiveness of possible public health interventions (e.g. water treatment, antibiotics, vaccines). We find that vaccine interventions are more effective than water treatment or antibiotic administration when direct household transmission is present. Summarizing, we position these results within the landscape of existing models of cholera, and speculate on its implications for epidemiology and public health.


Model dynamics: sum total values
In S1 Table we  Basic reproductive ratio -R 0 The numeric value for the basic reproductive ratio R 0 of V. cholera referenced in the main text, and used as part of our sensitivity analysis, was derived following the methods of  No direct infection η = 0 1.55 · 10 7 1.04 · 10 7 8.99 · 10 11 direct infection η = nominal value 7.74 · 10 7 5.24 · 10 7 4.53 · 10 12 direct infection & 0.5x water consumption 3.20 · 10 7 2.88 · 10 7 2.51 · 10 12 S1 Table. In S1 Table we present the summed total number of counts for all nine dynamic cases of this model considered in Fig 3. Of note, the values reported in column 3 of S1 Table are that of the sum total amount of bacteria across both reservoirs for each of the three model runs.
Firstly, we construct the arrays t = (t 0 , t 1 , ..., t m ) and σ = (σ 0 , σ 1 , ..., σ m ). The i th element of t, denoted t i , is defined to be the rate term associated with the flow of new infection into the i th compartment. That is, the flow of infection between two infected compartments is not included in t. The i th element of σ, denoted σ i , is defined to be the sum of all other rate terms associated with flows into or out of the i th compartment. That is, the total rate of change of the i th compartment is given by t i + σ i . In calculating R 0 , it suffices to restrict the index i in t i and σ i to infected compartments only. In the case of V. cholerae, these are the compartments I, A, W L , and W H . Below we present the elements of t and σ at the disease free equilibrium.
We calculate the corresponding T and Σ matrices. These are m × m matrices (in this case m = 4) defined by, where X j is the j th agent, selected from the m agents associated with t and σ, and x 0 is the disease-free equilibrium of the model. Calculating these derivatives for the V. cholera model one finds, Next we calculate the inverse of the matrix Σ. For our case this turns out to be, We then calculate the matrix product −T · Σ −1 .
Due to its analytic complexity, an analytic expression for R 0 was not explicitly determined. Instead the characteristic polynomial of −T · Σ −1 from which the maximum eigenvalue of this expression, was numerically determined.

Jacobian ODE system analysis & related values
Here we present both a symbolic and numeric form of the Jacobian Matrix of the V. cholerea ODE system, the numeric form being at the disease-free equilibrium. The eigenvalues for the disease-free Jacobian are predominantly non-positive but include 2 positive values. As with any set of ordinary differential equations, the Jacobian Matrix represents a linear approximation of the flow of the system in phase space at the selected point. The eigenvectors give the direction of flow and the sign of the associated eigenvalues indicates whether the flow is toward or away from the selected point. Given that the Jacobian Matrix is computed at the disease-free equilibrium and there exists positive eigenvalues, we can say that the DFE is an unstable equilibrium (given the nominal chosen model parameters) and the system tends not to equilibriate to the DFE point in phase space [2,3]. This is consistent with an R 0 value greater than one.