Fig. 1.

Flow diagram of the nested model. Within-host model: the number of bacteria produced at time $\tau$ with resistance level x is $\frac{1}{{\left(1,+,{\int }_{-\infty }^{+\infty },b_i^{\vartheta },(\tau ,x),\text{d},x\right)}^{\kappa }}{\int }_{-\infty }^{+\infty }J(x-y)p(y)b_i^{\vartheta }(\tau ,y)\text{d}y$, where $J(x-y)$ is the probability for a bacterial with resistance level $y\in \mathbb{R}$ to mutate towards a level $x\in \mathbb{R}$ and p(y) is the bacterial intrinsic growth rate. Bacterial cells with resistance level x, within an individual with immune system level i are cleared either by the immune system at rate ${\mu }_i(x)$ or by the antimicrobial efficiency at rate k(x). Between-host model: susceptible individuals are recruited at a constant rate ${\mathrm{\Lambda }}_i$. $I_i^T(t,\tau )$ and $I_i^U(t,\tau )$ are respectively treated and untreated infected individuals at time t, which are infected since time $\tau$. The force of infection in the whole population at time t is $\lambda (t)={\sum }_{i\in \mathcal{I}}{\int }_0^{\infty }[{\beta }_i^T(\tau )I_i^T(t,\tau )+{\beta }_i^U(\tau )I_i^U(t,\tau )]\text{d}\tau$, with ${\beta }_i^{\vartheta }(\tau )$ the disease transmission rate of an infected individual $\tau$-time post infection. At the time t, new infections occur at rate $\lambda (t)S_i(t)$, and are either treated with a probability $q_i^T$ or untreated with a probability $q_i^U=1-q_i^T$. The natural death rate of individuals is ${\mu }_h$. If infected since time $\tau$, the loss rate is ${\alpha }_i^{\vartheta }(\tau )$. Untreated individuals, and infected since time $\tau$ start the treatment at rate ${\omega }_T^U(\tau )$ while treated individuals stop the treatment at rate ${\omega }_U^T(\tau )$