Partition dimension of COVID antiviral drug structures

Ali Al Khabyah; Muhammad Kamran Jamil; Ali N. A. Koam; Aisha Javed; Muhammad Azeem; Ali Al Khabyah; Muhammad Kamran Jamil; Ali N. A. Koam; Aisha Javed; Muhammad Azeem

doi:10.3934/mbe.2022471

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 10: 10078-10095. doi: 10.3934/mbe.2022471

Previous Article Next Article

Research article

Partition dimension of COVID antiviral drug structures

1.
Department of Mathematics, College of Science, Jazan University, New Campus, Jazan 2097, Saudi Arabia
2.
Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University Lahore, Pakistan
3.
Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan

Received: 13 April 2022 Revised: 09 June 2022 Accepted: 10 June 2022 Published: 15 July 2022

In November 2019, there was the first case of COVID-19 (Coronavirus) recorded, and up to 3 $^{rd }$ of April 2020, 1,116,643 confirmed positive cases, and around 59,158 dying were recorded. Novel antiviral structures of the SARS-COV-2 virus is discussed in terms of the metric basis of their molecular graph. These structures are named arbidol, chloroquine, hydroxy-chloroquine, thalidomide, and theaflavin. Partition dimension or partition metric basis is a concept in which the whole vertex set of a structure is uniquely identified by developing proper subsets of the entire vertex set and named as partition resolving set. By this concept of vertex-metric resolvability of COVID-19 antiviral drug structures are uniquely identified and helps to study the structural properties of structure.

Keywords:

Citation: Ali Al Khabyah, Muhammad Kamran Jamil, Ali N. A. Koam, Aisha Javed, Muhammad Azeem. Partition dimension of COVID antiviral drug structures[J]. Mathematical Biosciences and Engineering, 2022, 19(10): 10078-10095. doi: 10.3934/mbe.2022471

Related Papers:

[1]	Yves Achdou, Ziad Kobeissi . Mean field games of controls: Finite difference approximations. Mathematics in Engineering, 2021, 3(3): 1-35. doi: 10.3934/mine.2021024
[2]	Benoît Perthame, Edouard Ribes, Delphine Salort . Career plans and wage structures: a mean field game approach. Mathematics in Engineering, 2019, 1(1): 38-54. doi: 10.3934/Mine.2018.1.38
[3]	Piermarco Cannarsa, Rossana Capuani, Pierre Cardaliaguet . C^1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games. Mathematics in Engineering, 2019, 1(1): 174-203. doi: 10.3934/Mine.2018.1.174
[4]	Pablo Blanc, Fernando Charro, Juan J. Manfredi, Julio D. Rossi . Games associated with products of eigenvalues of the Hessian. Mathematics in Engineering, 2023, 5(3): 1-26. doi: 10.3934/mine.2023066
[5]	Simone Paleari, Tiziano Penati . Hamiltonian lattice dynamics. Mathematics in Engineering, 2019, 1(4): 881-887. doi: 10.3934/mine.2019.4.881
[6]	Mario Pulvirenti . On the particle approximation to stationary solutions of the Boltzmann equation. Mathematics in Engineering, 2019, 1(4): 699-714. doi: 10.3934/mine.2019.4.699
[7]	Franco Flandoli, Eliseo Luongo . Heat diffusion in a channel under white noise modeling of turbulence. Mathematics in Engineering, 2022, 4(4): 1-21. doi: 10.3934/mine.2022034
[8]	Giacomo Canevari, Arghir Zarnescu . Polydispersity and surface energy strength in nematic colloids. Mathematics in Engineering, 2020, 2(2): 290-312. doi: 10.3934/mine.2020015
[9]	Zheming An, Nathaniel J. Merrill, Sean T. McQuade, Benedetto Piccoli . Equilibria and control of metabolic networks with enhancers and inhibitors. Mathematics in Engineering, 2019, 1(3): 648-671. doi: 10.3934/mine.2019.3.648
[10]	Jérôme Droniou, Jia Jia Qian . Two arbitrary-order constraint-preserving schemes for the Yang–Mills equations on polyhedral meshes. Mathematics in Engineering, 2024, 6(3): 468-493. doi: 10.3934/mine.2024019

Abstract

The authors are honored to participate in this special issue and express their admiration to Professor Italo Capuzzo-Dolcetta for his mathematical work. The first author met Prof. Dolcetta for the first time at the end of his Ph.D. thesis more than 20 years ago during an extended visit to Roma. Prof. Dolcetta mentorship at that time influenced his career deeply and he is particularly grateful for this opportunity.

1. Introduction

Mean-field games (MFG) is a tool to study the Nash equilibrium of infinite populations of rational agents. These agents select their actions based on their state and the statistical information about the population. Here, we study a price formation model for a commodity traded in a market under uncertain supply, which is a common noise shared by the agents. These agents are rational and aim to minimize the average trading cost by selecting their trading rate. The distribution of the agents solves a stochastic partial differential equation. Finally, a market-clearing condition characterizes the price.

We let $(\Omega, {\mathcal{F}}, ({\mathcal{F}}_t)_{0{\leqslant} t}, \mathbb{P})$ be a complete filtered probability space such that $({\mathcal{F}}_t)_{0{\leqslant} t}$ is the standard filtration induced by $t\mapsto W_t$ , the common noise, which is a one-dimensional Brownian motion. We consider a commodity whose supply process is described by a stochastic differential equation; that is, we are given a drift $b^S:[0, T]\times {\mathbb{R}}^2\to {\mathbb{R}}$ and volatility $\sigma^S:[0, T]\times {\mathbb{R}}^2\to {\mathbb{R}}^+_0$ , which are smooth functions, and the supply $Q_s$ is determined by the stochastic differential equation

$\begin{equation} dQ_s = b^S(Q_s, \varpi_s, s)ds+ \sigma^S(Q_s, \varpi_s, s)dW_s \quad \mbox{ in } [0, T] \end{equation}$

(1.1)

with the initial condition $\bar{q}$ . We would like to determine the drift $b^P:[0, T]\times {\mathbb{R}}^2\to {\mathbb{R}}$ , the volatility $\sigma^P:[0, T]\times {\mathbb{R}}^2\to {\mathbb{R}}^+_0$ , and $\bar{w}$ such that the price $\varpi_s$ solves

$\begin{equation} d\varpi_s = b^P(Q_s, \varpi_s, s)ds+\sigma^P(Q_s, \varpi_s, s)dW_s \quad \mbox{ in } [0, T] \end{equation}$

(1.2)

with initial condition $\bar{w}$ and such that it ensures a market clearing condition. It may not be possible to find $b^P$ and $\sigma^P$ in a feedback form. However, for linear dynamics, as we show here, we can solve quadratic models, which are of great interest in applications.

Let ${\bf X}_s$ be the quantity of the commodity held by an agent at time $s$ for $t {\leqslant} s {\leqslant} T$ . This agent trades this commodity, controlling its rate of change, ${\bf v}$ , thus

$\begin{equation} d {\bf X}_s = {\bf v}(s) ds \; \; \mbox{ in } [t, T]. \end{equation}$

(1.3)

At time $t$ , an agent who holds $x$ and observes $q$ and $w$ chooses a control process ${\bf v}$ , progressively measurable with respect to $\mathcal{F}_t$ , to minimize the expected cost functional

$\begin{equation} J(x, q, w, t; {\bf v}) = { \mathbb{E}} \left[ \int_t^T L( {\bf X}_s, {\bf v}(s))+\varpi_s {\bf v}(s) ds + \Psi \left( {\bf X}_ T, Q_T, \varpi_T\right)\right], \end{equation}$

(1.4)

subject to the dynamics (1.1), (1.2), and (1.3) with initial condition ${\bf X}_t = x$ , and the expectation is taken w.r.t. $\mathcal{F}_r$ . The Lagrangian, $L$ , takes into account costs such as market impact or storage, and the terminal cost $\Psi$ stands for the terminal preferences of the agent.

This control problem determines a Hamilton-Jacobi equation addressed in Section 2.1. In turn, each agent selects an optimal control and uses it to adjust its holdings. Because the source of noise in $Q_t$ is common to all agents, the evolution of the probability distribution of agents is not deterministic. Instead, it is given by a stochastic transport equation derived in Section 2.2. Finally, the price is determined by a market-clearing condition that ensures that supply meets demand. We study this condition in Section 2.3.

Mathematically, the price model corresponds to the following problem.

Problem 1. Given a Hamiltonian, $H: {\mathbb{R}}^2\to {\mathbb{R}}$ , $H\in C^\infty$ , a commodity's supply initial value, $\bar q\in {\mathbb{R}}$ , supply drift, $b^S: {\mathbb{R}}^2\times [0, T]\to {\mathbb{R}}$ , and supply volatility, $\sigma^S: {\mathbb{R}}^2\times [0, T]\to {\mathbb{R}}$ , a terminal cost, $\Psi: {\mathbb{R}}^3\to {\mathbb{R}}$ , $\Psi\in C^\infty({\mathbb{R}}^3)$ , and an initial distribution of agents, $\bar m\in C^\infty_c({\mathbb{R}})\cap {\mathcal{P}}({\mathbb{R}})$ , find $u: {\mathbb{R}}^3\times [0, T]\to {\mathbb{R}}$ , $\mu\in C([0, T]\times \Omega; {\mathcal{P}}({\mathbb{R}}^3))$ , $\bar w\in {\mathbb{R}}$ , the price at $t = 0$ , the price drift $b^P: {\mathbb{R}}^2\times[0, T] \to {\mathbb{R}}$ , and the price volatility $\sigma^P: {\mathbb{R}}^2\times [0, T]\to {\mathbb{R}}$ solving

$\begin{equation} \begin{cases} -u_t+H(x, w+u_x) = b^S u_q +b^P u_w +\tfrac{1}{2}(\sigma^S)^2u_{qq}+\sigma^S\sigma^P u_{qw}+\tfrac{1}{2}(\sigma^P)^2u_{ww}\\ d \mu_t = \left(\left(\mu\tfrac{(\sigma^S)^2 }{2}\right)_{qq}+(\mu\sigma^S\sigma^P )_{qw}+\left(\mu\tfrac{(\sigma^P)^2 }{2}\right)_{ww}- \operatorname{div} (\mu {\bf b} )\right) dt- \operatorname{div}(\mu {\boldsymbol \sigma}) dW_t\\ \int_{ {\mathbb{R}}^3}q+D_pH(x, w+u_x(x, q, w, t))\mu_t(dx\times dq\times dw) = 0, \mathit{\mbox{a.e.}}\omega \in \Omega, \; 0{\leqslant} t {\leqslant} T, \end{cases} \end{equation}$

(1.5)

and the terminal-initial conditions

$\begin{equation} \begin{cases} u(x, q, w, T) = \Psi(x, q, w)\\ \mu_0 = \bar m \times \delta_{\bar q} \times \delta_{\bar w}, \end{cases} \end{equation}$

(1.6)

where ${\bf b} = (-D_pH(x, w+u_x), b^S, b^P)$ , ${\boldsymbol \sigma} = (0, \sigma^S, \sigma^P)$ , and the divergence is taken w.r.t. $(x, q, w)$ .

Given a solution to the preceding problem, we construct the supply and price processes

$Q_t = \int_{ {\mathbb{R}}^3} q \; \mu_t(dx\times dq\times dw)$

and

$\varpi_t = \int_{ {\mathbb{R}}^3} w \; \mu_t(dx\times dq\times dw),$

which also solve

$\begin{cases} dQ_t& = b^S(Q_t, \varpi_t, t)dt+ \sigma^S(Q_t, \varpi_t, t)dW_t \quad \mbox{ in } [0, T] \\ d\varpi_t& = b^P(Q_t, \varpi_t, t)dt+\sigma^P(Q_t, \varpi_t, t)dW_t \quad \mbox{ in } [0, T] \end{cases}$

with initial conditions

$\begin{equation} \begin{cases} Q_0 = \bar{q}\\ \varpi_0 = \bar{w} \end{cases} \end{equation}$

(1.7)

and satisfy the market-clearing condition

$Q_t = \int_{ {\mathbb{R}}} -D_pH(x, \varpi_t + u_x(x, Q_t, \varpi_t, t)) \mu_t(dx).$

In ^[10], the authors presented a model where the supply for the commodity was a given deterministic function, and the balance condition between supply and demand gave rise to the price as a Lagrange multiplier. Price formation models were also studied by Markowich et al. ^[18], Caffarelli et al. ^[2], and Burger et al. ^[1]. The behavior of rational agents that control an electric load was considered in ^[16,17]. For example, turning on or off space heaters controls the electric load as was discussed in ^[13,14,15]. Previous authors addressed price formation when the demand is a given function of the price ^[12] or that the price is a function of the demand, see, for example ^[5,6,7,8,11]. An $N$ -player version of an economic growth model was presented in ^[9].

Noise in the supply together with a balance condition is a central issue in price formation that could not be handled directly with the techniques in previous papers. A probabilistic approach of the common noise is discussed in Carmona et al. in ^[4]. Another approach is through the master equation, involving derivatives with respect to measures, which can be found in ^[3]. None of these references, however, addresses problems with integral constraints such as (1.7).

Our model corresponds to the one in ^[10] for the deterministic setting when we take the volatility for the supply to be 0. Here, we study the linear-quadratic case, that is, when the cost functional is quadratic, and the dynamics (1.1) and (1.2) are linear. In Section 3.2, we provide a constructive approach to get semi-explicit solutions of price models for linear dynamics and quadratic cost. This approach avoids the use of the master equation. The paper ends with a brief presentation of simulation results in Section 4.

2. The model

In this section, we derive Problem 1 from the price model. We begin with standard tools of optimal control theory. Then, we derive the stochastic transport equation, and we end by introducing the market-clearing (balance) condition.

2.1. Hamilton-Jacobi equation and verification theorem

The value function for an agent who at time $t$ holds an amount $x$ of the commodity, whose instantaneous supply and price are $q$ and $w$ , is

$\begin{equation} u(x, q, w, t) = \inf\limits_{ {\bf v}} J(x, q, w, t; {\bf v}) \end{equation}$

(2.1)

where $J$ is given by (1.4) and the infimum is taken over the set ${\mathcal{A}}\left((t, T]\right)$ of all functions $v:[t, T]\to {\mathbb{R}}$ , progressively measurable w.r.t. $(\mathcal{F}_s)_{t{\leqslant} s {\leqslant} T}$ . Consider the Hamiltonian, $H$ , which is the Legendre transform of $L$ ; that is, for $p\in {\mathbb{R}}$ ,

$\begin{equation} H(x, p) = \sup\limits_{v\in {\mathbb{R}}} [-pv-L (x, v)]. \end{equation}$

(2.2)

Then, from standard stochastic optimal control theory, whenever $L$ is strictly convex, if $u$ is $C^2$ , it solves the Hamilton-Jacobi equation in ${\mathbb{R}}^3 \times [0, T)$

$\begin{equation} -u_t+H(x, w+u_x) -b^Pu_w -b^S u_q - \tfrac{(\sigma^P)^2}{2}u_{ww}- \tfrac{(\sigma^S)^2}{2}u_{qq}- \sigma^P\sigma^Su_{wq} = 0 \end{equation}$

(2.3)

with the terminal condition

$\begin{equation} u(x, q, w, T) = \Psi \left(x, q, w\right). \end{equation}$

(2.4)

Moreover, as the next verification theorem establishes, any $C^2$ solution of (2.3) is the value function.

Theorem 2.1(Verification). Let $\tilde u:[0, T]\times {\mathbb{R}}^3 \to {\mathbb{R}}$ be a smooth solution of (2.3) with terminal condition (2.4). Let $({\bf X}^*, Q, \varpi)$ solve (1.3), (1.1) and (1.2), where ${\bf X}^*$ is driven by the $(\mathcal{F}_t)_{0{\leqslant} t}$ -progressively measurable control

$\begin{equation*} {\bf v}^*(s): = -D_pH( {\bf X}^*_s, \varpi_s+ \tilde{u}_x ( {\bf X}^*_s, Q_s, \varpi_s, s)). \end{equation*}$

Then

1). ${\bf v}^*$ is an optimal control for (2.1)

2). $\tilde u = u$ , the value function.

2.2. Stochastic transport equation

Theorem 2.1 provides an optimal feedback strategy. As usual in MFG, we assume that the agents are rational and, hence, choose to follow this optimal strategy. This behavior gives rise to a flow that transports the agents and induces a random measure that encodes their distribution. Here, we derive a stochastic PDE solved by this random measure. To this end, let $u$ solve (2.3) and consider the random flow associated with the diffusion

$\begin{equation} \begin{cases} d {\bf X}_s = -D_pH( {\bf X}_s, \varpi_s+u_x( {\bf X}_s, Q_s, \varpi_s, s)) ds\\ dQ_s = b^S(Q_s, \varpi_s, s)ds+ \sigma^S(Q_s, \varpi_s, s)dW_s \\ d\varpi_s = b^P(Q_s, \varpi_s, s)ds+\sigma^P(Q_s, \varpi_s, s)dW_s \end{cases} \end{equation}$

(2.5)

with initial conditions

$\begin{equation*} \begin{cases} {\bf X}_0 = x\\ Q_0 = \bar{q}\\ \varpi_0 = \bar{w}. \end{cases} \end{equation*}$

That is, for a given realization $\omega\in\Omega$ of the common noise, the flow maps the initial conditions $(x, \bar{q}, \bar{w})$ to the solution of (2.5) at time $t$ , which we denote by $\left({\bf X}_t^\omega (x, \bar{q}, \bar{w}), Q_t^\omega (\bar{q}, \bar{w}), \varpi_t^\omega (\bar{q}, \bar{w})\right)$ . Using this map, we define a measure-valued stochastic process $\mu_t$ as follows:

Definition 2.2. Let $\omega\in \Omega$ denote a realization of the common noise $W$ on $0{\leqslant} s {\leqslant} T$ . Given a measure $\bar{m}\in {\mathcal{P}}({\mathbb{R}})$ and initial conditions $\bar{q}, \bar{w}\in {\mathbb{R}}$ take $\bar \mu \in {\mathcal{P}}({\mathbb{R}}^3)$ by $\bar \mu = \bar{m}\times \delta_{\bar{q}}\times \delta_{\bar{w}}$ and define a random measure $\mu_t$ by the mapping $\omega \mapsto \mu^\omega_t \in {\mathcal{P}}({\mathbb{R}}^3)$ , where $\mu_t^\omega$ is characterized as follows:

for any bounded and continuous function $\psi: {\mathbb{R}}^3\to {\mathbb{R}}$

$\begin{align*} &\int_{ {\mathbb{R}}^3} \psi(x, q, w) \mu_t^\omega(dx\times dq\times dw) \\ & = \int_{ {\mathbb{R}}^3} \psi \left( {\bf X}_t^\omega (x, q, w), Q_t^\omega(q, w), \varpi_t^\omega(q, w) \right)\bar{\mu}(dx\times dq \times dw). \end{align*}$

Remark 2.3. Because $\bar \mu = \bar{m}\times \delta_{\bar{q}}\times \delta_{\bar{w}}$ , we have

$\begin{align*} &\int_{ {\mathbb{R}}^3} \psi \left( {\bf X}_t^\omega (x, q, w), Q_t^\omega(q, w), \varpi_t^\omega(q, w) \right)\bar{\mu}(dx\times dq \times dw) \\ & = \int_{ {\mathbb{R}}} \psi \left( {\bf X}_t^\omega (x, \bar{q}, \bar{w}), Q_t^\omega(\bar{q}, \bar{w}), \varpi_t^\omega(\bar{q}, \bar{w}) \right)\bar{m}(dx). \end{align*}$

Moreover, due to the structure of (2.5),

$\mu_t^\omega = ( {\bf X}_t^\omega(x, \bar{q}, \bar{w}) \# \bar{m})\times \delta_{Q_t^\omega(\bar{q}, \bar{w})}\times \delta_{\varpi^\omega_t(\bar{q}, \bar{w})}.$

Definition 2.4. Let $\bar{\mu}\in {\mathcal{P}}({\mathbb{R}}^3)$ and write

$\begin{align*} {\bf b}(x, q, w, s)& = (-D_pH(x, w+u_x(x, q, w, s)), b^S(q, w, s), b^P(q, w, s)), \\ {\boldsymbol \sigma}(q, w, s)& = (0, \sigma^S(q, w, s), \sigma^P(q, w, s)). \end{align*}$

A measure-valued stochastic process $\mu = \mu(\cdot, t) = \mu_t(\cdot)$ is a weak solution of the stochastic PDE

$\begin{equation} d\mu_t = \left(- \operatorname{div} (\mu {\bf b} )+\left(\mu\tfrac{(\sigma^S)^2 }{2}\right)_{qq}+(\mu\sigma^S\sigma^P )_{qw}+\left(\mu\tfrac{(\sigma^P)^2 }{2}\right)_{ww}\right) dt- \operatorname{div}(\mu {\boldsymbol \sigma}) dW_t, \end{equation}$

(2.6)

with initial condition $\bar{\mu}$ if for any bounded smooth test function $\psi: {\mathbb{R}}^3\times[0, T]\to {\mathbb{R}}$

$\begin{align} &\int_{ {\mathbb{R}}^3} \psi(x, q, w, t)\mu_t(dx\times dq \times dw) = \int_{ {\mathbb{R}}^3} \psi(x, q, w, 0)\bar{\mu}(dx\times dq \times dw) \end{align}$

(2.7)

$\begin{align} &+\int_0^t\int_{ {\mathbb{R}}^3}\partial_t\psi+ D\psi \cdot {\bf b} +\tfrac{1}{2} \operatorname{tr} \left( {\boldsymbol \sigma}^T {\boldsymbol \sigma} D^2\psi \right)\mu_s(dx \times dq \times dw) ds \end{align}$

(2.8)

$\begin{align} & + \int_0^t\int_{ {\mathbb{R}}^3} D\psi \cdot {\boldsymbol \sigma} \mu_s(dx \times dq \times dw) dW_s, \end{align}$

(2.9)

where the arguments for ${\bf b}$ , ${\boldsymbol \sigma}$ and $\psi$ are $(x, q, w, s)$ and the differential operators $D$ and $D^2$ are taken w.r.t. the spatial variables $x, q, w$ .

Theorem 2.5. Let $\bar{m}\in {\mathcal{P}}({\mathbb{R}})$ and $\bar{q}, \bar{w}\in {\mathbb{R}}$ . The random measure from Definition 2.2 is a weak solution of the stochastic partial differential equation (2.6) with initial condition $\bar{\mu} = \bar{m}\times \delta_{\bar{q}}\times \delta_{\bar{w}}$ .

Proof. Let $\psi: {\mathbb{R}}^3\times [0, T]\to {\mathbb{R}}$ be a bounded smooth test function. Consider the stochastic process $s \mapsto \int_{ {\mathbb{R}}^3}\psi(x, q, w, s)\mu^\omega_s(dx\times dq \times dw)$ . Let

$( {\bf X}_t(x, \bar{q}, \bar{w}), Q_t(\bar{q}, \bar{w}), \varpi_t(\bar{q}, \bar{w}))$

be the flow induced by (2.5). By the definition of $\mu^\omega_t$ ,

$\begin{align*} &\int_{ {\mathbb{R}}^3}\psi(x, q, w, t)\mu_t^\omega(dx\times dq\times dw)-\int_{ {\mathbb{R}}^3}\psi(x, q, w, 0)\bar{\mu}(dx\times dq\times dw) \\ & = \int_{ {\mathbb{R}}} \left[\psi( {\bf X}^\omega_t(x, \bar{q}, \bar{w}), Q^\omega_t(\bar{q}, \bar{w}), \varpi^\omega_t(\bar{q}, \bar{w}), t)-\psi(x, \bar{q}, \bar{w}, 0)\right]\bar{m} (dx). \end{align*}$

Then, applying Ito's formula to the stochastic process

$s\mapsto \int_{ {\mathbb{R}}} \psi( {\bf X}_s(x, \bar{q}, \bar{w}), Q_s(\bar{q}, \bar{w}), \varpi_s(\bar{q}, \bar{w}), s)\bar{m}(dx),$

the preceding expression becomes

$\begin{align*} &\int_0^t d \Big( \int_{ {\mathbb{R}}} \psi( {\bf X}_s(x, \bar{q}, \bar{w}), Q_s(\bar{q}, \bar{w}), \varpi_s(\bar{q}, \bar{w}), s)\bar{m}(dx) \Big) \\ & = \int_0^t \int_{ {\mathbb{R}}}\left[D_t\psi+D\psi \cdot {\bf b}+ \tfrac{1}{2} \operatorname{tr} ( {\boldsymbol \sigma}^T {\boldsymbol \sigma} D^2\psi )\right]\bar{m}(dx)ds \\ &+ \int_0^t \int_{ {\mathbb{R}}} D\psi\cdot {\boldsymbol \sigma}\bar{m}(dx) dW_s \\ & = \int_0^t\int_{ {\mathbb{R}}^3}\left[D_t\psi+ D\psi \cdot {\bf b} + \tfrac{1}{2} \operatorname{tr} ( {\boldsymbol \sigma}^T {\boldsymbol \sigma} D^2\psi) \right]\mu_s(dx\times dq\times dw)ds \\ &+\int_0^t\int_{ {\mathbb{R}}^3} D\psi\cdot {\boldsymbol \sigma} \mu_s(dx\times dq\times dw) dW_s, \end{align*}$

where arguments of ${\bf b}$ , ${\boldsymbol \sigma}$ and the partial derivatives of $\psi$ in the integral with respect to $\bar{m}(dx)$ are $({\bf X}_s(x, \bar{q}, \bar{w}), Q_s(\bar{q}, \bar{w}), \varpi_s(\bar{q}, \bar{w}), s)$ , and in the integral with respect to $\mu_t(dx\times dq \times dw)$ are $(x, q, w, t)$ . Therefore,

$\begin{align*} &\int_{ {\mathbb{R}}^3}\psi(x, q, w, t)\mu_t^\omega(dx\times dq\times dw)-\int_{ {\mathbb{R}}^3}\psi(x, q, w, 0)\bar{\mu}(dx\times dq\times dw) \\ & = \int_0^t\int_{ {\mathbb{R}}^3}\left[D_t\psi+ D\psi \cdot {\bf b} + \tfrac{1}{2} \operatorname{tr} (D^2\psi:( {\boldsymbol \sigma}, {\boldsymbol \sigma})) \right]\mu^\omega_s(dx\times dq\times dw)ds\\ &+\int_0^t\int_{ {\mathbb{R}}^3} D\psi\cdot {\boldsymbol \sigma} \mu^\omega_s(dx\times dq\times dw) dW_s. \end{align*}$

Hence, (2.7) holds.

2.3. Balance condition

The balance condition requires the average trading rate to be equal to the supply. Because agents are rational and, thus, use their optimal strategy, this condition takes the form

$\begin{align} Q_t& = \int_{ {\mathbb{R}}^3} -D_pH(x, w + u_x(x, q, w, t)) \mu^\omega_t(dx\times dq \times dw), \end{align}$

(2.10)

where $\mu_t^\omega$ is given by Definition 2.2. Because $Q_t$ satisfies a stochastic differential equation, the previous can also be read in differential form as

$\begin{align} b^S(Q_t, \varpi_t, t)dt+ \sigma^S(Q_t, \varpi_t, t)dW_t = d\int_{ {\mathbb{R}}^3} -D_pH(x, w + u_x(x, q, w, t)) \mu^\omega_t(dx\times dq \times dw). \end{align}$

(2.11)

The former condition determines $b^P$ and $\sigma^P$ . In general, $b^P$ and $\sigma^P$ are only progressively measurable with respect to $(\mathcal{F}_t)_{0{\leqslant} t}$ and not in feedback form. In this case, the Hamilton–Jacobi (2.3) must be replaced by either a stochastic partial differential equation or the problem must be modeled by the master equation. However, as we discuss next, in the linear-quadratic case, we can find $b^P$ and $\sigma^P$ in feedback form.

3. Potential-free linear-quadratic price model

Here, we consider a price model for linear dynamics and quadratic cost. The Hamilton-Jacobi equation admits quadratic solutions. Then, the balance equation determines the dynamics of the price, and the model is reduced to a first-order system of ODE.

Suppose that $L(x, v) = \tfrac{c}{2}v^2$ and, thus, $H(x, p) = \tfrac{1}{2c}p^2$ . Accordingly, the corresponding MFG model is

$\begin{equation} \begin{cases} -u_t + \tfrac{1}{2c}(w+u_x)^2 -b^Pu_w -b^S u_q - \tfrac{1}{2}(\sigma^P)^2u_{ww}- \tfrac{1}{2}(\sigma^S)^2u_{qq}- \sigma^P\sigma^Su_{wq} = 0 \\ d \mu_t = \left(\left(\mu\tfrac{(\sigma^S)^2 }{2}\right)_{qq}+(\mu\sigma^S\sigma^P )_{qw}+\left(\mu\tfrac{(\sigma^P)^2 }{2}\right)_{ww}- \operatorname{div} (\mu {\bf b} )\right) dt- \operatorname{div}(\mu {\boldsymbol \sigma}) dW_t \\ Q_t = -\tfrac{1}{c} \varpi_t +\int_{ {\mathbb{R}}} -\tfrac{1}{c} u_x(x, q, w, t) \mu_t^\omega(dx\times dq \times dw). \end{cases} \end{equation}$

(3.1)

Assume further that $\Psi$ is quadratic; that is,

$\Psi(x, q, w) = c_0+c_1^1x+c_1^2q + c_1^3w + c_2^1x^2 + c_2^2xq + c_2^3xw + c_2^4q^2 + c_2^5qw + c_2^6w^2.$

3.1. Balance condition

Let

$\Pi_t = \int_{ {\mathbb{R}}^3} u_x(x, q, w, t) \mu_t(dx\times dq\times dw).$

The balance condition is $Q_t = -\tfrac{1}{c} \left(\varpi_t+\Pi_t\right)$ . Furthermore, Definition 2.2 provides the identity

$\Pi_t = \int_{ {\mathbb{R}}} u_x( {\bf X}_t^*(x, \bar{q}, \bar{w}), Q_t(\bar{q}, \bar{w}), \varpi_t(\bar{q}, \bar{w}), t) \bar{m}(dx).$

Lemma 3.1. Let $({\bf X}^*, Q, \varpi)$ solve (1.3), (1.1) and (1.2) with ${\bf v} = {\bf v}^*$ , the optimal control, and initial conditions $\bar{q}, \bar{w}\in {\mathbb{R}}$ . Let $u\in C^3({\mathbb{R}}^3\times[0, T])$ solve the Hamilton-Jacobi equation (2.3). Then

$\begin{align} d\Pi_t = & \int_{ {\mathbb{R}}} \left(u_{xq} \sigma^S + u_{xw} \sigma^P \right)\bar{m}(dx)dW_t, \end{align}$

(3.2)

where the arguments for the partial derivatives of $u$ are $({\bf X}_t^*(x, \bar{q}, \bar{w}), Q_t(\bar{q}, \bar{w}), \varpi_t(\bar{q}, \bar{w}), t)$ .

Proof. By Itô's formula, the process $t\mapsto u_x({\bf X}^*_t, Q_t, \varpi_t, t)$ solves

$\begin{align} &d\big(u_x( {\bf X}^*_t, Q_t, \varpi_t, t)\Big) \\ & = \Big( u_{xt} + u_{xx} {\bf v}^* + u_{xq} b^S + u_{xw} b^P + u_{xqq} \tfrac{1}{2} (\sigma^S)^2 + u_{xqw} \sigma^S \sigma^P + u_{xww} \tfrac{1}{2} (\sigma^P)^2 \Big) dt + \\ &+ \Big( u_{xq}\sigma^S + u_{xw} \sigma^P \Big) dW_t, \end{align}$

(3.3)

with ${\bf v}^*(t) = -\tfrac{1}{c}(\varpi_t+u_x({\bf X}^*_t, Q_t, \varpi_t, t))$ . By differentiating the Hamilton-Jacobi equation, we get

$\begin{equation*} -u_{tx}+\tfrac{1}{c}(\varpi_t+u_x)u_{xx} -b^Pu_{wx} -b^S u_{qx} - \tfrac{(\sigma^P)^2}{2}u_{wwx}- \tfrac{(\sigma^S)^2}{2}u_{qqx}- \sigma^P\sigma^Su_{wqx} = 0. \end{equation*}$

Substituting the previous expression in (3.3), we have

$\begin{align*} &d\Big( \int_{ {\mathbb{R}}} u_x( {\bf X}^*_t(x, \bar{q}, \bar{w}), Q_t(\bar{q}, \bar{w}), \varpi_t(\bar{q}, \bar{w}), t)\bar{m}(dx)\Big) \\ & = \int_{ {\mathbb{R}}} \left(\tfrac{1}{c}(\varpi_t+u_x)u_{xx} +u_{xx} {\bf v}^* \right)\bar{m}(dx)dt +\int_{ {\mathbb{R}}} \left(u_{xq} \sigma^S + u_{xw} \sigma^P \right)\bar{m}(dx)dW_t. \end{align*}$

The preceding identity simplifies to

$\begin{gather*} \int_{ {\mathbb{R}}} \left(u_{xq} \sigma^S + u_{xw} \sigma^P \right)\bar{m}(dx)dW_t. \end{gather*}$

Using Lemma 3.1, we have

$-c dQ_t = \int_ {\mathbb{R}}\left(u_{xq} \sigma^S + u_{xw} \sigma^P \right)\bar{m}(dx)dW_t +d\varpi_t;$

that is,

$\begin{gather*} -cb^S dt - c\sigma^S dW_t = \left( \sigma^S \int_ {\mathbb{R}} u_{xq}\bar{m}(dx) + \sigma^P \int_ {\mathbb{R}} u_{xw}\bar{m}(dx) \right)dW_t + d\varpi_t\\ = b^P dt + \left(\sigma^S \int_ {\mathbb{R}} u_{xq}\bar{m}(dx) + \sigma^P \int_ {\mathbb{R}} u_{xw}\bar{m}(dx) + \sigma^P\right)dW_t. \end{gather*}$

Thus,

$\begin{gather} b^P = -cb^S, \\ \sigma^P = -\sigma^S \dfrac{c+ \int_ {\mathbb{R}} u_{xq}\bar{m}(dx)}{1+ \int_ {\mathbb{R}} u_{xw}\bar{m}(dx)}. \end{gather}$

(3.4)

3.2. Quadratic solutions to the Hamilton-Jacobi equation

If $u$ is a second-degree polynomial with time-dependent coefficients, then

$\int_ {\mathbb{R}} u_{xq}( {\bf X}_t^*(x, \bar{q}, \bar{w}), Q_t(\bar{q}, \bar{w}), \varpi_t(\bar{q}, \bar{w}), t)\bar{m}(dx)$

and

$\int_ {\mathbb{R}} u_{xw}( {\bf X}_t^*(x, \bar{q}, \bar{w}), Q_t(\bar{q}, \bar{w}), \varpi_t(\bar{q}, \bar{w}), t)\bar{m}(dx)$

are deterministic functions of time. Accordingly, $b^P$ and $\sigma^P$ are given in feedback form by (3.4), thus, consistent with the original assumption. Here, we investigate the linear-quadratic case that admits solutions of this form.

Now, we assume that the dynamics are affine; that is,

$\begin{equation} \begin{cases} b^P(t, q, w) = b_0^P(t)+q b_1^P(t)+w b_2^P(t)\\ b^S(t, q, w) = b_0^S(t)+q b_1^S(t)+w b_2^S(t)\\ \sigma^P(t, q, w) = \sigma_0^P(t)+q \sigma _1^P(t)+w \sigma _2^P(t)\\ \sigma^S(t, q, w) = \sigma_0^S(t) +q \sigma _1^S(t)+w \sigma _2^S(t). \end{cases} \end{equation}$

(3.5)

Then, (3.4) gives

$\begin{align*} b^P_0& = -cb_0^S , &\sigma^P_0 & = -\sigma_0^S \dfrac{c+ \int_ {\mathbb{R}} u_{xq}\bar{m}(dx)}{1+ \int_ {\mathbb{R}} u_{xw}\bar{m}(dx)}\\ b^P_1& = -cb_1^S , &\sigma^P_1 & = -\sigma_1^S \dfrac{c+ \int_ {\mathbb{R}} u_{xq}\bar{m}(dx)}{1+ \int_ {\mathbb{R}} u_{xw}\bar{m}(dx)}\\ b^P_2& = -cb_2^S , &\sigma^P_2 & = -\sigma_2^S \dfrac{c+ \int_ {\mathbb{R}} u_{xq}\bar{m}(dx)}{1+ \int_ {\mathbb{R}} u_{xw}\bar{m}(dx)}. \end{align*}$

Because all the terms in the Hamilton-Jacobi equation are at most quadratic, we seek for solutions of the form

$\begin{align*} u(t, x, q, w) = & a_0(t) + a_1^1(t)x + a_1^2(t)q + a_1^3(t)w\\ & + a_2^1(t)x^2 + a_2^2(t)xq + a_2^3(t)xw + a_2^4(t)q^2 + a_2^5(t)qw + a_2^6(t)w^2, \end{align*}$

where $a_i^j:[0, T]\to {\mathbb{R}}$ . Therefore, the previous identities reduce to

$\begin{align} b^P_0& = -cb_0^S , &\sigma^P_0 & = -\sigma_0^S \dfrac{c+ a_2^2 }{1+ a_2^3} \\ b^P_1& = -cb_1^S , &\sigma^P_1 & = -\sigma_1^S \dfrac{c+ a_2^2 }{1+ a_2^3} \\ b^P_2& = -cb_2^S , &\sigma^P_2 & = -\sigma_2^S \dfrac{c+ a_2^2 }{1+ a_2^3}. \end{align}$

(3.6)

Using (3.6) and grouping coefficients in the Hamilton-Jacobi PDE, we obtain the following ODE system

$\begin{align*} \dot{a}_2^1 = &\tfrac{2 \left(a_2^1\right)^2}{c} \\ \dot{a}_2^2 = &\tfrac{c^2 a_2^3 b_1^S-c a_2^2 b_1^S+2 a_2^1 a_2^2}{c} \\ \dot{a}_2^3 = &\tfrac{c^2 a_2^3 b_2^S-c a_2^2 b_2^S+2 a_2^1+2 a_2^1 a_2^3}{c} \\ \dot{a}_1^1 = &\tfrac{c^2 a_2^3 b_0^S-c a_2^2 b_0^S+2 a_1^1 a_2^1}{c} \\ \dot{a}_2^4 = &c a_2^5 b_1^S-2 a_2^4 b_1^S+\tfrac{a_2^5 \left(a_2^2+c\right) \left(\sigma _1^S\right)^2}{a_2^3+1}-\tfrac{1}{4} \left(\tfrac{4 a_2^6 \left(a_2^2+c\right)^2 \left(\sigma _1^S\right)^2}{\left(a_2^3+1\right)^2}+4 a_2^4 \left(\sigma _1^S\right)^2\right)+\tfrac{\left(a_2^2\right)^2}{2 c} \\ \dot{a}_2^5 = &2 c a_2^6 b_1^S+c a_2^5 b_2^S-a_2^5 b_1^S-2 a_2^4 b_2^S-\tfrac{1}{2} \left(\tfrac{4 a_2^6 \left(a_2^2+c\right)^2 \sigma _1^S \sigma _2^S}{\left(a_2^3+1\right)^2}+4 a_2^4 \sigma _1^S \sigma _2^S\right)+\tfrac{2 a_2^5 \left(a_2^2+c\right) \sigma _1^S \sigma _2^S}{a_2^3+1}+\tfrac{a_2^2 \left(a_2^3+1\right)}{c} \\ \dot{a}_2^6 = &2 c a_2^6 b_2^S-a_2^5 b_2^S-\tfrac{1}{4} \left(\tfrac{4 a_2^6 \left(a_2^2+c\right)^2 \left(\sigma _2^S\right)^2}{\left(a_2^3+1\right)^2}+4 a_2^4 \left(\sigma _2^S\right)^2\right)+\tfrac{a_2^5 \left(a_2^2+c\right) \left(\sigma _2^S\right)^2}{a_2^3+1}+\tfrac{\left(a_2^3+1\right)^2}{2 c} \\ \dot{a}_0 = &c a_1^3 b_0^S-a_1^2 b_0^S+\tfrac{a_2^5 \left(a_2^2+c\right) \left(\sigma _0^S\right)^2}{a_2^3+1}-\tfrac{1}{2} \left(\tfrac{2 a_2^6 \left(a_2^2+c\right)^2 \left(\sigma _0^S\right)^2}{\left(a_2^3+1\right)^2}+2 a_2^4 \left(\sigma _0^S\right)^2\right)+\tfrac{\left(a_1^1\right)^2}{2 c} \\ \dot{a}_1^2 = &c a_2^5 b_0^S+c a_1^3 b_1^S-2 a_2^4 b_0^S-a_1^2 b_1^S+\tfrac{2 a_2^5 \left(a_2^2+c\right) \sigma _0^S \sigma _1^S}{a_2^3+1}-\tfrac{1}{2} \left(\tfrac{4 a_2^6 \left(a_2^2+c\right)^2 \sigma _0^S \sigma _1^S}{\left(a_2^3+1\right)^2}+4 a_2^4 \sigma _0^S \sigma _1^S\right)+\tfrac{a_1^1 a_2^2}{c} \\ \dot{a}_1^3 = &2 c a_2^6 b_0^S+c a_1^3 b_2^S-a_2^5 b_0^S-a_1^2 b_2^S-\tfrac{1}{2} \left(\tfrac{4 a_2^6 \left(a_2^2+c\right)^2 \sigma _0^S \sigma _2^S}{\left(a_2^3+1\right)^2}+4 a_2^4 \sigma _0^S \sigma _2^S\right)+\tfrac{2 a_2^5 \left(a_2^2+c\right) \sigma _0^S \sigma _2^S}{a_2^3+1}+\tfrac{a_1^1 \left(a_2^3+1\right)}{c} , \end{align*}$

with terminal conditions

$\begin{align*} a_0(T)& = \Psi(0, 0, 0) = c_0 & a_1^1(T)& = D_{x}\Psi(0, 0, 0) = c_1^1\\ a_1^2(T)& = D_{q}\Psi(0, 0, 0) = c_1^2 & a_1^3(T)& = D_{w}\Psi(0, 0, 0) = c_1^3\\ a_2^1(T)& = \tfrac{1}{2} D_{xx}\Psi(0, 0, 0) = c_2^1 & a_2^2(T)& = D_{xq}\Psi(0, 0, 0) = c_2^2\\ a_2^3(T)& = D_{xw}\Psi(0, 0, 0) = c_2^3 & a_2^4(T)& = \tfrac{1}{2} D_{qq}\Psi(0, 0, 0) = c_2^4\\ a_2^5(T)& = D_{qw}\Psi(0, 0, 0) = c_2^5 & a_2^6(T)& = \tfrac{1}{2} D_{ww}\Psi(0, 0, 0) = c_2^6. \end{align*}$

While this system has a complex structure, it admits some simplifications. For example, the equation for $a_2^1$ is independent of other terms and has the solution

$a_2^1(t) = \tfrac{c c_2^1}{c+2c_2^1(T-t)}.$

Moreover, we can determine $a_2^2$ and $a_2^3$ from the linear system

$\tfrac{d}{dt} \begin{bmatrix} a_2^2\\ a_2^3 \end{bmatrix} = \begin{bmatrix} -b_1^S + \tfrac{2}{c} a_2^1 & c b_1^S\\ -b_2^S & c b_2^S +\tfrac{2}{c}a_2^1 \end{bmatrix} \begin{bmatrix} a_2^2\\ a_2^3 \end{bmatrix} + \begin{bmatrix} 0\\ \tfrac{2}{c}a_2^1 \end{bmatrix}.$

Lemma 3.1 takes the form

$\begin{gather*} d\Pi_t = \Big(a_2^2(t) \sigma^S(Q_t, \varpi_t, t) + a_2^3 (t)\sigma^P (Q_t, \varpi_t, t) \Big)dW_t. \end{gather*}$

Therefore,

$\begin{gather*} \Pi_t = \Pi_0 + \int_0^t \Big(a_2^2(r) \sigma^S(Q_r, \varpi_r, r) + a_2^3 (r)\sigma^P (Q_r, \varpi_r, r) \Big)dW_r \end{gather*}$

where

$\Pi_0 = a_1^1(0)+2a_2^1(0)\int_{ {\mathbb{R}}}x\bar{m}(dx)+a_2^2(0)\bar{q}+a_2^3(0)\bar{w}.$

Replacing the above in the balance condition at the initial time, that is $\bar{w} = -c\bar{q}-\Pi_0$ , we obtain the initial condition for the price

$\begin{equation} \bar{w} = \tfrac{-1}{1+a_2^3(0)}\Big(a_1^1(0) + 2a_2^1(0) \int_{ {\mathbb{R}}}x\bar{m}(dx) + (a_2^2(0)+c) \bar{q}\Big). \end{equation}$

(3.7)

where $a_1^1$ can be obtained after solving for $a_2^1$ , $a_2^2$ and $a_2^3$ .

Now, we proceed with the price dynamics using the balance condition. Under linear dynamics, we have

$\begin{align*} Q_t& = -\tfrac{1}{c} \left(\varpi_t+\Pi_0\right) \\ & -\tfrac{1}{c} \int_0^t a_2^2(r) \Big(\sigma_0^S(r) +Q_r \sigma _1^S(r)+\varpi_r \sigma _2^S(r) \Big) + a_2^3 (r)\Big(\sigma_0^P(r)+Q_r \sigma _1^P(r)+\varpi_r \sigma _2^P(r) \Big) dW_r. \end{align*}$

Thus, replacing the price coefficients for (3.6), we obtain

$\begin{align*} d\varpi_t = &-c\left( b_0^S(t)+ b_1^S(t)Q_t+ b_2^S(t)\varpi_t\right) dt \\ &- \tfrac{c+a_2^2(t)}{1+a_2^3(t)} \left(\sigma_0^S(t)+\sigma_1^S(t) Q_t+\sigma_2^S(t) \varpi_t\right)dW_t, \\ dQ_t = &b^S dt + \sigma^S dW_t, \end{align*}$

which determines the dynamics for the price.

4. Simulation results

In this section, we consider the running cost corresponding to $c = 1$ ; that is,

$L( {\bf v}) = \tfrac{1}{2} {\bf v}^2$

and terminal cost at time $T = 1$

$\Psi(x) = (x-\alpha)^2.$

We take $\bar{m}$ to be a normal standard distribution; that is, with zero-mean and unit variance. We assume the dynamics for the normalized supply is mean-reverting

$dQ_t = (1-Q_t) dt + Q_t dW_t,$

with initial condition $\bar{q} = 1$ . Therefore, the dynamics for the price becomes

$d\varpi_t = -(1-Q_t) dt - \tfrac{1+a_2^2}{1+a_2^3} Q_t dW_t,$

with initial condition $\bar{w}$ given by (3.7), and $a_2^2$ and $a_2^3$ solve

$\begin{align*} \dot{a}_2^2 = & -a_2^3 + a_2^2 (1+2 a_2^1) \\ \dot{a}_2^3 = & 2 a_2^1(1+a_2^3), \end{align*}$

with terminal conditions $a_2^2(1) = 0$ and $a_2^3(1) = 0$ . We observe that the coefficient multiplying $Q_t$ in the volatility of the price is now time-dependent.

For a fixed simulation of the supply, we compute the price for different values of $\alpha$ . Agents begin with zero energy average. The results are displayed in Figure 1. As expected, the price is negatively correlated with the supply. Moreover, as the storage target increases, prices increase, which reflects the competition between agents who, on average, want to increase their storage.

Figure 1. Supply vs. Price for the values

$\alpha = 0$ ,

$\alpha = 0.1$ ,

$\alpha = 0.25$ ,

$\alpha = 0.5$ .

DownLoad: Full-Size Img PowerPoint

Acknowledgments

The authors were partially supported by KAUST baseline funds and KAUST OSR-CRG2017-3452.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, et al., Clinical features of patients infected with 2019 novel coronavirus in wuhan, china, Lancet, 395 (2020), 497–506. https://doi.org/10.1016/S0140-6736(20)30183-5 doi: 10.1016/S0140-6736(20)30183-5
[2]	M. Wang, R. Cao, L. Zhang, X. Yang, J. Liu, M. Xu, et al., Remdesivir and chloroquine effectively inhibit the recently emerged novel coronavirus (2019-nCoV) in vitro, Cell Res., 30 (2020), 269–271. https://doi.org/10.1038/s41422-020-0282-0 doi: 10.1038/s41422-020-0282-0
[3]	D. Zhou, S. Dai, Q. Tong, COVID-19: a recommendation to examine the effect of hydroxychloroquine in preventing infection and progression, J. Antimicrob. Chemother., 75 (2020), 1667–1670. https://doi.org/10.1093/jac/dkaa114 doi: 10.1093/jac/dkaa114
[4]	J. Lung, Y. Lin, Y. Yang, Y. Chou, L. Shu, Y. Cheng, et al., The potential chemical structure of anti-SARS-CoV-2 RNA-dependent RNA polymerase, J. Med. Virol., 92 (2020), 693–697. https://doi.org/10.1002/jmv.25761 doi: 10.1002/jmv.25761
[5]	J. S. Morse, T. Lalonde, S. Xu, W. R. Liu, Learning from the past: Possible urgent prevention and treatment options for severe acute respiratory infections caused by 2019-nCoV, ChemBioChem, 21 (2020), 730–738. https://doi.org/10.1002/cbic.202000047 doi: 10.1002/cbic.202000047
[6]	X. Xu, P. Chen, J. Wang, J. Feng, H. Zhou, X. Li, et al., Evolution of the novel coronavirus from the ongoing wuhan outbreak and modeling of its spike protein for risk of human transmission, Sci. China Life Sci., 63 (2020), 457–460. https://doi.org/10.1007/s11427-020-1637-5 doi: 10.1007/s11427-020-1637-5
[7]	T. K. Warren, R. Jordan, M. K. Lo, A. S. Ray, R. L. Mackman, V. Soloveva, et al., Therapeutic efficacy of the small molecule GS-5734 against ebola virus in rhesus monkeys, Nature, 531 (2016), 381–385. https://doi.org/10.1038/nature17180 doi: 10.1038/nature17180
[8]	A. Savarino, L. D. Trani, I. Donatelli, R. Cauda, A. Cassone, New insights into the antiviral effects of chloroquine, Lancet Infect. Dis., 6 (2006), 67–69. https://doi.org/10.1016/S1473-3099(06)70361-9 doi: 10.1016/S1473-3099(06)70361-9
[9]	Y. Yan, Z. Zou, Y. Sun, X. Li, K. F. Xu, Y. Wei, et al., Anti-malaria drug chloroquine is highly effective in treating avian influenza a h5n1 virus infection in an animal model, Cell Res., 23 (2013), 300–302. https://doi.org/10.1038/cr.2012.165 doi: 10.1038/cr.2012.165
[10]	Johnson & Johnson is already ramping up production on its fanxiexian_myfh1 billion coronavirus vaccine. Available from: https://www.forbes.com/sites/thomasbrewster/2020/03/30/johnson–johnson-is-already-ramping-up-production-on-its-1-billion-coronavirus-vaccine/?sh=2a66d09aaa66.
[11]	Z. F. Yang, L. P. Bai, W. Huang, X. Li, S. Zhao, N. Zhong, et al., Comparison of in vitro antiviral activity of tea polyphenols against influenza a and b viruses and structure–activity relationship analysis, Fitoterapia, 93 (2014), 47–53. https://doi.org/10.1016/j.fitote.2013.12.011 doi: 10.1016/j.fitote.2013.12.011
[12]	P. Chowdhury, M. Sahuc, Y. Rouillé, C. Rivière, N. Bonneau, A. Vandeputte, et al., Theaflavins, polyphenols of black tea, inhibit entry of hepatitis c virus in cell culture, PLOS One, 13 (2018), e0198226. https://doi.org/10.1371/journal.pone.0198226 doi: 10.1371/journal.pone.0198226
[13]	A. Ali, W. Nazeer, M. Munir, S. M. Kang, M-polynomials and topological indices of zigzagand rhombic benzenoid systems, Open Chem., 16 (2018), 122–135. https://doi.org/10.1515/chem-2018-0010 doi: 10.1515/chem-2018-0010
[14]	M. K. Jamil, M. Imran, K. A. Sattar, Novel face index for benzenoid hydrocarbons, Mathematics, 8 (2020), 312. https://doi.org/10.3390/math8030312 doi: 10.3390/math8030312
[15]	M. K. Siddiqui, M. Naeem, N. A. Rahman, M. Imran, Computing topological indices of certain networks, J. Optoelectron. Adv. Mater., 18 (2016), 9–10.
[16]	M. Nadeem, M. Azeem, H. A. Siddiqui, Comparative study of zagreb indices for capped, semi-capped, and uncapped carbon nanotubes, Polycyclic Aromat. Compd., 2021 (2020), 1–18. https://doi.org/10.1080/10406638.2021.1890625 doi: 10.1080/10406638.2021.1890625
[17]	M. F. Nadeem, M. Imran, H. M. A. Siddiqui, M. Azeem, A. Khalil, Y. Ali, Topological aspects of metal-organic structure with the help of underlying networks, Arabian J. Chem., 14 (2021), 103157. https://doi.org/10.1016/j.arabjc.2021.103157 doi: 10.1016/j.arabjc.2021.103157
[18]	A. N. A. Koam, A. Ahmad, M. E. Abdelhag, M. Azeem, Metric and fault-tolerant metric dimension of hollow coronoid, IEEE Access, 9 (2021), 81527–81534. https://doi.org/10.1109/ACCESS.2021.3085584 doi: 10.1109/ACCESS.2021.3085584
[19]	A. Ahmad, A. N. A. Koam, M. H. F. Siddiqui, M. Azeem, Resolvability of the starphene structure and applications in electronics, Ain Shams Eng. J., 13 (2022), 101587. https://doi.org/10.1016/j.asej.2021.09.014 doi: 10.1016/j.asej.2021.09.014
[20]	M. Azeem, M. F. Nadeem, Metric-based resolvability of polycyclic aromatic hydrocarbons, Eur. Phys. J. Plus, 136 (2021), 395. https://doi.org/10.1140/epjp/s13360-021-01399-8 doi: 10.1140/epjp/s13360-021-01399-8
[21]	Z. Hussain, M. Munir, M. Choudhary, S. M. Kang, Computing metric dimension and metric basis of $2d$ lattice of alpha-boron nanotubes, Symmetry, 10 (2018), 300. https://doi.org/10.3390/sym10080300 doi: 10.3390/sym10080300
[22]	S. Imran, M. K. Siddiqui, M. Hussain, Computing the upper bounds for the metric dimension of cellulose network, Appl. Math. E-notes, 19 (2019), 585–605.
[23]	A. N. A. Koam, A. Ahmad, Barycentric subdivision of cayley graphs with constant edge metric dimension, IEEE Access, 8 (2020), 80624–80628. https://doi.org/10.1109/ACCESS.2020.2990109 doi: 10.1109/ACCESS.2020.2990109
[24]	X. Liu, M. Ahsan, Z. Zahid, S. Ren, Fault-tolerant edge metric dimension of certain families of graphs, AIMS Math., 6 (0202), 1140–1152. http://dx.doi.org/2010.3934/math.2021069
[25]	J. B. Liu, Z. Zahid, R. Nasir, W. Nazeer, Edge version of metric dimension anddoubly resolving sets of the necklace graph, Mathematics, 6 (2018), 243. https://doi.org/10.3390/math6110243 doi: 10.3390/math6110243
[26]	H. Raza, Y. Ji, Computing the mixed metric dimension of a generalized petersengraph $p (n, 2)$ , Front. Phys., 8 (2020), 211. https://doi.org/10.3389/fphy.2020.00211 doi: 10.3389/fphy.2020.00211
[27]	M. F. Nadeem, M. Azeem, A. Khalil, The locating number of hexagonal möbius ladder network, J. Appl. Math. Comput., 66 (2021), 149–165. https://doi.org/10.1007/s12190-020-01430-8 doi: 10.1007/s12190-020-01430-8
[28]	M. Ahsan, Z. Zahid, S. Zafar, A. Rafiq, M. Sarwar Sindhu, M. Umar, Computing the edge metric dimension of convex polytopes related graphs, J. Math. Comput. Sci., 22 (2021), 174–188. http://dx.doi.org/10.22436/jmcs.022.02.08 doi: 10.22436/jmcs.022.02.08
[29]	A. Ahmad, M. Baca, S. Sultan, Minimal doubly resolving sets of necklace graph, Math. Rep., 20 (2018), 123–129.
[30]	T. Vetrik, A. Ahmad, Computing the metric dimension of the categorial product of graphs, Int. J. Comput. Math., 94 (2017), 363–371. https://doi.org/10.1080/00207160.2015.1109081 doi: 10.1080/00207160.2015.1109081
[31]	A. Ahmad, S. Sultan, On minimal doubly resolving sets of circulant graphs, Acta Mech. Slovaca, 20 (2017), 6–11. https://doi.org/10.21496/ams.2017.002 doi: 10.21496/ams.2017.002
[32]	A. Ahmad, M. Imran, O. Al-Mushayt, S. A. H. Bokhary, On the metric dimension of barcycentric subdivision of cayley graphs $cay(z_{n}\oplus z_{m})$ , Miskolc Math. Notes, 16 (2015), 637–646. https://doi.org/10.18514/MMN.2015.1192 doi: 10.18514/MMN.2015.1192
[33]	J. B. Liu, M. F. Nadeem, M. Azeem, Bounds on the partition dimension of convex polytopes, Comb. Chem. Throughput Screening, 25 (2020), 547–553. https://doi.org/10.2174/1386207323666201204144422 doi: 10.2174/1386207323666201204144422
[34]	M. Azeem, M. Imran, M. F. Nadeem, Sharp bounds on partition dimension of hexagonal mobius ladder, J. King Saud Univ. Sci., 34 (2022), 101779. https://doi.org/10.1016/j.jksus.2021.101779 doi: 10.1016/j.jksus.2021.101779
[35]	N. Mehreen, R. Farooq, S. Akhter, On partition dimension of fullerene graphs, AIMS Math., 3 (2018), 343–352. http://dx.doi.org/10.3934/Math.2018.3.343 doi: 10.3934/Math.2018.3.343
[36]	A. Shabbir, M. Azeem. On the partition dimension of tri-hexagonal alpha-boron nanotube, IEEE Access, 9 (2021), 55644–55653. https://doi.org/10.1109/ACCESS.2021.3071716 doi: 10.1109/ACCESS.2021.3071716
[37]	M. K. Siddiqui, M. Imran, Computing the metric and partition dimension of h-naphtalenic and vc5c7 nanotubes, J. Optoelectron. Adv. Mater., 17 (2015), 790–794.
[38]	H. M. A. Siddiqui, M. Imran, Computing metric and partition dimension of 2-dimensional lattices of certain nanotubes, J. Comput. Theor. Nanosci., 11 (2014), 2419–2423. https://doi.org/10.1166/jctn.2014.3656 doi: 10.1166/jctn.2014.3656
[39]	E. C. M. Maritz, T. Vetrík, The partition dimension of circulant graphs, Quaestiones Math., 41 (2018), 49–63. https://doi.org/10.2989/16073606.2017.1370031 doi: 10.2989/16073606.2017.1370031
[40]	Z. Hussain, S. Kang, M. Rafique, M. Munir, U. Ali, A. Zahid, et al., Bounds for partition dimension of m-wheels, Open Phys., 17 (2019), 340–344. https://doi.org/10.1515/phys-2019-0037 doi: 10.1515/phys-2019-0037
[41]	Amrullah, E. Baskoro, R. Simanjuntak, S. Uttunggadewa, The partition dimension of a subdivision of a complete graph, Procedia Comput. Sci., 74 (2015), 53–59. https://doi.org/10.1016/j.procs.2015.12.075 doi: 10.1016/j.procs.2015.12.075
[42]	C. Wei, M. F. Nadeem, H. M. A. Siddiqui, M. Azeem, J. B. Liu, A. Khalil, On partition dimension of some cycle-related graphs, Mathematical Problems in Engineering, 2021 (2021), 4046909. https://doi.org/10.1155/2021/4046909 doi: 10.1155/2021/4046909
[43]	J. Santoso, Darmaji, The partition dimension of cycle books graph, J. Phys. Conf. Ser., 974 (2018), 012070.
[44]	Darmaji, R. Alfarisi, On the partition dimension of comb product of path and complete graph, in AIP Conference Proceedings, (2017), 020038. https://doi.org/10.1063/1.4994441
[45]	A. Nadeem, A. Kashif, S. Zafar, Z. Zahid, On 2-partition dimension of the circulant graphs, J. Intell. Fuzzy Syst., 40 (2021), 9493–9503. https://doi.org/10.3233/JIFS-201982 doi: 10.3233/JIFS-201982
[46]	P.J. Slater, Leaves of trees, Proceeding of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congr. Numerantium, 14 (1975), 549–559.
[47]	F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Comb., 2 (1976), 191–195.
[48]	G. Chartrand, E. Salehi, P. Zhang, The partition dimension of graph, Aequationes Math., 59 (2000), 45–54. https://doi.org/10.1007/PL00000127 doi: 10.1007/PL00000127
[49]	G. Chartrand, L. Eroh, M. A. O. Johnson, R. Ortrud, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105 (2000), 99–113. https://doi.org/10.1016/S0166-218X(00)00198-0 doi: 10.1016/S0166-218X(00)00198-0
[50]	S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., 70 (1996), 217–229.
[51]	A. Sebö, E. Tannier, On metric generators of graphs, Math. Oper. Res., 29 (2004), 383–393. https://doi.org/10.1287/moor.1030.0070 doi: 10.1287/moor.1030.0070
[52]	M. F. Nadeem, M. Hassan, M. Azeem, S. Ud-Din Khan, M. R. Shaik, M. A. F. Sharaf, et al., Application of resolvability technique to investigate the different polyphenyl structures for polymer industry, J. Chem., 2021 (2021), 6633227. https://doi.org/10.1155/2021/6633227 doi: 10.1155/2021/6633227
[53]	J. Caceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, C. Seara, et al., On the metric dimension of cartesian product of graphs, SIAM J. Discrete Math., 21 (2007), 423–441. https://doi.org/10.1137/050641867 doi: 10.1137/050641867
[54]	Z. Beerliova, F. Eberhard, T. Erlebach, A. Hall, M. Hoffmann, M. Mihalak, et al., Network discovery and verification, IEEE J. Selected Areas in Commun., 24 (2006), 2168–2181. https://doi.org/10.1109/JSAC.2006.884015 doi: 10.1109/JSAC.2006.884015
[55]	V. Chvatal, Mastermind, Combinatorica, 3 (1983), 325–329. https://doi.org/10.1007/BF02579188 doi: 10.1007/BF02579188
[56]	R. A. Melter, I. Tomescu, Metric bases in digital geometry, Comput. Visual Graphics Image Process., 25 (1984), 113–121. https://doi.org/10.1016/0734-189X(84)90051-3 doi: 10.1016/0734-189X(84)90051-3
[57]	C. Hernando, M. Mora, P. J. Slater, D. R. Wood, Fault-tolerant metric dimension of graphs, Convexity Discrete Struct., 5 (2008), 81–85.
[58]	J. Wei, M. Cancan, A. Rehman, M. Siddiqui, M. Nasir, M. Younas, et al., On topological indices of remdesivir compound used in treatment of corona virus (COVID 19), Polycyclic Aromat. Compd., 2021 (2021), 1–19. https://doi.org/10.1080/10406638.2021.1887299 doi: 10.1080/10406638.2021.1887299
[59]	S. Mondal, N. De, A. Pal, Topological indices of some chemical structures applied for the treatment of COVID-19 patients, Polycyclic Aromat. Compd., 42 (2022), 1–15. https://doi.org/10.1080/10406638.2020.1770306 doi: 10.1080/10406638.2020.1770306

This article has been cited by:

1.	Masaaki Fujii, Akihiko Takahashi, Strong Convergence to the Mean Field Limit of a Finite Agent Equilibrium, 2022, 13, 1945-497X, 459, 10.1137/21M1441055
2.	Masaaki Fujii, Akihiko Takahashi, Strong Convergence to the Mean-Field Limit of A Finite Agent Equilibrium, 2021, 1556-5068, 10.2139/ssrn.3905899
3.	Diogo Gomes, Julian Gutierrez, Ricardo Ribeiro, A Random-Supply Mean Field Game Price Model, 2023, 14, 1945-497X, 188, 10.1137/21M1443923
4.	Yuri Ashrafyan, Tigran Bakaryan, Diogo Gomes, Julian Gutierrez, 2022, The potential method for price-formation models, 978-1-6654-6761-2, 7565, 10.1109/CDC51059.2022.9992621
5.	Diogo Gomes, Julian Gutierrez, Mathieu Laurière, Machine Learning Architectures for Price Formation Models, 2023, 88, 0095-4616, 10.1007/s00245-023-10002-8
6.	Matt Barker, Pierre Degond, Ralf Martin, Mirabelle Muûls, A mean field game model of firm-level innovation, 2023, 33, 0218-2025, 929, 10.1142/S0218202523500203
7.	Khaled Aljadhai, Majid Almarhoumi, Diogo Gomes, Melih Ucer, A mean-field game model of price formation with price-dependent agent behavior, 2024, 1982-6907, 10.1007/s40863-024-00465-0
8.	Diogo Gomes, Julian Gutierrez, Mathieu Laurière, 2023, Machine Learning Architectures for Price Formation Models with Common Noise, 979-8-3503-0124-3, 4345, 10.1109/CDC49753.2023.10383244
9.	Masaaki Fujii, Masashi Sekine, Mean-Field Equilibrium Price Formation With Exponential Utility, 2023, 1556-5068, 10.2139/ssrn.4420441
10.	Yuri Ashrafyan, Diogo Gomes, A Fully-Discrete Semi-Lagrangian Scheme for a Price Formation MFG Model, 2025, 2153-0785, 10.1007/s13235-025-00620-y
11.	Masaaki Fujii, Masashi Sekine, Mean-field equilibrium price formation with exponential utility, 2024, 24, 0219-4937, 10.1142/S0219493725500017
12.	Masaaki Fujii, Masashi Sekine, Mean Field Equilibrium Asset Pricing Model with Habit Formation, 2025, 1387-2834, 10.1007/s10690-024-09507-1

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)