We are interested in dispersive equations of the Korteweg-de Vries type of the form `u_t + f(u)_x - (-\Delta)^{\alpha/2}u_x = 0`,
for `\alpha \geq 1`. For example, one can cite the Korteweg-de Vries equation when `\alpha = 2` and the Benjamin-Ono equation when `\alpha = 1`.
These equations have been used, among others, to describe the wave propagation in shallow water and can be derived from Newton's second law.
Because of the nonlinearity `f(u)_x` depending on the spatial derivative, and thus the difficulty to prove the Lispchitz property,
to study the initial value problem, the global existence and the continuity with respect to the initial datum, remains difficult.
Theorem: Let `\alpha \geq 1` and `u_0 \in H^{\alpha/2}(\mathbb{R})`. Then the initial value problem is globally well posed in `H^{s}(\mathbb{R}), s \geq \alpha/2`.
Moreover pour all `t \in \mathbb{R}`, `||u(t)||_{\alpha/2} = ||u_0(t)||_{\alpha/2}, ||u(t)||_s \leq C e^{e^|t|}`, while
for `|t| \leq 1/||u_0(t)||_s, ||u(t)||_s \leq C ||u_0(t)||_s`.
with J. He, Remark on the well-posedness of weakly dispersive equations, ESAIM Proc. Surveys 64, (2018), 111-120.link
with N. Bedjaoui and J. M.C. Correia, Well-posedness of the generalized Korteweg-de Vries-Burgers equation with nonlinear dispersion and nonlinear dissipation, Int. J. Pure Math. 2,(2015), 38-46.hal
with Y. Zhang, Comparison of solutions of Boussinesq systems, Adv Pure Appl Anal. 5, 2, (2014), 101-115.hal
Comparison between shallow water models in 2-dimension (in french), M2AN Math. Model. Numer. Anal.s 41, 3, (2007), 513-542. hal
When the solution is defined in a bounded domain, change his point of view may be useful.
When `f(u) = u^2/2`, there exists a locally invertible change of variables `v = \Lambda u = u - \frac{1}{2}\sum_{k\neq k1 \in \mathbb{Z}^*} \frac{e^{ikx} {ik}/(1+|k|^\alpha)}{\sigma(k_1)+\sigma(k-k_1)-\sigma(k)}\hat u(k_1) \hat u (k-k_1)`,
where `\sigma(k) = {ik}/(1+k^\alpha)` is the dispersion relation, and `v` is solution of the normal form
`v_t + \tilde{f}(v)_x - (-\Delta)^{\alpha/2}v_x = 0`. Here `\tilde{f}` is a trilinear function that allows to extend the existence of solution.
This method is based on the theory of Poincaré's normal forms.
Theorem: Let `s > \alpha/2`. There exists `\varepsilon > 0` such that if `|| u_0 || \leq \varepsilon`, then the unique solution
`u \in \mathcal{C}([-T^2,T^2]; H^s(\mathbb{T}))` with `T \~ 1/\varepsilon` satisfies for `|t| \leq T^2, ||u(t)||_s \leq C ||u_0(t)||_s.`
Asymptotic behavior of small solution of the Benjamin-Ono equations with time-dependant coefficients, J. Appl. Anal. 21, 1, (2015) 9–23. hal
Continuation of time bounds for a regularized Boussinesq system, Acta Appl. Math., 117, (2012), 1-13. hal
Long time bounds for the periodic Benjamin-Ono-BBM equation, Nonlin. Anal. TMA 71, 10, (2009), 5010-5021. hal
Another important question concerns the uniqueness of the solutions. It can be put in the following form:
let `u(x,t)=0` for all `(t,x) \in I \times \Omega` with `I \times \Omega \subset [0,T] \times \mathbb{R}`, then `u = 0?`
When `I = [0,T]` and `\Omega` is compact, an adequate analytical development has been proposed by Bourgain. Thanks to the Paley-Wiener theorem,
it is possible to write `\hat u (\xi + i \sigma, t) = e^{i(\xi + i \sigma)^{\alpha+1}} \hat u_0 (\xi + i \sigma).`
Adding a decay condition on the initial data `u_0` allows to conclude positively.
Theorem: Let `u \in \mathcal{C}([0,T];H^s)` with `s\geq 4` such that for all `t \in [0,T], u(t)` is compactly supported. Then `u` is identically null.
This assumption can be weakened by taking `I \times \Omega` an open set of `[0,T] \times \mathbb{R}`. It is then necessary to determine a Carleman inequality
`\int |u|^2 e^{\psi(x,t)} dx dt \leq \frac{C}{\tau^{\alpha+1}} \int |u_t + f(u)_x - (-\Delta)^{\alpha/2}u_x|^2 e^{\psi(x,t)} dx dt,`
where `\psi` is a convex function. Thus, when the right-hand-side is bounded, it is sufficient to `\tau \to \infty` so that u vanishes.
Theorem: Let `u \in \mathcal{C}([0,T];H^s)` with `s\geq 4` such that `u=0`in an open set of `[0,T] \times \mathbb{R}`. Then `u`is identically null.
with A. Esfahani, Carleman estimates and unique continuation property for N-dimensional Benjamin-Bona-Mahony equations, J. Int. Eq Applications, 33, 4 (2021), 443-450. link
A note on Carleman estimates and Unique continuation property for the Boussinesq system, Comm. Math. Anal. 15, 2, (2013), 29-38. hal
Carleman estimates and Unique continuation property for the Kadomtsev-Petviashvili equations, Appl. Anal. 92, 12, (2012), 2526-2535. hal
Unique continuation property for Boussinesq-type systems, Comm. Math. Anal., 9, 1, (2010), 121-127. hal
Unique continuation property for the KP-BBM equation, Diff. Int. Eq. 22, 3-4, (2009), 393-399. hal
1.2 Decay of the solution of nonlinear dispersive equations
One method is to use dispersive (or Strichartz) inequalities of the type `||u(t)||_\infty \leq {||u_0||_{L^1 \cap H^s}}/(1+|t|)^{1/{\alpha+1}}.`
This reflects the decay of the solutions, due to dispersion, i.e. the separation of the different Fourier modes that move at distinct speeds.
Such an inequality is deduced from the dispersion relation. The solution of the equation linearized around `0` is written as an oscillating integral (via the Fourier transform)
`u(x,t) = 1/{2\pi} \int e^{i(\xi^{\alpha+1}t + x \xi)} \hat u_0{\xi} d\xi,`
and a stationary phase lemma allows us to conclude.
Theorem: Let `\rho \geq \alpha +2.` Then for `u_0 \in L^p \cap H^s` small enough,
there exists a unique global in time solution `u \in \mathcal{C}(\mathbb{R};L^p \cap H^s)` and for all `t \in \mathbb{R}`,
`||u(t)||_q \leq {||u_0||_{L^p \cap H^s}}/(1+|t|)^{{1-2/q}/{\alpha+1}}.`
Spectral methods are very suitable for the discretization of such equations, nonlinearity and derivatives being easily treated.
To preserve the conservation laws, high order schemes are required. The resulting nonlinear system is then solved using a fixed point method.
To avoid periodic boundary conditions while preserving conservation of mass and energy, we have recently developed compact schemes.
with N. Bedjaoui and R. Kumar, Asymptotic behavior of solution of Whitham-Broer-Kaup type equations with negative dispersion, J. Applied Analysis, 28, 1, (2022), 109-119. link
Asymptotic behavior of small solution of the Benjamin-Ono equations with time-dependant coefficients, J. Appl. Anal. 21, 1, (2015) 9–23. hal
On the decay in time of solutions of the generalized regularized Boussinesq system, Adv. Nonlin. Stud. 10, (2010), 387-349. hal
with F. Hamidouche et S. Mefire, Numerical study of the solutions of the 3D generalized Kadomtsev-Petviashvili equations for long times, CiCP Comm. Comput. Phys. 6, 5 (2009), 1022-1062. hal
On the decay in time of solutions of some generalized regularized long waves equations, CPAA Comm. Pure Appl. Ana. 7, 3, (2008), 513-532. hal
We are interested in the stochastic oscillating integral `u(w, x,t) = 1/{2\pi} \int e^{i(\sigma(\xi) W_t + x \xi)} \hat u_0{\xi} d\xi.`
If `\sigma` is the symbol of the operator `A`, then this function is a solution of the linear stochastic PDE `du + Au \circ dW = 0`
or in Itô form `du + Au dW - 1/2 A^2u dt = 0.`
Following the methods developed in the deterministic case above, we established a stationary phase lemma that yields an average decay result.
If `\sigma(\xi) = \xi/(1+\xi^2)`, we obtain global well-posedness with a nonlinearity `A u^{p+1}`.
Theorem: Let `\rho \geq 9`. Then for `u_0 \in L^2 \cap H^4` small enough,
there exists a unique global in time solution with path in `\mathcal{C}(\mathbb{R}; H^1)` and for all `t \in \mathbb{R}`,
`\mathbb{E}(||u(t)||_\infty) \leq {C(||u_0||)}/(1+|t|)^{1/6}.`
This stochastic PDE can be discretized by spectral methods. To preserve mass and energy, a midpoint-in-time scheme has been chosen.
Note that in general the order of convergence of a scheme is lower for a stochastic equation than for its deterministic version.
Because of the speed of convergence of the Stratonovich integral, one expects a (strong) order `1/2`.
A proper writing of the global error and the consistency error allows to show that the order is `1`.
Theorem: Let `T>0` and `u_0 \in H^1`. For `N>0` large enough and `\delta t = T/N`, we have for all `n\leq N`,
`\lim_{C \to \infty} \mathbb{P} ( Sup_{\delta t} {||u(t_n)-u_n||_{H^1}}/{\delta t} \geq C ) = 0.`
with S. Dumont and O. Goubet, Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion, In memory of Ezzeddine Zahrouni DCDS 8 (2021) 2877-2891. link
with G. Fenger and O. Goubet, Numerical analysis of the midpoint scheme for the generalized Benjamin-Bona-Mahony equation with white noise dispersion, CiCP Comm. Comput. Phys. 26, 5 (2019), 1397-1414. link
with M. Chen and O. Goubet, Generalized regularized long wave equation with white noise dispersion, Stochastic PDE 5, 3, (2017), 319-342. link
1.3 Analysis of damped nonlinear dispersive equations
The dissipation in the equations reflects the damping of waves due for example to the Coriolis force, or the mangrove forests...
From a mathematical point of view, the dissipation can be written `u_t + f(u)_x - \delta (-\Delta)^{\alpha/2}u_x + \varepsilon \mathcal{A}u = 0,`
with `\langle \mathcal{A}u, u \rangle > 0`. Thus, multiplying the equation by u and integrating in space, we obtain
`d/{dt} ||u||^2 = - \varepsilon \langle \mathcal{A}u, u \rangle < 0,`
and the `L^2-`norm decreases whereas it was preserved in the purely dispersive case above. The study of the space
`\mathcal{H} := \{ u:\mathbb{R} \to \mathbb{R}, \langle \mathcal{A}u, u \rangle < + \infty \}`
is then of primary importance. Relevant choices of damping, e.g. `\hat{\mathcal{A} u} = -\gamma(\xi) \hat u(\xi)`,
provide continuous and compact injection properties in `L^\infty`. In particular it allows `\mathcal{H}` to be an algebra.
This may clearly define the nonlinearity `f(u)_x.`
Theorem: Let `s>0`. If `1/\gamma^s \in L^1(\mathbb{R}),` then `\mathcal{H}_{\gamma^s} ` is continuously injected in `L^\infty(\mathbb{R})`.
If `\gamma_1(\xi) \geq \gamma_2(\xi),` then `\mathcal{H}_{\gamma_1^s}` is compactly injected in `\mathcal{H}_{\gamma_2^s}.`
If `1/\gamma^s \in L^1(\mathbb{R}),` and for all `\xi, \xi_1 \in \mathbb{R}, \sqrt{\gamma(\xi)} \leq \sqrt{\gamma(\xi-\xi_1)} + \sqrt{\gamma(\xi_1)},`
then `\mathcal{H}_{\gamma^s} ` is an algebra.
From then on, we can build other operators that, in addition to damp the `L^2-`norm, stabilize the energy. We were interested
more specifically on operators with memory as `\mathcal{A}u = - \int_0^{\infty} \mu(s) u_{x x}(t-s) ds.`
Theorem: Let `\mu` being such that there exists `\delta >0`, for almost every `s\in \mathbb{R}^+` `\mu'(s) +\delta \mu(s) \leq 0.`
Then there exists `\omega >0` and a positive increasing function `\mathcal{Q}`, depending on `\mu` and the domain, such that
`||u(t)||^2 \leq \mathcal{Q}(u_0) e^{-\omega t}.`
with F. Dell'Oro, Benjamin-Bona-Mahony equations with memory and Rayleigh friction, App. Math. Optimization 83 (2021) 813-831. link
with F. Dell’Oro, O. Goubet and V. Pata, A semidiscrete scheme for evolution equations with memory, DCDS A 39, 10, (2019), 5637-5658. link
with F. Dell’Oro, O. Goubet and V. Pata, Global attractors for the Benjamin-Bona-Mahony equation with memory, Indiana University Mathematics Journal 69, 3, (2020), 749-783. arxiv
with J.P. Chehab and P. Garnier, Long-time behavior of solutions of a BBM equation with generalized damping, DCDS-B 20, 7, (2015) 1897-1915.hal
1.4 Dispersive-dissipative perturbations of hyperbolic conservation law
Formally, when `\varepsilon` and `\delta` go `0`, the equation becomes the hyperbolic conservation law
`u_t + f(u)_x = 0,`
for which there is a unique entropy solution. It is possible to prove this convergence as soon as the nonlinearity
`f` is convex and the dissipation is large compared to the dispersion (`\delta ≪ \varepsilon`).
This allows to attenuate the oscillations due to the dispersion. Consider `u_t + f(u)_x = \varepsilon u_{x x} - \delta u_{x x x}.`
Three cases are possible :
If `\delta` is large compared to `\varepsilon`, threre is no convergence.
If `\delta= o(\varepsilon^2)`, the sequence converges to the entropy solution.
If `\delta = O(\varepsilon^2)`, the sequence converges but not necessary to the entropy solution.
We obtained similar result for a nonlinear dispersion `g(u_{x x})_x`. Theorem: Let `varepsilon>0` and `f:\mathbb{R} \to \mathbb{R}` a convex function with `f''(u) \leq C (1+|u|^\beta)`.
If `g(u) = -u^2, \delta = o(\varepsilon^{5/2})` and `0\leq \beta < 1/2`.
If `g(u) = -|u|, \delta = o(\varepsilon^2)` and `0\leq \beta < 3`.
If `g(u) = -|u|^n` with `1< n < 2`, and `\delta = o(\varepsilon^{{3n-1}/2}, \varepsilon^{{5n-1}/{2(2n-1)}})` and `0\leq \beta < {4-n}/n^2`.
Then the sequence converges to the entropy solution.
with N. Bedjaoui, J. M.C. Correia, On a limit of perturbed conservation laws with saturating diffusion and non-positive dispersion, Z. Angew. Math. Phys. (2020), 20 pages. link
with N. Bedjaoui, J. M.C. Correia, Convergence of a family of perturbed conservation laws with diffusion and non-positive dispersion, Nonlinear Analysis 192 (2020), 15 pages. link
with N. Bedjaoui and J. M.C. Correia, On a limit of perturbed conservation laws with diffusion and non-positive dispersion, Comm. Math. Sci. 14, 6, (2016), 1501-1516. hal
2. Advection-reaction-diffusion for biological invasions
2.1 Parabolic-Hyperbolic coupling for sugar transport in phloem
Münch (1930) introduced the movement of sap in the phloem as a hydrostatic pressure flow. Sugar concentration is
high in the leaves where solutes are synthesized and loaded into the phloem. Water is drawn by osmosis, leading to a high
turgor pressure that pushes the sap down (toward the roots). Sucrose distribution (C) via phloem pressure (P) is described
by a system of nonlinear transport PDEs as
`e/E \partial_t P - \nabla \cdot ( e/\mu k \nabla P ) - L_R (\psi - P + RTC) -V_s U = 0, e \partial_t C - \nabla \cdot ( e/\mu C k \nabla P ) - U = 0.`
This system of equations presents both theoretical and numerical difficulties. On the one hand, it is a nonlinear system.
And on the other hand, the parabolic equation does not offer any regularization. The well-posedness remains difficult, e.g. with a vanishing viscosity method.
We can establish the positivity of the solutions by a maximum principle. Theorem: Let `(P_0,C_0) \in W^{2,\infty} \times W^{1,\infty}` nonnegative. Then for all time `t>0` and
a.e. `x in \Omega, P(t,x) \geq 0, C(t,x) \geq 0.`
A trick is to use the variable `log(C)`, allowing to transform the second equation into a linear equation.
Finite element method is used to simulate the equations.
with D. Sellier, Diurnal dynamics of phloem loading: theoretical consequences for transport efficiency and flow characteristics, Tree Phys. 39, 2, (2019), 300-311. link
with D. Sellier, A surfacel model of the nonlinear non-steady-state phloem transport, Math. Bios. Eng. 14, 4, (2017), 1055-1069.hal
2.2 Advection-reaction-diffusion for fungus invasions
We derived a mixed PDE-ODE model to describe short and long-range spores dispersal `(U_S, U_L)` in interaction with the plant growth.
While the ODE provides the plant population dynamic (its growth and disease statut), Fick's law gives the dispersal as
`\partial_t U_S = D_S \Delta U_S - \delta_S U_s + \gamma f; \partial_t U_L = D_L \Delta U_L - \delta_L U_L + \gamma (1-f) + V\cdot \nabla U_L.`
Theorem: Assume `[U_S] = [U_L]= 0,` and `[D_S \nabla U_S] = [U_L \nabla U_L]= 0,` then the system is globally well-posed.
Moreover, `(U_S,U_L,S,L,I,R,T)` converges to the disease free equilibrium `(0,0,0,0,0,R^\star,T^\star)` with `R^\star(x) +T^\star(x) = k(x).`
The main difficulty in the numerical solution is to preserve the row structure of a parcel.
For this purpose, the parcel is meshed in a Cartesian grid along the rows and
a finite volume scheme is implemented.
with F.M. Hamelin, Y. Aigu, S.E. Strelkov, M.A. Lewis, Host diversification may split epidemic spread into two successive fronts advancing at different speeds, Bull. Math. Biol. 84, 68 (2022), 23 pages. link
with A. Calonnec, J.B. Burie and M. Langlais, Modelling of powdery mildew spread over a spatially heterogeneous growing grapevine, IOBC/wprs Bulletin 105, (2014) 137-148. link
with A. Calonnec, J.B. Burie and M. Langlais, How changes in the dynamic of crop susceptibility and cultural practices can be used to better control the spread of a fungal pathogen at the plot scale?, Ecological Modelling 290, (2014), 178-191. hal
Vinoid, software to simulate the growth of a vine in interaction with powdery mildew, taking into account climate, cultural practices and treatments based on discrete elements, written in C and python.
Agrosafe, software to simulate the airborne spread of a pathogen in a heterogeneous plot for growth based on a finite volume method, written in C++ and python.
2.3 Advection-reaction-diffusion for pest invasions
Because insect pest populations are known to be strongly influenced by landscape characteristics,
experimenting with crop protection strategies must be done at the agro-ecosystem. At this scale,
the levers of action consist of modifications of the landscape, for example by crop rotations, field size and geometry...
The quality and distribution of resources used by a given species can be very heterogeneous in space and time.
Landscape and population can be mediated by foraging.
Foraging can be composed of two processes: resource perception and energy supply management.
We derive a multiscale model that is highly sensitive to landscape structure and demographic composition.
Adult pests `(A)` move across the landscape according to these complex perception mechanisms (energetic `U`, youngness `Y`) as
`\partial_t A = \nabla \cdot (d \nabla A) - v \cdot \nabla A - c \partial_U A + f(Y, A);` where the youngness is deduced from
the life cycle of the pest and given by an integro-differential equation.
Theorem: Let `Y_0 \in L_x^\infty(\Omega)` and `A_0 = A_0(x,U)` be in `L_x^\infty(\Omega) \times L_U^\infty([0,1])` nonnegative.
Then there exists a unique solution `(A,Y) \in L_t^\infty([0,T] ; L_x^\infty(\Omega) \times L_U^\infty([0,1]) ) \times L_t^\infty([0,T] ; L_x^\infty(\Omega)).`
The main innovation of our approach consists in taking into account an energy dimension `(U)` which manages the displacement potential.
A finite volume method on a Cartesian mesh from a GIS is used. Then in silico predictions and
Morris sensitivity analysis were performed.
with S. Poggi, M. Sergent, M. Plantegenest, R. Le Cointe and Y. Bourhis, Dynamic role of grasslands as sources of soil-dwelling insect pests: new insights from in silico experiments for pest management strategies, Ecological Modelling 440 (2021) 109378. link
with Y. Bourhis, S. Poggi, R. Le Cointe, A.-M. Cortesero and N. Parisey, Foraging the landscape grip for population dynamics—a mechanistic model applied to crop protection, Ecological Modelling 354, (2017), 26-36. link
with Y. Bourhis, S. Poggi, A.-M. Cortesero, A. Le Rallec and N. Parisey, Perception-based foraging for competing resources: assessing pest population dynamics at the landscape scale from heterogeneous resource distribution, Ecological Modelling 312, (2015), 211-221.hal
3. Reaction-diffusion and ODE for Medicine
3.1 Therapeutic targeting of HCC and HNSCC.
Sorafenib was originally developed as an inhibitor of the oncogenic RAF family of kinases.
RAF kinases are essential regulators of the RAS-RAF-MEK-ERK pathway, a critical transduction
pathway that relays trophic and mitogenic signals in eukaryotic cells. Compared to normal cells,
cancer cells often show activation of this transduction pathway.
As this cascade is an important target of sorafenib in HCC cells, we explored its regulation
using mathematical modeling based on Michaelis-Menten kinetics
`\frac{d[pERK]}{dt} = \frac{ (V_{E,1} + V_{E,2}[pMEK])([ERK_{total}]-[pERK]) }{ K_{E,1} + ([ERK_{total}]-[pERK])}
- \frac{V_{E,3}[pERK]}{K_{E,2}+[pERK]}.`
We analyzed the dynamic regulation of the core components of the RAF-MEK-ERK cascade
in three human HCC cell lines with heterogeneous responses to sorafenib. In silico predictions and
Morris and Sobol sensitivity analysis were performed to highlight an unexpected mode of action of sorafenib.
with M. Lottin, S. Soudet, J. Fercot, F. Racine, J. Demagny, J. Bettoni, D. Chatelain, M.-A. Sevestre, Y. MAMMERI, M. Lamuraglia, A. Galmiche, Z. Saidak, Molecular landscape of the coagulome of oral squamous cell carcinoma, Cancers 14, 2 (2022) 460. link
with Z. Saidak, A.-S. Giacobbi, M. Chenda Morisse and A. Galmiche, La modélisation mathématique, un outil essentiel pour l’étude du ciblage thérapeutique des tumeurs solides, médecine/sciences 33, 12, (2017), 1055-1062. link
with Z. Saidak, A.S. Giacobbi C. Louandre, C. Sauzay and A. Galmiche, Mathematical modelling unveils the essential role of cellular phosphatases in the inhibition of RAF-MEK-ERK signalling by sorafenib in hepatocellular carcinoma cells, Cancer letters 392, (2017), 1-8. link
with T. Colin, M.-C. Durrieu, J. Joie, Y. Lei, C. Poignard and O. Saut, Modelling of the migration of endothelial cells on bioactive micropatterned polymers, Math. Bios. Eng. 10, 4, (2013), 997-1015. hal
3.2 COVID-19
Many mathematical models have been proposed to assist governments as an early warning device on the size of the epidemic, the speed at which it will spread and the effectiveness of control measures.
Most of the models are SIR (discrete or continuous) and few take into account spatial spread.
We developed a reaction-diffusion model to describe the spread of COVID-19 virus. The proposed model provides
an explicit description of the physical environment by taking into account the average daily movements of susceptible, exposed and asymptomatic individuals.
Theorem: Let `0\leq S_0, E_0, I_{a,0}, I_{s,0},R_0 \leq N_0` be the initial datum. Then
there exists a unique global weak solution `(S, E, I_a, I_s,R) \in L^\infty_t (\mathbb{R}_+; L^\infty(\Omega))^5`.
The solution is nonnegative, bounded by `N_0`, and converges a.e to the disease free equilibrium.
Moreover, if `\mathcal{R}_0 := \omega_0 ( \frac{\beta_e}{\delta} + \frac{(1-p)\beta_a}{\gamma} + \frac{p\beta_s}{\gamma+\mu} )\frac{S_0}{N_0} > 1,`
then the disease `(E,I_a,I_s)` initially grows exponentially.
with J. Arcede, R. Caga-anan, R. Namoco, I. Gonzales, Z. Lachica, M.A Mata, A modeling strategy for novel pandemics using monitoring data: the case of early COVID-19 pandemic in Northern Mindanao, Philippines, SciEnggJ 15, 1, (2022) 35-46. link
with R. Caga-anana, M.N. Razab, G.S.G. Labradora, E.B. Metilloa, and P. del Castillo, Effect of vaccination to COVID-19 disease progression and herd immunity, Comput. Math. Biophys 9 (2021), 262-272. link
with J.P. Lanoix, J.L. Schmit and M. Lefranc, Which features of an outpatient treatment for COVID-19 would be most important for pandemic control? A modelling study, J. Roy. Soc. Interface 18, 182 (2021) 20210319. link
with J. Arcede and R. Basañez, Hybrid modeling of COVID-19 spatial propagation over island country, In Srinivas R., Kumar R., Dutta M. (eds) Advances in Computational Modeling and Simulation. Lecture Notes in Mechanical Engineering (2022). Springer, Singapore. link
A reaction-diffusion system to better comprehend the unlockdown: Application of SEIR-type model with diffusion to the spatial spread of COVID-19 in France, Comput. Math. Biophys. 8 (2020) 102-113. link
with J. Arcede, R. Caga-Anan and C. Mentuda, Accounting for symptomatic and asymptomatic in a SEIR-type model of COVID-19, Math. Mod. Nat. Pheno. (2020), 13 pages. link
4. Reaction-diffusion with stiff sources for batteries performance
We derived models for lithium-sulfur (and lithium-oxide) battery cells, supported by a multi-scale description of the
composite microstructure. At the macroscopic scale, the equations are deduced from the conservation law of chemical species,
which is written for the concentration `C_i` of the species `i` as
`\partial_t C_i + \nabla \cdot (\vec{u} C_i + \vec{F_i}) = S_i`,
where `\vec{u}` is the fluid velocity, `S_i` is a stiff source term and `\vec{F_i}` is the electrochemical potential given by the Nernst-Planck equation
`\vec{F_i} = - D_i \nabla C_i + M_i z_i e \vec{E} C_i.`
The term on the right is the electromigration due to the electric field. The originality of our work consists in updating at each iteration
the chemical diffusion defined as the diffusion (constant) in a homogeneous material multiplied by the ratio between porosity and tortuosity.
These last two quantities are calculated at the macroscopic scale using a kinetic Monte Carlo method.
We developed a finite volume approach with a mesh size of the order of a micron that preserves the species while accessing the microscopic scale.
with G. Arora, R. Kumar, Homotopy Perturbation and Adomian Decomposition Methods for Condensing Coagulation and Lifshitz-Slyzov Models, Int J Geomath 14, 4 (2023), 18 pages. link
with M. Maiza, D.A. Nguyen, N. Legrand, P. Desprez and A.A. Franco, Evaluating the impact of transport inertia on the electrochemical response of lithium ion battery single particle models, J. Power Sources 423, (2019), 263-270. link
with C. Gaya, Y. Yin, A. Torayev and A.A. Franco, Investigation of bi-porous electrodes for lithium oxygen batteries, Electrochimica Acta 279, (2018), 118-127. link
with Y. Yin, A. Torayev, C. Gaya and A.A. Franco,Linking the Performances of Li-O Batteries to Discharge Rate, Electrode and Electrolyte Properties through the Nucleation Mechanism of Li, The Journal of Physical Chemistry Part C 121, 36, (2017), 19577-19585. link
with J.P. Chehab and A.A. Franco, Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation, DCDS-S 10, 1, (2017), 87-100. link
with V. Thangavel, K.-H. Xue, M. Quiroga, A. Mastouri, C. Guéry, P. Johansson, M. Morcrette and A.A. Franco, A microstructurally resolved model for Li-S batteries assessing the impact of the cathode design on the discharge performance, Journal of The Electrochemical Society 163, 14, (2016), 26 pages. link
LiS, software to simulate Performance calculation of Lithium-Sulfur batteries solved by a finite volume method.