A Boundary Value Approach to Optimization with an Application to Salmonella Competition

Abstract

We develop a novel optimization framework to study strategies in ecological competition processes. The optimization method uses theory from dynamical systems describing the asymptotic behavior of a bistable system based on initial conditions, which we implement using a numerical boundary value problem. As an application of our method, we develop a model of the competition between Salmonella Typhimurium and the host’s native microflora, which constantly and densely inhabit the intestinal lining of most mammals. S. Typhimurium invades the gut in two distinct phenotypic populations, one virulent and one avirulent, though the avirulent bacteria have the ability to activate a virulence factor and thereby “switch” into the virulent population. Counterintuitively, some studies have found that the combined population of S. Typhimurium gains an environmental advantage over the commensal microbiota after the virulent subpopulation provokes the body’s inflammatory defenses. Our model represents the competition between the commensal microbiota, the avirulent salmonella, and the virulent salmonella populations and incorporates a simple representation of the immune response. We use our model to predict optimal strategies that would favor salmonella in its competition with the commensal bacteria.

Appendix

To determine the minimal initial salmonella population size necessary to outcompete the commensals, we rescale time by $t=\tau s$ to transform system (4) to the following boundary value problem:

$$\begin{aligned} U^{\prime }= & {} \tau (U(g_U-\alpha (C+V)- \gamma U+\sigma _U f(V) - \kappa M - d_r)) \nonumber \\ V^{\prime }= & {} \tau (V(g_V-\alpha (C+U)- \gamma V+\sigma _V f(V) - (\kappa -r_V)M)+d_rU) \nonumber \\ C^{\prime }= & {} \tau (C(g_C-\alpha (V+U)- \gamma C+\sigma _C f(V) - \kappa M))\nonumber \\ P^{\prime }= & {} aM-\mu _P P\\ D^{\prime }= & {} g_D-(b_P P+b_V V +\rho V+\mu _D)D\nonumber \\ M^{\prime }= & {} \tau ((b_PP(M)+b_VV)D(V,M)-(\delta _VV+\mu _M)M)\nonumber \\ d_r^{\prime }= & {} 0\nonumber \\ f(V)= & {} \frac{\delta V}{1+V},\nonumber \end{aligned}$$

(7)

where $^{\prime }=\frac{\hbox {d}}{\hbox {d}s}$, and

$$\begin{aligned} U(0)= & {} xS_0\nonumber \\ V(0)= & {} (1-x)S_0\nonumber \\ C(0)= & {} g_C/\gamma \\ P(0)= & {} 0\nonumber \\ D(0)= & {} g_D/\mu _D\nonumber \\ M(0)= & {} 0\nonumber \\ C(1)= & {} U(1)+V(1)\nonumber . \end{aligned}$$

(8)

where $\tau$ is a large positive number that we choose.

Here we consider $d_r$ as a stationary variable instead of a parameter. Since we are minimizing $S_0$, it might seem more natural to allow $S_0$ to be a stationary variable and leave $d_r$ as a parameter that we can vary. Solving such a boundary value problem and continuing the solution over varied $d_r$ in AUTO results in solutions as in Fig. 9a, where the minimum value of $S_{\mathrm{nec}}$ appears as a local minimum. Unfortunately, bifurcation continuation methods cannot continue along minima, as minima are not bifurcations. Treating $d_r$ as a variable allows us to identify folds in the solution of the boundary value problem (7)–(8) where the derivative of $d_r$ with respect to $S_0$ becomes unbounded. Such a fold is shown in Fig. 9b. Solving the boundary value problem in this setting therefore allows us to continue the solution over a new parameter; in particular, we can continue the solution in x to determine $S_{\mathrm{min}}=\min _{d_r} S_{\mathrm{nec}}(d_r,x)$.

Fig. 9

Minimal salmonella population sizes necessary in successful invasions derived using boundary conditions (8). a The $S_0$ value obtained by solving system (4) together with $S_0^{\prime }=0$ and boundary conditions (8) over a range of $d_r$ values. The bifurcation continuation software AUTO cannot continue along minima, and so this solution cannot be continued along a new parameter. b The $d_r$ value obtained by solving system (7) with boundary conditions (8) for varied $S_{\mathrm{nec}}\equiv S_0$. Here, the minimum value of $S_{\mathrm{nec}}$ with respect to $d_r$ is given by a fold bifurcation, which allows us to continue to track the minimum as x is varied in AUTO (Color figure online)

We seek the critical value of $S_0$ so that beginning with any initial salmonella population size below this value results in the commensals outcompeting the salmonella, while initial salmonella population sizes above this level yield successful invasion. Our method to determine this value is based on the description in Sect. 2. After rescaling time, solutions to the boundary value problem (7)–(8) for fixed $S_0$ and x return a value of $d_r$ for which we have $C(1)=U(1)+V(1)$, where C, U, and V are now considered functions of rescaled time s. Consequently, we are really searching for the value of $d_r$ for which $S_0=S_{\mathrm{nec}}(d_r,x)$. If $S_0$ is any larger for the fixed x and $d_r$, then $U(s_0)+V(s_0)=C(s_0)$ for some $0\le s_0< 1$, and if $S_0$ is any smaller, then the combined population $U+V$ will not match C before $s=1$, and possibly never will. Analogous to the treatment of system (2) in Sect. 2, we can approximate the critical value of $S_0$ within arbitrary precision by taking $\tau$ sufficiently large. We take $\tau =20$, which seems to be large enough to provide an accurate approximation of the critical $S_0$, as seen in Fig. 10.

Fig. 10

Justification for the choice of $\tau$. The value of $d_r$ in the solution to system (7) with boundary conditions (8) remains near its asymptotic value around and beyond $\tau =20$. Here, $x=0.5$ and $S_0=0.652$ (Color figure online)

Similarly, in order to minimize $S_{\mathrm{nec}}(d_r,x)$ over x for fixed $d_r$, we consider the following system:

$$\begin{aligned} U^{\prime }= & {} \tau (U(g_U-\alpha (C+V)- \gamma U+\sigma _U f(V) - \kappa M - d_r)) \\ V^{\prime }= & {} \tau (V(g_V-\alpha (C+U)- \gamma V+\sigma _V f(V) - (\kappa -r_V)M)+d_rU) \nonumber \\ C^{\prime }= & {} \tau (C(g_C-\alpha (V+U)- \gamma C+\sigma _C f(V) - \kappa M))\nonumber \\ P^{\prime }= & {} aM-\mu _P P\nonumber \\ D^{\prime }= & {} g_D-(b_P P+b_V V +\rho V+\mu _D)D\nonumber \\ M^{\prime }= & {} \tau ((b_PP(M)+b_VV)D(V,M)-(\delta _VV+\mu _M)M)\nonumber \\ x^{\prime }= & {} 0\nonumber \\ f(V)= & {} \frac{\delta V}{1+V},\nonumber \end{aligned}$$

(9)

The technique to solve the boundary value problem (9) with boundary conditions (8) is identical to that described above, with the roles of $d_r$ and x switched.

To add the constraint that the final proportion of avirulent salmonella is equal to the initial proportion of avirulent salmonella (or any fraction $\beta$ of the initial proportion, although we set $\beta =1$ here), we restrict our $(d_r,x)$ parameter values to only those such that $x=U^*/(U^*+V^*)$, where $(U,V,C,M)=(U^*,V^*,0,M^*)$, $U^*,V^*>0$ is the steady state corresponding to a successful invasion. We find such parameter pairs by continuing this steady state along $d_r$ and setting $x=U^*/(U^*+V^*)=U^*(d_r)/(U^*(d_r)+V^*(d_r))$ for each fixed $d_r$. Since we do not have an analytical form for this steady state, we load the $d_r$ and corresponding $x=U^*/(U^*+V^*)$ values into XPPAUT as functions of a new parameter q, so that $(d_r(q),x(q))$ preserves the equality $x_{\mathrm{init}}=x_{\mathrm{final}}$ for all q. We then solve the system

$$\begin{aligned} U^{\prime }= & {} \tau (U(g_U-\alpha (C+V)- \gamma U+\sigma _U f(V) - \kappa M - d_r)) \nonumber \\ V^{\prime }= & {} \tau (V(g_V-\alpha (C+U)- \gamma V+\sigma _V f(V) - (\kappa -r_V)M)+d_rU) \nonumber \\ C^{\prime }= & {} \tau (C(g_C-\alpha (V+U)- \gamma C+\sigma _C f(V) - \kappa M))\nonumber \\ P^{\prime }= & {} aM-\mu _P P\\ D^{\prime }= & {} g_D-(b_P P+b_V V +\rho V+\mu _D)D\nonumber \\ M^{\prime }= & {} \tau ((b_PP(M)+b_VV)D(V,M)-(\delta _VV+\mu _M)M)\nonumber \\ S_0^{\prime }= & {} 0\nonumber \\ f(V)= & {} \frac{\delta V}{1+V},\nonumber \end{aligned}$$

(10)

with boundary conditions (8) in XPPAUT. Here there is no advantage to treating x or $d_r$ as a stationary parameter in place of $S_0$, as our parameter space $(d_r,x)=(d_r,x(d_r))$ is now one dimensional, and consequently, we need only vary a single parameter to find the global minimum $S_{\mathrm{min}}$. We continue the solution of the boundary value problem in AUTO over varied q, from which we can extract the corresponding $d_r(q)$ and x(q) for each q, and the result is plotted in Fig. 8a, b.