Nonlinear Dynamics of Complex SystemsView this Special Issue
Flow, Wind, and Stochastic Connectivity Modeling Infectious Diseases
We study in this paper the trends of the evolution of different infections using a SIR flow (first-order ODE system), completed by a differential inclusion, a geodesic motion in a gyroscopic field of forces, and a stochastic SIR perturbation of the flow (Itô ODE system). We are interested in mathematical analysis, bringing new results on studied evolutionary models: infection flow together with a differential inclusion, bounds of evolution, dual description of disease evolution, log-optimal and rapid path, epidemic wind (geometric dynamics), stochastic equations of evolution, and stochastic connectivity. We hope that the paper will be a guideline for strategizing optimal sociopolitical countermeasures to mitigate infectious diseases.
All topics in this paper are based on dynamics induced by flows and differential inclusions, dynamical systems of geometric origin, nonholonomic dynamical systems, and stochastic differential equations. Their combination reflects the mathematical complexity of the studied problems.
The mathematical literature that helped us to do this study is classified as follows: stochastic modeling of geometric structures [1–4], infectious disease flow [5–7], differential inclusions , geometric dynamics on Riemannian manifolds [9, 10], nonholonomic optimization [11, 12], and nonholonomic spaces .
The original results can be summarized by properties of infectious disease flow, Maple simulations for COVID-19 in Romania, infectious disease differential inclusion, bounds of disease evolution, dual description of disease evolution, epidemic wind generated by the flow and the geometry of the space, and computation of optimal striking time for stochastic connectivity.
The Maple simulations for COVID-19, with Romanian data, are more suggestive than what could be done according to the model related to the works [3, 5–7]. Our results on the asymptotic behavior of infectious disease flow and bounds of disease evolution via extremum problems are finer than those presented by M. W. Hirsch, S. Smale, R. L. Devaney in their book , Chapter 11. The Pfaff evolution, the epidemic wind, and finding the optimal striking time for stochastic connectivity via an extremum problem are totally original ideas, suggested by our recent papers. Particularly, the Pfaff evolution underlines that we can study the infectious disease system like a Carnot group.
Although the epidemic differential inclusion and epidemic wind appear to have been created ad hoc, it explains the pandemic spread in the sense that any two points on the globe can be joined by an epidemic trajectory. This is not true if we stop with the explanations only at the epidemic flow. The same idea is underlined by stochastic connectivity.
1.1. Disease Infection Data
The evolution of disease infections in each region has been modeled recently via a stochastic susceptible-infected-recovered (SSIR) model [3, 5–7] with the following data: Evolution parameter: is the daily-time parameter. States: (1) denotes the total susceptible population at time ; (2) denotes the number of active infections at time ; (3) denotes the total number of recoveries and deaths at time t; (4) denote the change in the states at time ; (5) is an incremental Wiener process (Brownian motion), which models the randomness in the evolution. Constant parameters: (1) (measured by ) is a constant denoting the growth rate, which factors the rise in the number of infections, due to interactions between susceptible and infected populations. This parameter is a lumped constant which is meant to account for (a) the population size, (b) reproduction number of infectious diseases, and (c) exposure factor (which depends on mobility, precautionary measures, etc.); (2) (measured by ) is the rate of outcomes, that is, the rate at which the infections are neutralized, which may be due to recovery or death. It is assumed that recovered persons would not spread the infections again (at least for a window of a month); (3) is a parameter used to model the randomness in the evolution, which may cause local deviations from the typical (exponential) trends; (4) is the population of the region, and and are the initial number of susceptible individuals and active infections.
2. Infectious Disease Flow
The susceptible-infected-recovered (SIR) model, with three different states, was selected [3,5 –7] to describe the evolution of different infections in a region of the world. On , the infectious disease Cauchy problem is
Since the initial differential system is equivalent to the symmetric systemautomatically two first integrals and appear, and hence the general solution is (spiral curve in a plane). In other words, the infectious disease nonlinear differential system is compartmental and proves the mass conservation property [5–7]. The family of field surfaces has the general equation , where is an arbitrary function. The vortex lines of a SIR vector field are plane curves.
Given an initial condition summing to , it follows and . Also, the previous differential system has a straight line of equilibrium points (particularly, and are two equilibrium points). The general theory shows that a nonisolated equilibrium point can be stable but not asymptotically stable.
2.1. Open Problems
(1)Investigate whether there are monomial connections on of components so that the SIR vector field is convex with respect to .(2)Convolution is a very powerful technique in applications. Transforming the usual product into a convolution (product), let us replace the initial Cauchy problem with a convolution problemwhere the convolution (product) is defined by . Applying the Laplace transform, study the solution of this convolution problem.
2.2. Maple Simulations for COVID-19
We denote assume values and (hypothetical data used for research purpose). Romanian media, 06.05.2020: = Romania population = 19,410 million; x0 = quarantined + nonquarantined = 19,389; y0 = infected = 13,837; z0 = recovered 5.454 + deceased 858 = 6,312 (real data normalized by 1,000).
The graph admits a limit point ; the graph has a maximum point and a limit point . The graph has a limit point . Of course, . See Figures 1–3 .(i)with(DEtools): phaseportrait(, [x(t), y(t), z(t)], t = 0 .. 100, [[x (0) = 19.389, y(0) = 13.837, z(0) = 6.312]], stepsize = 0.5e − 1, scene = [t, x(t)], linecolour = , method = classical [foreuler]);(ii)phaseportrait(, [x(t), y(t), z(t)], t = 0, ..., 100, [[x(0) = 19.389, y(0) = 13.837, z(0) = 6.312]], stepsize = 0.5e − 1, scene = [t, y(t)], linecolour = ), method = classical[foreuler]);(iii)phaseportrait(, [x(t), y(t), z(t)], t = 0, ..., 100, [[x(0) = 19.389, y(0) = 13.837, z(0) = 6.312]], stepsize = 0.5e − 1, scene = [t, z(t)], linecolour = ), method = classical[foreuler]).
Remark 1. Let us animate the surface with respect to the parameter (Figure 4).(i)with(plots);(ii)animate(plot3d, , ).
2.3. Parametrization by S
It would be more natural to parameterize the previous general solution by “raw material” S; namely,where and . If the social constants and the state are so that , then there are infections; otherwise, there are not. So, trying to influence the transmission constants we can limit the number of infections. In fact, the state exists, if and only if and then .
If we start with the equilibrium points , we find
The more interesting case holds for . Then, the ODE for becomesor changing the variable as ,
The solution of this ODE is written in the form
The denominator of the integrand has three roots . Using the qualitative approximationwe find the implicit solution
The constant is determined from the initial condition . The initial condition would give us the equilibrium point .
Let us observe that for . Hence, the final state appears for .
We fix . From the graphs and , in Figure 5, we can read the following: when is decreasing from to , the function is increasing from 0 to a maximum for and then decreasing again to 0. In other words, when the epidemic goes out, that is, again, and the variable stops at , then we talk about an uncontaminated population.
Maple simulation (Figure 5):(i)plot(, x = 0.. 6, color = [”red”, ”green”]).
2.4. Asymptotic Behavior of Infectious Disease Flow
Theorem 1. For positive initial conditions , the limit value exists and is .
Proof. Suppose .
Because is monotonically decreasing () and nonnegative, it has a limit , with .
Since is monotonically decreasing () and nonnegative, it has a limit , with .
On the other hand, is monotonically increasing () and nonnegative. But shows that has a limit . For large enough, we have . Then, the differential system consisting of the first equation and third equation converges to the linear system corresponding to the linearization at the point ; that is,The matrix of this linear system has the eigenvalues and . Consequently, render this system unstable, and then the trajectory diverges. This contradicts .
Suppose . Similarly, it follows the existence of , satisfying , , and .
Theorem 2. For the infectious disease flow, one has (see equilibrium point).
Proof. We use the second differential equation . Let be the primitive of function . Since , we find . At limit, when , we must have (see equilibrium point).
Remark 2. Let , , and be the primitives of functions , , and . Then, .
Theorem 3. Let , , and be the primitives of functions , , and . For positive initial conditions , the limit valuesare related by
Proof. Let us consider the first differential equationIt follows , and hence .
Integrating the third differential equation, we findSince represent the evolution of disease infections, the initial conditions , and are perfectly suited for biological applications. Suppose these initial conditions. The monotony of the state functions is described by the signum of derivatives: either which show that is decreasing and and are increasing or which show that are decreasing and is increasing .
2.5. Covering All the Manifold by a Differential Inclusion
As any flow, the infectious disease flow does not cover all the manifold , and so there are pairs of points that cannot be joined by a flow trajectory. The natural question arises: what mathematical construction allows us to cover all the manifold?
The infectious disease vector field determines an orthogonal distribution generated by two linearly independent vector fields and , orthogonal to .
The three vector fields determine the differential inclusionwhich can be used to understand and suitably interpret the spreading of the disease, in the sense that any two points on can be joined by a piecewise solution of the differential inclusion.
3. Bounds of Disease Evolution
Let us select the best values of state variables when we evolve along the solution of the disease flow. The aim is to manage correctly a pandemic since the values of certain parameters can be chosen subject to some conditions expressing their ranges and interrelationships. The choice determines the values of a number of other variables on which the desirability of the end result depends, such as cost, weight, speed, bandwidth, and reliability.
3.1. Extrema Constrained by Equalities
A basic problem we discuss at the beginning of this paper is as follows: “find subject to .”
The constraints satisfy the condition of nondegenerate constraint qualification.
To solve this problem, we attach the Lagrange function
Sincethe critical points of the function are given by the algebraic system
It follows the critical points
On the other hand,
The associated matrixwith respect to , is negative definite. The function is concave. Hence, all critical points are maximum points. The maximum value of objective function is
The introduction of Lagrange multipliers as additional variables looks artificial but it makes it possible to apply to the constrained-extremum problem the same first-order condition used in the free-extremum problem (but for more complex function ). Note also that have a certain specific meaning: if the solution is regarded asthen the marginal variations are
Maple Simulations: we denote . Let us find the extrema of some functions constrained by the spiral curve.(i)with (optimization); The Minimize command automatically selects the most appropriate solver.(ii)Minimize(, , assume = nonnegative); (iii)Minimize(, , assume = nonnegative); (iv)Minimize(, , assume = nonnegative); (v)Minimize(, , assume = nonnegative); (vi)Minimize(, , assume = nonnegative); (vii)Minimize(, , assume = nonnegative);
3.2. Extrema Constrained by Field Lines
To simplify, we use standard notations in mathematics . Then, infectious disease Cauchy problem on is written
To find bounds of significant functions connected to this flow, we use the technique of optimization of an objective function constrained by a field line .
Let us start with finding the maximum for the total susceptible population : findwith the restriction.
Since the component is a decreasing function, the maximum is reached at the starting point .
Let us show that we do not have constrained critical points that produce an extremum. We set the critical point condition . In this case, and hence .
We eliminate , since this condition leads to an equilibrium point of the dynamical system. Also, the solution is not convenient. Indeed, a critical point of the form cancels the expression whose sign at the point would decide the property of extremum: , that is, .
Since , the function is convex on the subset .
Proposition 1. The maximum of the total susceptible population is reached at the starting point .
Let us find now the maximum for the number of active infections : determine constrained by .
We set the critical point condition . In this case, . It follows the relation . The convenient solution (critical point) is .
The sufficient condition is equivalent to or .
Theorem 4. Suppose that on an evolution field line there exists a point satisfying
Then, the number of active infections has an upper bound at this point.
To find the minimum for the number of recoveries and deaths , we use the problem: determine constrained by .
Since the function is increasing, the minimum is reached at the starting point .
Let us show that we do not have constrained critical points that produce an extremum. We set the critical point condition . In this case, . It follows the relation . We have no convenient solution (critical point) since leads to an equilibrium point of the dynamical system.
On the subset , the function is convex. Indeed, we have .
Proposition 2. The minimum number of recoveries and deaths is reached at the starting point .
4. Dual Description of Disease Evolution
A vector field determines a flow (collinearity condition) and a Pfaff equation (orthogonality condition) . By duality, the nonlinear ODEs in infectious disease flow are transformed into an infectious disease Pfaff equation
Since , this Pfaff equation is noncompletely integrable (it represents a nonholonomic surface, i.e., a collection of integral curves). The integral curves are orthogonal to infectious disease field lines. Since any two points in are joined by an integral curve of this Pfaff equation, the dual evolution of infectious disease shows that all parts of the world will be infected.
Simplifying by , we obtain an equivalent infectious disease Pfaff equation
The normal vector field to this distribution is . Two independent vector fields tangent to the distribution are
It followsand hence the vector fields determine a Carnot group (in a future paper we shall study the infectious disease system like a Carnot group).
Proposition 3. For , one has
Proof. For , we consider the curve . Integrating the Pfaff form and selecting in a convenient way, it follows
4.1. Log-Optimal and Rapid Path
To find bounds of significant functions for disease distribution, we can use the technique of optimization of an objective function with nonholonomic constraints [11, 12]. One of these functions is the “volume” ( product) of disease states.
To find bounds for the function (logarithm of “volume”), we use the problem “determine constrained by the Pfaff equation of evolution.”
The critical point condition is
Consequently, the critical point set is described bythat is,subject to the condition . This algebraic system has a solution since the curve\is transversal to the plane . Indeed, the tangent vector to the curve has the componentsthe normal vector to the plane has the components , and the scalar product is different from zero.
5. Epidemic Wind
So far, predictive mathematical models for epidemics were treated as flows. Now, we add a more complex idea, namely, to look at the evolution of an epidemic as a wind created ad hoc by the epidemic flow and the “geometry of the world.” These are fundamental to understand the course of the epidemics and to plan effective control strategies for answering the question: how can we explain an exponentially growing number of patients all over the world who were diagnosed with COVID-19?
The time has come for us to treat the epidemics like winds (geometric dynamics and geodesic motion in a gyroscopic field of forces) [9, 10], producing chaotic dynamics. The geometric dynamics is generated by primordial data: flow and geometry of the space.
The infectious disease autonomous flow on is
On the Riemannian manifold , the flow and the metric determine the least squares autonomous Lagrangian
We attach an integral action
Explicitly, the epidemic wind is described by the second-order differential system
Adding all three ODEs, we obtain an ODE whose last term is . Furthermore, the last second-order ODE is equivalent to the first-order ODE
Theorem 5. The geometric dynamics (wind) represented by previous second-order ODEs is decomposable into the infectious disease flow and transversal to flow spiral trajectories.
Proof. We give the proof in generic coordinates. The subset of solutions corresponding to the initial values are solutions reducible to solutions of the infectious disease flow. The subset of solutions corresponding to the initial values , are transversal to the solutions of the infectious disease flow. The converse is also true.
Based on the existence and uniqueness theorem, each solution of any second-order prolongation of the first-order ODE system has the property: implies .
In generic coordinates, the attached Hamiltonian to the wind iswhere (kinetic energy). The maximal solutions of infectious disease wind are split into three categories of curves: (1) curves characterized by (flow trajectories); (2) curves satisfying ; (3) curves characterized by . The transversal curves in category (2) can have the images throughout, but the curves in category (3) have the images only in the set .
The solutions of infectious disease wind are highly sensitive to initial conditions. In other words, small differences in initial conditions, such as those due to rounding errors in numerical computation, can yield widely diverging outcomes for infectious disease wind, rendering long-term prediction of its behavior impossible in general. A single field line and an infinity of transverse curves start from a fixed point.
In a flow, the starting point is fixed, but the endpoint is the one that results. In geometric dynamics, the initial conditions are in the form of point-direction point-endpoint The solution we find optimizes the objective function of the smallest squares (the best approximation of flow in the sense of a convenient Riemannian metric). Small disturbances of the initial direction or of the endpoint can produce dramatic changes in the solution, which highlight the complexity of the problem.
Remark 3. In our sense, any wind is strongly dependent on the Riemannian manifold . The best selection of the Riemannian manifold adapted to infectious disease wind is after constant curvature: (1) curvature 0, Euclidean manifold , used in the previous explanations; (2) curvature , hyperbolic manifold ; (2) curvature 1, sphere , with the metric induced by the Euclidean metric on .
6. Stochastic Connectivity
Stochastic differential equations are widely used to model epidemic infections, molecular dynamics, biophysical dynamics, climate dynamics, engineering systems, and so on, under random fluctuations.
Let us write the flow in Pfaff terminology and let us replace the parameter by a control . We use , as independent Wiener processes. Starting from the (nonlinear control system) infectious disease Cauchy problemon , a stochastic perturbation is defined by stochastic differential equation system,where the functions are drift coefficients and , are diffusion coefficients.
If and , then the stochastic perturbation satisfies ; that is, we have again a first integral .
Suppose that the control is piecewise smooth and has values in a bounded and closed set , where will be selected at the end of this section by an extremum problem, determining the optimal striking time. The set of such controls, denoted by , is called the set of admissible controls.
We explore how stochastic noise can be used to find connectivity properties generated by the underlying deterministic infectious disease dynamics and randomness.
Definition 1. A strong solution of this stochastic differential system with the initial condition is an adapted continuous process, such that, for all , it satisfies the stochastic integral systemHowever, there is a number of subtle points involved: first, the existence of the integrals requires some degree of regularity on and on the functions , (matrix); in particular, it must be the case that, for all , with probability one, .
Second, the solution is required to exist for all with probability one.
Properties of Itô integral are that, for all , we have
Definition 2. A weak solution of the SDE with the initial condition is a continuous stochastic process defined on some probability space , such that, for some Wiener processes and some admissible filtration , the process is adapted and satisfies the associated stochastic integral system.
The drift coefficients are uniformly Lipschitz functions. The basic result, due to Itô, is that, for uniformly Lipschitz functions , the SDE has strong solutions and that, for each initial value , the solution is unique.
Without loss of generality, we consider that the amplitudes of error are constants; that is, .
Definition 3. Let be a feedback bounded control. A stochastic process , which satisfies the SDE system, is called an admissible stochastic process.
Theorem 6. Let and be two points in the set . Denote as the Euclidean sphere of radius , centered at . Then, for any and , there exists a striking time and an admissible stochastic process , satisfying the boundary conditionssuch that
Proof. We start with the nonnegative random variable . Markov inequality shows that