# Flow, Wind, and Stochastic Connectivity Modeling Infectious Diseases

Authors: C. Udriste; I. Tevy; A. S. Rasheed
Journal: Complexity
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

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 [8], geometric dynamics on Riemannian manifolds [9, 10], nonholonomic optimization [11, 12], and nonholonomic spaces [13].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 [6], 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:t is the daily-time parameter.States: (1)St denotes the total susceptible population at time t; (2) It denotes the number of active infections at time t; (3) Rt denotes the total number of recoveries and deaths at time t; (4) dSt,dIt,and dRt denote the change in the states at time t; (5) dWt is an incremental Wiener process (Brownian motion), which models the randomness in the evolution.Constant parameters: (1)β (measured by β=people×time−1) 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) γ (measured by γ=time−1) 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) σ is a parameter used to model the randomness in the evolution, which may cause local deviations from the typical (exponential) trends; (4) Ptotal is the population of the region, and S0 and I0 are the initial number of susceptible individuals and active infections.


## 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:t is the daily-time parameter.States: (1)St denotes the total susceptible population at time t; (2) It denotes the number of active infections at time t; (3) Rt denotes the total number of recoveries and deaths at time t; (4) dSt,dIt,and dRt denote the change in the states at time t; (5) dWt is an incremental Wiener process (Brownian motion), which models the randomness in the evolution.Constant parameters: (1)β (measured by β=people×time−1) 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) γ (measured by γ=time−1) 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) σ is a parameter used to model the randomness in the evolution, which may cause local deviations from the typical (exponential) trends; (4) Ptotal is the population of the region, and S0 and I0 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 ℝ+3, the infectious disease Cauchy problem is(1)S˙t=−βStIt,I˙t=βSt−γIt,R˙t=γIt,S0=S0,I0=I0,R0=R0,S0+I0+R0=Ptotal.Since the initial differential system is equivalent to the symmetric system(2)dS−βS=dIβS−γ=dRγ,automatically two first integrals Seβ/γR=c1 and S+I+R=c2 appear, and hence the general solution is Seβ/γR=c1,S+I+R=c2 (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 ϕSeβ/γR,S+I+R=c, where ϕc1,c2 is an arbitrary C1 function. The vortex lines of a SIR vector field are plane curves.Given an initial conditionS0,I0,R0 summing to Ptotal, it follows c1=S0eβ/γR0 and c2=Ptotal=S0+I0+R0. Also, the previous differential system has a straight line of equilibrium points I=0,S+R=Ptotal (particularly, S=0,I=0,R=Ptotal and S=γ/β,I=0,R=Ptotal−γ/β 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 ℝ++3 of components(3)Γijk=cijkx1a1x2a2x3a3,cijk>0,xi∈ℝ++,ai∈ℝ,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 problem(4)S˙t=−βSt∗It,I˙t=βSt−γ∗It,R˙t=γIt,S0=S0,I0=I0,R0=R0,S0+I0+R0=Ptotal,where the convolution (product) is defined by f∗gt=∫0tfτgt−τdτ. Applying the Laplace transform, study the solution of this convolution problem.


### 2.2. Maple Simulations for COVID-19

We denoteS=x,I=y,R=z,t=daysassume values β=7.5×10−3 and γ=0.06 (hypothetical data used for research purpose). Romanian media, 06.05.2020: Ptotal  = 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 grapht,xt admits a limit point x=x¯=limt⟶∞xt≥0; the graph t,yt has a maximum point and a limit point y=y¯=limt⟶∞yt=0. The graph t,zt has a limit point z=z¯=limt⟶∞zt≥0. Of course, x¯+z¯=Ptotal. See Figures 1–3 .(i)
with(DEtools):phaseportrait([Dxt=−7.5∗10−3∗xt∗yt,Dyt=7.5∗10−3∗xt−0.6e−1∗yt,Dtt=0.6e−1∗yt], [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 =  sin1/2∗t∗Pi, method = classical [foreuler]);(ii)
phaseportrait([Dxt=−7.5∗10−3∗xt∗yt,Dyt=7.5∗10−3∗xt−0.6e−1∗yt,Dtt=0.6e−1∗yt], [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 =  sin1/2∗t∗Pi), method = classical[foreuler]);(iii)
phaseportrait([Dxt=−7.5∗10−3∗xt∗yt,Dyt=7.5∗10−3∗xt−0.6e−1∗yt,Dtt=0.6e−1∗yt], [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 =  sin1/2∗t∗Pi), method = classical[foreuler]).Figure 1
Susceptible population evolutiont,xt.Figure 2
Active infection evolutiont,yt.Figure 3
Number of recoveries and deaths’ evolutiont,zt.Remark 1.
Let us animate the surfaceSeβ/γR=c1 with respect to the parameter a=β/γ (Figure 4).(i)
with(plots);(ii)
animate(plot3d,x∗expa∗z,x=1,…,10,z=1,…,10, a=0.125,…,1).Figure 4
Animate SR first integral.


### 2.3. Parametrization by S

It would be more natural to parameterize the previous general solution by “raw material” S; namely,(5)S=S,I=γβlnc2S−S,R=−γβlnc1S,where c1S<1<c2S and γ/βlnc2/c1=Ptotal. If the social constants β,γ and the state S are so that I>0, then there are infections; otherwise, there are not. So, trying to influence the transmission constants β,γ we can limit the number of infections. In fact, the state I exists, if and only if S1≤S≤S2 and then IS1=IS2=0.If we start with the equilibrium pointsS0=Ptotal,I0=0,andR0=0, we find(6)I=γβlnSS0+S0−S,R=−γβlnSS0,S<S0.The more interesting case holds forS0>γ/β. Then, the ODE for St becomes(7)S˙t=βSt2−βS0St−γStlnStS0,or changing the variable as St=S0ut,(8)u˙t=βS0ut2−ut−γutlnut,ut<1.The solution of this ODE is written in the form(9)t+c=∫duβS0u2−u−γulnu.The denominator of the integrand has three roots0,u0,and1. Using the qualitative approximation(10)t+c=k∫duuu−u0u−1,u0<u<1,we find the implicit solution(11)u1/u01−u1/1−u0u−u01/u01−u0=cet/k.The constantc is determined from the initial condition u0=1−ε. The initial condition u0=1 would give us the equilibrium point ut≡1.Let us observe thatut⟶u0 for t⟶∞. Hence, the final state I=0,S=S1 appears for t⟶∞.We fixβ=1,γ=3,S0=5. From the graphs S,IS and S,FS=S2−5S−3SlnS/5, in Figure 5, we can read the following: when S is decreasing from S0 to S1, the function I is increasing from 0 to a maximum for S=β/γ and then decreasing again to 0. In other words, when the epidemic goes out, that is, I=0 again, and the variable S stops at S1>0, then we talk about an uncontaminated population.Figure 5
IntersectionIS∩FS.Maple simulation (Figure5):(i)
plot([3∗ln1/5∗x+5−x,x2−5∗x−3∗x∗ln1/5∗x], x = 0.. 6, color = [”red”, ”green”]).


### 2.4. Asymptotic Behavior of Infectious Disease Flow

Theorem 1.
For positive initial conditionsS0,I0,R0, the limit value S¯+I¯=limt⟶∞St+It exists and is Ptotal−R¯.Proof.
SupposeSt≤γ/β.
BecauseSt is monotonically decreasing (S˙t=−βStIt≤0) and nonnegative, it has a limit S¯=limt⟶∞St, with 0≤S¯≤γ/β.
SinceIt is monotonically decreasing (I˙t=βSt−γIt≤0) and nonnegative, it has a limit I¯=limt⟶∞It, with I¯≥0.
On the other hand,Rt is monotonically increasing (R˙t=γIt≥0) and nonnegative. But St+It+Rt=Ptotal shows that Rt has a limit R¯≥0. For t large enough, we have St≈S¯,It≈I¯,andRt≈R¯. Then, the differential system consisting of the first equation and third equation converges to the linear system corresponding to the linearization at the point S¯,I¯; that is,(12)S˙=−βI¯St−S¯−βS¯It−I¯,R˙t=γI¯It−I¯.
The matrix of this linear system has the eigenvaluesλ1=−βI¯<0 and λ2=γI¯>0. Consequently, S¯,I¯ render this system unstable, and then the trajectory St,It,Rt diverges. This contradicts St+It+Rt=Ptotal.
SupposeSt>γ/β. Similarly, it follows the existence of S¯,I¯,R¯, satisfying γ/β≤S¯, I¯≥0, and R¯≥0.Theorem 2.
For the infectious disease flow, one hasI¯=limt⟶∞It=0 (see equilibrium point).Proof.
We use the second differential equationdIt/It=d/dtlnIt=βSt−γ. Let S be the primitive of function St. Since ∫0bβSt−γdt=lnIb/I0, we find βSb−S0−γb=lnIb/I0. At limit, when b⟶∞, we must have I¯=I∞=0 (see equilibrium point).Remark 2.
LetS, ℐ, and ℛ be the primitives of functions St, It, and Rt. Then, St+ℐt+ℛt=tPtotal.Theorem 3.
LetS, ℐ, and ℛ be the primitives of functions St, It, and Rt. For positive initial conditions S0,I0,R0, the limit values(13)S¯=limt⟶∞St,ℐ¯=limt⟶∞ℐt,R¯=limt⟶∞Rtare related by(14)−βℐ¯−ℐ0=lnS¯S0,R¯−R0=γℐ¯−ℐ0.Proof.
Let us consider the first differential equation(15)dStSt=ddtlnSt=−βItdt.
It follows−β∫0∞Itdt=lnS¯/S0, and hence −βℐ¯−ℐ0=lnS¯/S0.
Integrating the third differential equation, we find(16)R¯−R0=γℐ¯−ℐ0.
SinceS,I,R represent the evolution of disease infections, the initial conditions S0≥0,I0≥0,R0≥0, and S0+I0+R0=Ptotal are perfectly suited for biological applications. Suppose these initial conditions. The monotony of the state functions St,It,Rt is described by the signum of derivatives: either S˙≤0,I˙>0,R˙≥0 which show that St is decreasing and It and Rt are increasing ∀t>0 or S˙≤0,I˙≤0,R˙≥0 which show that St,It are decreasing and Rt is increasing ∀t>0.


### 2.5. Covering All the Manifold by a Differential Inclusion

As any flow, the infectious disease flow does not cover all the manifoldℝ+3, 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 fieldX=I−βS,βS−γ,γ determines an orthogonal distribution generated by two linearly independent vector fields Y=βS−γ,βS,0 and Z=0,−γ,βS−γ, orthogonal to X.The three vector fieldsX,Y,Z determine the differential inclusion(17)ddtSIR∈X,SpY,Z,which can be used to understand and suitably interpret the spreading of the disease, in the sense that any two points on ℝ+3 can be joined by a piecewise solution of the differential inclusion.


## 2.1. Open Problems

(1)
Investigate whether there are monomial connections∇ on ℝ++3 of components(3)Γijk=cijkx1a1x2a2x3a3,cijk>0,xi∈ℝ++,ai∈ℝ,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 problem(4)S˙t=−βSt∗It,I˙t=βSt−γ∗It,R˙t=γIt,S0=S0,I0=I0,R0=R0,S0+I0+R0=Ptotal,where the convolution (product) is defined by f∗gt=∫0tfτgt−τdτ. Applying the Laplace transform, study the solution of this convolution problem.


## 2.2. Maple Simulations for COVID-19

We denoteS=x,I=y,R=z,t=daysassume values β=7.5×10−3 and γ=0.06 (hypothetical data used for research purpose). Romanian media, 06.05.2020: Ptotal  = 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 grapht,xt admits a limit point x=x¯=limt⟶∞xt≥0; the graph t,yt has a maximum point and a limit point y=y¯=limt⟶∞yt=0. The graph t,zt has a limit point z=z¯=limt⟶∞zt≥0. Of course, x¯+z¯=Ptotal. See Figures 1–3 .(i)
with(DEtools):phaseportrait([Dxt=−7.5∗10−3∗xt∗yt,Dyt=7.5∗10−3∗xt−0.6e−1∗yt,Dtt=0.6e−1∗yt], [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 =  sin1/2∗t∗Pi, method = classical [foreuler]);(ii)
phaseportrait([Dxt=−7.5∗10−3∗xt∗yt,Dyt=7.5∗10−3∗xt−0.6e−1∗yt,Dtt=0.6e−1∗yt], [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 =  sin1/2∗t∗Pi), method = classical[foreuler]);(iii)
phaseportrait([Dxt=−7.5∗10−3∗xt∗yt,Dyt=7.5∗10−3∗xt−0.6e−1∗yt,Dtt=0.6e−1∗yt], [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 =  sin1/2∗t∗Pi), method = classical[foreuler]).Figure 1
Susceptible population evolutiont,xt.Figure 2
Active infection evolutiont,yt.Figure 3
Number of recoveries and deaths’ evolutiont,zt.Remark 1.
Let us animate the surfaceSeβ/γR=c1 with respect to the parameter a=β/γ (Figure 4).(i)
with(plots);(ii)
animate(plot3d,x∗expa∗z,x=1,…,10,z=1,…,10, a=0.125,…,1).Figure 4
Animate SR first integral.


## 2.3. Parametrization by S

It would be more natural to parameterize the previous general solution by “raw material” S; namely,(5)S=S,I=γβlnc2S−S,R=−γβlnc1S,where c1S<1<c2S and γ/βlnc2/c1=Ptotal. If the social constants β,γ and the state S are so that I>0, then there are infections; otherwise, there are not. So, trying to influence the transmission constants β,γ we can limit the number of infections. In fact, the state I exists, if and only if S1≤S≤S2 and then IS1=IS2=0.If we start with the equilibrium pointsS0=Ptotal,I0=0,andR0=0, we find(6)I=γβlnSS0+S0−S,R=−γβlnSS0,S<S0.The more interesting case holds forS0>γ/β. Then, the ODE for St becomes(7)S˙t=βSt2−βS0St−γStlnStS0,or changing the variable as St=S0ut,(8)u˙t=βS0ut2−ut−γutlnut,ut<1.The solution of this ODE is written in the form(9)t+c=∫duβS0u2−u−γulnu.The denominator of the integrand has three roots0,u0,and1. Using the qualitative approximation(10)t+c=k∫duuu−u0u−1,u0<u<1,we find the implicit solution(11)u1/u01−u1/1−u0u−u01/u01−u0=cet/k.The constantc is determined from the initial condition u0=1−ε. The initial condition u0=1 would give us the equilibrium point ut≡1.Let us observe thatut⟶u0 for t⟶∞. Hence, the final state I=0,S=S1 appears for t⟶∞.We fixβ=1,γ=3,S0=5. From the graphs S,IS and S,FS=S2−5S−3SlnS/5, in Figure 5, we can read the following: when S is decreasing from S0 to S1, the function I is increasing from 0 to a maximum for S=β/γ and then decreasing again to 0. In other words, when the epidemic goes out, that is, I=0 again, and the variable S stops at S1>0, then we talk about an uncontaminated population.Figure 5
IntersectionIS∩FS.Maple simulation (Figure5):(i)
plot([3∗ln1/5∗x+5−x,x2−5∗x−3∗x∗ln1/5∗x], x = 0.. 6, color = [”red”, ”green”]).


## 2.4. Asymptotic Behavior of Infectious Disease Flow

Theorem 1.
For positive initial conditionsS0,I0,R0, the limit value S¯+I¯=limt⟶∞St+It exists and is Ptotal−R¯.Proof.
SupposeSt≤γ/β.
BecauseSt is monotonically decreasing (S˙t=−βStIt≤0) and nonnegative, it has a limit S¯=limt⟶∞St, with 0≤S¯≤γ/β.
SinceIt is monotonically decreasing (I˙t=βSt−γIt≤0) and nonnegative, it has a limit I¯=limt⟶∞It, with I¯≥0.
On the other hand,Rt is monotonically increasing (R˙t=γIt≥0) and nonnegative. But St+It+Rt=Ptotal shows that Rt has a limit R¯≥0. For t large enough, we have St≈S¯,It≈I¯,andRt≈R¯. Then, the differential system consisting of the first equation and third equation converges to the linear system corresponding to the linearization at the point S¯,I¯; that is,(12)S˙=−βI¯St−S¯−βS¯It−I¯,R˙t=γI¯It−I¯.
The matrix of this linear system has the eigenvaluesλ1=−βI¯<0 and λ2=γI¯>0. Consequently, S¯,I¯ render this system unstable, and then the trajectory St,It,Rt diverges. This contradicts St+It+Rt=Ptotal.
SupposeSt>γ/β. Similarly, it follows the existence of S¯,I¯,R¯, satisfying γ/β≤S¯, I¯≥0, and R¯≥0.Theorem 2.
For the infectious disease flow, one hasI¯=limt⟶∞It=0 (see equilibrium point).Proof.
We use the second differential equationdIt/It=d/dtlnIt=βSt−γ. Let S be the primitive of function St. Since ∫0bβSt−γdt=lnIb/I0, we find βSb−S0−γb=lnIb/I0. At limit, when b⟶∞, we must have I¯=I∞=0 (see equilibrium point).Remark 2.
LetS, ℐ, and ℛ be the primitives of functions St, It, and Rt. Then, St+ℐt+ℛt=tPtotal.Theorem 3.
LetS, ℐ, and ℛ be the primitives of functions St, It, and Rt. For positive initial conditions S0,I0,R0, the limit values(13)S¯=limt⟶∞St,ℐ¯=limt⟶∞ℐt,R¯=limt⟶∞Rtare related by(14)−βℐ¯−ℐ0=lnS¯S0,R¯−R0=γℐ¯−ℐ0.Proof.
Let us consider the first differential equation(15)dStSt=ddtlnSt=−βItdt.
It follows−β∫0∞Itdt=lnS¯/S0, and hence −βℐ¯−ℐ0=lnS¯/S0.
Integrating the third differential equation, we find(16)R¯−R0=γℐ¯−ℐ0.
SinceS,I,R represent the evolution of disease infections, the initial conditions S0≥0,I0≥0,R0≥0, and S0+I0+R0=Ptotal are perfectly suited for biological applications. Suppose these initial conditions. The monotony of the state functions St,It,Rt is described by the signum of derivatives: either S˙≤0,I˙>0,R˙≥0 which show that St is decreasing and It and Rt are increasing ∀t>0 or S˙≤0,I˙≤0,R˙≥0 which show that St,It are decreasing and Rt is increasing ∀t>0.


## 2.5. Covering All the Manifold by a Differential Inclusion

As any flow, the infectious disease flow does not cover all the manifoldℝ+3, 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 fieldX=I−βS,βS−γ,γ determines an orthogonal distribution generated by two linearly independent vector fields Y=βS−γ,βS,0 and Z=0,−γ,βS−γ, orthogonal to X.The three vector fieldsX,Y,Z determine the differential inclusion(17)ddtSIR∈X,SpY,Z,which can be used to understand and suitably interpret the spreading of the disease, in the sense that any two points on ℝ+3 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: “findmaxfS,I,R=I subject to Seβ/γR=c1,S+I+R=c2.”The constraints satisfy the condition of nondegenerate constraint qualification.To solve this problem, we attach the Lagrange function(18)F=I+λSeβ/γR−c1+μS+I+R−c2.Since(19)dF=dI+λeβ/γRdS+Sβγeβ/γRdR+μdS+dI+dR,the critical points of the function F are given by the algebraic system(20)∂F∂S=λeβ/γR+μ=0,∂F∂I=1+μ=0,∂F∂R=λSβγeβ/γR+μ=0,Seβ/γR=c1,S+I+R=c2.It follows the critical points(21)S=γβ,R=−γβlnλ,I=c2−γβ−γβlnλ,λ=γβc1,μ=−1.On the other hand,(22)d2F=λ2βγeβ/γRdSdR+Sβγ2eβ/γRdR2.The associated matrix(23)0λβγeβ/γRλβγeβ/γRλSβγ2eβ/γR,with respect to dS,dR, is negative definite. The function F is concave. Hence, all critical points are maximum points. The maximum value of objective function I is(24)Imax=c2−γβ−γβlnλ.The introduction of Lagrange multipliersλ,μ 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 F). Note also that λ,μ have a certain specific meaning: if the solution is regarded as(25)S∗c1,c2,I∗c1,c2,R∗c1,c2,λ∗c1,c2,μ∗c1,c2,then the marginal variations are(26)∂∂c1IS∗c1,c2,I∗c1,c2,R∗c1,c2=λ∗c1,c2,∂∂c2IS∗c1,c2,I∗c1,c2,R∗c1,c2=μ∗c1,c2.Maple Simulations: we denoteS=x,I=y,R=z. 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(x, x∗expz=1,x+y+z=2, assume = nonnegative); 0.158594339562640796,x=HFloat0.1585943395626408,y=HFloat0.0,z=HFloat1.8414056604373585(iii)
Minimize(y, x∗expz=10,x+y+z=50, assume = nonnegative); 39.9999999999999858,x=HFloat10.0,y=HFloat39.999999999999986,z=HFloat0.0(iv)
Minimize(z, x∗expz=10,x+y+z=50, assume = nonnegative); 0.,x=HFloat1.0,y=HFloat0.9999999999999998,z=HFloat0.0(v)
Minimize(y, x∗expz=a,x+y+z=b, assume = nonnegative); 0.,a=HFloat1.1097167915216666,b=HFloat1.2345190270116868,x=HFloat0.5722702360103161,y=HFloat0.0,z=HFloat0.6622487910013707(vi)
Minimize(y, x∗exp10∗z=a,x+y+z=b, assume = nonnegative); 0.,a=HFloat1.4613527630961411,b=HFloat1.458647134440591,x=HFloat1.4584481765262327,y=HFloat0.0,z=HFloat1.9895791435813165e−4(vii)
Minimize(x2+y2+z2, x∗exp10∗z=1,x+y+z=2, assume = nonnegative); 1.99415568473971461,x=HFloat0.9413968463983702y=HFloat1.052564103608946,z=HFloat0.0060390499926835095


### 3.2. Extrema Constrained by Field Lines

To simplify, we use standard notations in mathematicsS=x1,I=x2,R=x3. Then, infectious disease Cauchy problem on ℝ+3 is written(27)x˙1t=−βx1tx2t,x˙2t=βx1t−γx2t,x˙3t=γx2t,x10=x01,x20=x02,x30=x03,x01+x02+x03=Ptotal.To find bounds of significant functions connected to this flow, we use the technique of optimization of an objective function constrained by a field linex=xt,x0,t∈0,∞,x0=x01,x02,x03.Let us start with finding the maximum for the total susceptible populationx1t: findmaxfx1,x2,x3=x1with the restrictionx=xt,x0.Since the componentx1t is a decreasing function, the maximum is reached at the starting point x0.Let us show that we do not have constrained critical points that produce an extremum. We set the critical point condition∇f,X=0. In this case, ∇f=1,0,0 and hence −βx1tx2t=0.We eliminatex2=0, since this condition leads to an equilibrium point of the dynamical system. Also, the solution x1=0 is not convenient. Indeed, a critical point of the form x¯=x¯1=0,x¯2>0,x¯3≥0 cancels the expression HessfX,Xx¯+∇f,DXXx¯ whose sign at the point x¯ would decide the property of extremum: ∇f,DXXx¯≡0, that is, βx1x2−β2x12+βγx1≡0.Sincex¨1t=βx1tx2tβx2t−x1t+γ, the function x1t is convex on the subset βx2t−x1t+γ≥0.Proposition 1.
The maximum of the total susceptible population is reached at the starting pointx0.Let us find now the maximum for the number of active infectionsx2t: determine maxfx1,x2,x3=x2 constrained by x=xt,x0.We set the critical point condition∇f,X=0. In this case, ∇f=0,1,0. It follows the relation βx1t−γx2t=0. The convenient solution (critical point) is x¯=x¯1=γ/β≥0,x¯2>0,x¯3≥0.The sufficient conditionHessfX,Xx¯+∇f,DXXx¯<0 is equivalent to ∇f,DXXx¯<0 or −β2x¯1x¯2+βx¯1−γ2<0.Theorem 4.
Suppose that on an evolution field line there exists a pointx¯ satisfying(28)β2γ2−12<β3γ3x¯2.Then, the number of active infections has an upper bound at this point.To find the minimum for the number of recoveries and deathsx3t, we use the problem: determine minfx1,x2,x3=x3 constrained by x=xt,x0.Since the functionx3t is increasing, the minimum is reached at the starting point x0.Let us show that we do not have constrained critical points that produce an extremum. We set the critical point condition∇f,X=0. In this case, ∇f=0,0,1. It follows the relation γx2t=0. We have no convenient solution (critical point) since x2=0 leads to an equilibrium point of the dynamical system.On the subsetx1≥γ/β, the function x3t is convex. Indeed, we have x¨3t=γβx1t−γx2t.Proposition 2.
The minimum number of recoveries and deaths is reached at the starting pointx0.


## 3.1. Extrema Constrained by Equalities

A basic problem we discuss at the beginning of this paper is as follows: “findmaxfS,I,R=I subject to Seβ/γR=c1,S+I+R=c2.”The constraints satisfy the condition of nondegenerate constraint qualification.To solve this problem, we attach the Lagrange function(18)F=I+λSeβ/γR−c1+μS+I+R−c2.Since(19)dF=dI+λeβ/γRdS+Sβγeβ/γRdR+μdS+dI+dR,the critical points of the function F are given by the algebraic system(20)∂F∂S=λeβ/γR+μ=0,∂F∂I=1+μ=0,∂F∂R=λSβγeβ/γR+μ=0,Seβ/γR=c1,S+I+R=c2.It follows the critical points(21)S=γβ,R=−γβlnλ,I=c2−γβ−γβlnλ,λ=γβc1,μ=−1.On the other hand,(22)d2F=λ2βγeβ/γRdSdR+Sβγ2eβ/γRdR2.The associated matrix(23)0λβγeβ/γRλβγeβ/γRλSβγ2eβ/γR,with respect to dS,dR, is negative definite. The function F is concave. Hence, all critical points are maximum points. The maximum value of objective function I is(24)Imax=c2−γβ−γβlnλ.The introduction of Lagrange multipliersλ,μ 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 F). Note also that λ,μ have a certain specific meaning: if the solution is regarded as(25)S∗c1,c2,I∗c1,c2,R∗c1,c2,λ∗c1,c2,μ∗c1,c2,then the marginal variations are(26)∂∂c1IS∗c1,c2,I∗c1,c2,R∗c1,c2=λ∗c1,c2,∂∂c2IS∗c1,c2,I∗c1,c2,R∗c1,c2=μ∗c1,c2.Maple Simulations: we denoteS=x,I=y,R=z. 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(x, x∗expz=1,x+y+z=2, assume = nonnegative); 0.158594339562640796,x=HFloat0.1585943395626408,y=HFloat0.0,z=HFloat1.8414056604373585(iii)
Minimize(y, x∗expz=10,x+y+z=50, assume = nonnegative); 39.9999999999999858,x=HFloat10.0,y=HFloat39.999999999999986,z=HFloat0.0(iv)
Minimize(z, x∗expz=10,x+y+z=50, assume = nonnegative); 0.,x=HFloat1.0,y=HFloat0.9999999999999998,z=HFloat0.0(v)
Minimize(y, x∗expz=a,x+y+z=b, assume = nonnegative); 0.,a=HFloat1.1097167915216666,b=HFloat1.2345190270116868,x=HFloat0.5722702360103161,y=HFloat0.0,z=HFloat0.6622487910013707(vi)
Minimize(y, x∗exp10∗z=a,x+y+z=b, assume = nonnegative); 0.,a=HFloat1.4613527630961411,b=HFloat1.458647134440591,x=HFloat1.4584481765262327,y=HFloat0.0,z=HFloat1.9895791435813165e−4(vii)
Minimize(x2+y2+z2, x∗exp10∗z=1,x+y+z=2, assume = nonnegative); 1.99415568473971461,x=HFloat0.9413968463983702y=HFloat1.052564103608946,z=HFloat0.0060390499926835095


## 3.2. Extrema Constrained by Field Lines

To simplify, we use standard notations in mathematicsS=x1,I=x2,R=x3. Then, infectious disease Cauchy problem on ℝ+3 is written(27)x˙1t=−βx1tx2t,x˙2t=βx1t−γx2t,x˙3t=γx2t,x10=x01,x20=x02,x30=x03,x01+x02+x03=Ptotal.To find bounds of significant functions connected to this flow, we use the technique of optimization of an objective function constrained by a field linex=xt,x0,t∈0,∞,x0=x01,x02,x03.Let us start with finding the maximum for the total susceptible populationx1t: findmaxfx1,x2,x3=x1with the restrictionx=xt,x0.Since the componentx1t is a decreasing function, the maximum is reached at the starting point x0.Let us show that we do not have constrained critical points that produce an extremum. We set the critical point condition∇f,X=0. In this case, ∇f=1,0,0 and hence −βx1tx2t=0.We eliminatex2=0, since this condition leads to an equilibrium point of the dynamical system. Also, the solution x1=0 is not convenient. Indeed, a critical point of the form x¯=x¯1=0,x¯2>0,x¯3≥0 cancels the expression HessfX,Xx¯+∇f,DXXx¯ whose sign at the point x¯ would decide the property of extremum: ∇f,DXXx¯≡0, that is, βx1x2−β2x12+βγx1≡0.Sincex¨1t=βx1tx2tβx2t−x1t+γ, the function x1t is convex on the subset βx2t−x1t+γ≥0.Proposition 1.
The maximum of the total susceptible population is reached at the starting pointx0.Let us find now the maximum for the number of active infectionsx2t: determine maxfx1,x2,x3=x2 constrained by x=xt,x0.We set the critical point condition∇f,X=0. In this case, ∇f=0,1,0. It follows the relation βx1t−γx2t=0. The convenient solution (critical point) is x¯=x¯1=γ/β≥0,x¯2>0,x¯3≥0.The sufficient conditionHessfX,Xx¯+∇f,DXXx¯<0 is equivalent to ∇f,DXXx¯<0 or −β2x¯1x¯2+βx¯1−γ2<0.Theorem 4.
Suppose that on an evolution field line there exists a pointx¯ satisfying(28)β2γ2−12<β3γ3x¯2.Then, the number of active infections has an upper bound at this point.To find the minimum for the number of recoveries and deathsx3t, we use the problem: determine minfx1,x2,x3=x3 constrained by x=xt,x0.Since the functionx3t is increasing, the minimum is reached at the starting point x0.Let us show that we do not have constrained critical points that produce an extremum. We set the critical point condition∇f,X=0. In this case, ∇f=0,0,1. It follows the relation γx2t=0. We have no convenient solution (critical point) since x2=0 leads to an equilibrium point of the dynamical system.On the subsetx1≥γ/β, the function x3t is convex. Indeed, we have x¨3t=γβx1t−γx2t.Proposition 2.
The minimum number of recoveries and deaths is reached at the starting pointx0.


## 4. Dual Description of Disease Evolution

A vector field determines a flow (collinearity condition) and a Pfaff equation (orthogonality condition) [9]. By duality, the nonlinear ODEs in infectious disease flow are transformed into an infectious disease Pfaff equation(29)ω=−βSIdS+βS−γIdI+γIdR=0.Sinceω∧dω=βγI2dS∧dI∧dR, 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 ℝ+3 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 byI, we obtain an equivalent infectious disease Pfaff equation(30)βSdS−βS−γdI−γdR=0.The normal vector field to this distribution isN=βS,−βS−γ,−γ. Two independent vector fields tangent to the distribution are(31)X=∂∂S+βSγ∂∂R,Y=∂∂I−βS−γγ∂∂R.It follows(32)X,Y=Z=−βγ∂∂R,X,Z=0,Y,Z=0,and hence the vector fields X,Y,Z determine a Carnot group (in a future paper we shall study the infectious disease system like a Carnot group).Proposition 3.
Fort∈0,b, one has(33)βS2b+2γRb≥βS20+2γR0.Proof.
Fort∈0,b, we consider the curve ct=St,It,Rt. Integrating the Pfaff form and selecting It in a convenient way, it follows(34)β2S2b−S20+γRb−R0=∫ctβS−γdI=∫0bβSt−γI˙tdt=∫0bβSt−γ2Itdt≥0.


### 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” (SIR product) of disease states.To find bounds for the functiongS,I,R=lnSIR (logarithm of “volume”), we use the problem “determine maxgS,I,R constrained by the Pfaff equation of evolution.”The critical point condition is(35)η=dSS+dII+dRR+λ−βSIdS+βSI−γIdI+γIdR≡0.It follows(36)λβSI=1S,λβSI−γI=−1I,λγI=−1R,or(37)λ=1S2I=−1βS−γI2=−1γRI.Consequently, the critical point set is described by(38)βS−γI=S2,S2=γR,S+I+R=Ptotal,that is,(39)S=12βI+β2I−4γI,R=βγS−1I,S+I+R=Ptotal,subject to the condition β2I−4γ≥0. This algebraic system has a solution since the curve\(40)S=12βI+β2I−4γI,I=I,R=βγS−1I,is transversal to the plane S+I+R=Ptotal. Indeed, the tangent vector to the curve has the components(41)T:∂S∂I=12β+β2I−2γβ2I−4γI,∂I∂I=1,∂R∂I=βγS−1the normal vector to the plane has the components N=1,1,1, and the scalar product T,N is different from zero.


## 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” (SIR product) of disease states.To find bounds for the functiongS,I,R=lnSIR (logarithm of “volume”), we use the problem “determine maxgS,I,R constrained by the Pfaff equation of evolution.”The critical point condition is(35)η=dSS+dII+dRR+λ−βSIdS+βSI−γIdI+γIdR≡0.It follows(36)λβSI=1S,λβSI−γI=−1I,λγI=−1R,or(37)λ=1S2I=−1βS−γI2=−1γRI.Consequently, the critical point set is described by(38)βS−γI=S2,S2=γR,S+I+R=Ptotal,that is,(39)S=12βI+β2I−4γI,R=βγS−1I,S+I+R=Ptotal,subject to the condition β2I−4γ≥0. This algebraic system has a solution since the curve\(40)S=12βI+β2I−4γI,I=I,R=βγS−1I,is transversal to the plane S+I+R=Ptotal. Indeed, the tangent vector to the curve has the components(41)T:∂S∂I=12β+β2I−2γβ2I−4γI,∂I∂I=1,∂R∂I=βγS−1the normal vector to the plane has the components N=1,1,1, and the scalar product T,N is different from zero.


## 5. Epidemic Wind

The geometric data of the world [9, 10] change the epidemic flow into an epidemic wind. This is a new idea that we are adding to the spread of infections.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 onℝ+3 is(42)S˙t=−βStIt,I˙t=βSt−γIt,R˙t=γIt.On the Riemannian manifoldℝ+3,δij, the flow and the metric determine the least squares autonomous Lagrangian(43)2L=S˙t+βStIt2+I˙t−βSt−γIt2+R˙t−γIt2.We attach an integral action(44)CS⋅,I⋅,R⋅=∫t0t1LSt,It,Rt,S˙t,I˙t,R˙tdt.A geometric dynamics [9, 10] appears, described by the Euler-Lagrange ODEs(45)∂L∂S−ddt∂L∂S˙=0,∂L∂I−ddt∂L∂I˙=0,∂L∂R−ddt∂L∂R˙=0.Explicitly, the epidemic wind is described by the second-order differential system(46)βItS˙t−I˙t+2βSt−γIt2−ddtS˙t+βStIt=0,βStS˙t−I˙t+γI˙t−γR˙t+2β2St2It2−βγStIt+γ2It−ddtI˙t−βStIt+γIt=0,ddtR˙t−γIt=0.Adding all three ODEs, we obtain an ODE whose last term is−S¨t+I¨t+R¨t. Furthermore, the last second-order ODE is equivalent to the first-order ODE(47)R˙t−γIt=c1.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 valuesxt0=x0,x˙t0=Xxt0 are solutions reducible to solutions of the infectious disease flow. The subset of solutions corresponding to the initial values xt0=x0,x˙t0=W≠λXxt0,λ>0, are transversal to the solutions of the infectious disease flow. The converse is also true.
Based on the existence and uniqueness theorem, each solutionx=xt of any second-order prolongation of the first-order ODE system has the property: x˙t0=Xxt0 implies x˙t=Xxt,∀t∈I.
In generic coordinates, the attached Hamiltonian to the wind is(48)H=12x˙−Xx,x˙+Xx=12x˙2−fx,where fx=1/2Xx2 (kinetic energy). The maximal solutions of infectious disease wind are split into three categories of curves: (1) curves characterized by H=0 (flow trajectories); (2) curves satisfying H=const>0; (3) curves characterized by H=const<0. The transversal curves in category (2) can have the images throughout, but the curves in category (3) have the images only in the set fx≥−H.
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 manifoldM,gijx. The best selection of the Riemannian manifold adapted to infectious disease wind is after constant curvature: (1) curvature 0, Euclidean manifold ℝ3,gijx=δij, used in the previous explanations; (2) curvature −1, hyperbolic manifold ℍ3,gijx=δij/x32; (2) curvature 1, sphere S3=x=x1,x2,x3,x4∈ℝ4|x2=1, with the metric induced by the Euclidean metric on ℝ4.


## 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 parameterβ by a control ut. We use Wat,t≥0,a=1,2,3, as independent Wiener processes. Starting from the (nonlinear control system) infectious disease Cauchy problem(49)dSt=−utStItdt,dIt=utSt−γItdt,dRt=γItdt,S0=S0,I0=I0,R0=R0,S0+I0+R0=Ptotal,on ℝ+3, a stochastic perturbation is defined by stochastic differential equation system,(50)dSt=−utStItdt+σa1t,St,It,Rt,utdWat,dIt=utSt−γItdt+σa2t,St,It,Rt,utdWat,dRt=γItdt−σa3t,St,It,Rt,utdWat,S0=S0,I0=I0,R0=R0,S0+I0+R0=Ptotal,where the functions utStIt,utStIt−γIt,γIt are drift coefficients and σait,St,It,Rt,ut,i=1,2,3;a=1,…,d, are diffusion coefficients.IfS0+I0+R0=Ptotal and σa1+σa2=σa3, then the stochastic perturbation satisfies dSt+dIt+dRt=0; that is, we have again a first integral St+It+Rt=Ptotal.Suppose that the controlut is piecewise smooth and has values in a bounded and closed set U=δ,m⊂0,∞, where m 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 U, 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 solutionct=St,It,Rt of this stochastic differential system with the initial condition S0,I0,R0 is an adapted continuous process, such that, for all t≥0, it satisfies the stochastic integral system(51)St−S0=−∫0tusSsIsds+∫0tσa1s,Ss,Is,Rs,usdWas,It−I0=∫0tusSs−γIsds+∫0tσa2s,Ss,Is,Rs,usdWas,Rt−R0=∫0tγIsds−∫0tσa3s,Ss,Is,Rs,usdWas,a.s.
However, there is a number of subtle points involved: first, the existence of the integrals requires some degree of regularity onct=St,It,Rt and on the functions ut, σ=σai (matrix); in particular, it must be the case that, for all t≥0, with probability one, ∫0tσs2ds<∞.
Second, the solution is required to exist for allt<∞ with probability one.
Properties of Itô integral are that, for allG,H∈L20,T, we have(52)E∫0TGdW=0,E∫0TGdW2=E∫0TG2dt,E∫0TGdW∫0THdW=E∫0TGHdt.Definition 2.
A weak solutionct=St,It,Rt of the SDE with the initial condition S0,I0,R0 is a continuous stochastic process defined on some probability space Ω,F,P, such that, for some Wiener processes Wat and some admissible filtration F, the process ct 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 functionsσ, the SDE has strong solutions and that, for each initial value c0=S0,I0,R0, the solution is unique.Without loss of generality, we consider that the amplitudes of error are constants; that is,σait,St,It,Rt,ut=σai.Definition 3.
Letut=uSt,It,Rt be a feedback bounded control. A stochastic process ct=St,It,Rt, which satisfies the SDE system, is called an admissible stochastic process.Theorem 6.
LetP=SP,IP,RP and Q=SQ,IQ,RQ be two points in the set 0,13⊂ℝ+3. Denote DQ,r as the Euclidean sphere of radius r, centered at Q. Then, for any ɛ>0 and r>0, there exists a striking time t<∞ and an admissible stochastic process ct, satisfying the boundary conditions(53)S0,I0,R0=SP,IP,RP,ESt,EIt,ERt=SQ,IQ,RQ,such that(54)ℙct∈DQ,r≥1−ɛ.Proof.
We start with the nonnegative random variablect−Q2. Markov inequality shows that(55)ℙct−Q2≥r2≤1r2Ect−Q2,or, equivalently,(56)ℙct−Q2≤r2≥1−1r2Ect−Q2.
The stochastic integral variant of the previous SDE system is(57)St=−∫0tusSsIsds+σa1Wat+SP,It=∫0tusSs−γIsds+σa2Wat+IP,Rt=∫0tγIsds−σa3Wat+RP.
It is well-known that the expectationE has the properties(58)EWat=0,EWat2=tEWatWbt=0,a≠b.
If two random variablesX,Y:Ω⟶ℝ are independent, then EXY=EXEY, provided that EX<∞ and EY<∞.
The boundary conditions produce(59)ESt=SQ⇒∫0tE−usSsIsds+SP=SQ,EIt=IQ⇒∫0tEusSs−γIsds+IP=IQ,ERt=RQ⇒∫0tEγIsds+RP=RQ.
Moreover,SP+IP+RP=1 implies SQ+IQ+RQ=1.
We use the advantages of Itô integral and the independence of Wiener processesWat. We find(60)Ect−Q2=ESt−SQ2+EIt−IQ2+ERt−RQ2=E−∫0tusSsIsds+σa1Wat+SP−SQ2+E∫0tusSs−γIsds+σa2Wat+IP−IQ2+E∫0tγIsds−σa3Wat+RP−RQ2=E∫0tusSsIsds∫0tusSsIsusds+σa1Wat2+SP−SQ2−2∫0tusSsIsdsσa1Wat−2∫0tusSsIsdsSP−SQ+2σa1WatSP−SQ+E∫0tusSs−γIsds∫0tusSs−γIsds+σa2Wat2+IP−IQ2+2∫0tusSs−γIsdsσa2Wat+2∫0tusSs−γIsdsIP−IQ+2σa2WatIP−IQ+E∫0tγIsds∫0tγIsds+σa3Wat2+RP−RQ2−2∫0tγIsdsσa3Wat−2∫0tγIsdsRP−RQ+2σa3WatRP−RQ.
This last expression is reduced to(61)E∫0tusSsIsds∫0tusSsIsds+t∑aσa1σa1−SP−SQ2+E∫0tusSs−γIsds∫0tusSs−γIsds+t∑aσa2σa2−IP−IQ2+E∫0tγIsds∫0tγIsds+t∑aσa3σa3−RP−RQ2=E∫0tusSsIsds2+E∫0tusSs−γIsds2+E∫0tγIsds2+tδij∑aσaiσaj−P−Q2.
On the other hand, the Cauchy-Schwarz inequalities(62)∫0tusSsIsds2≤t∫0tusSsIs2ds,∫0tusSs−γIsds2≤t∫0tusSs−γ2I2sds,∫0tγIsds2≤t∫0tγIs2ds,the properties of expectation, and the condition ut≤m produce the estimations(63)E∫0tusSsIsds2≤m2SQ2IQ2t2,E∫0tusSs−γIsds2≤mSQ−γ2IQ2t2,E∫0tγIsds2≤γ2IQ2t2.
Denoting(64)k12=m2SQ2IQ2,k22=mSQ−γ2IQ2,k32=γ2IQ2,σ2=δij∑aσaiσaj,we find(65)Ect−Q2≤k12+k22+k32t2+σ2t−P−Q2.
For anyϵ>0 and r>0, the equation(66)k12+k22+k32t2+σ2t−P−Q2−ɛr2=0,has a strictly positive solution(67)t+=−σ2+σ4+4k12+k22+k32Λ22k12+k22+k32,where Λ2=P−Q2+ɛr2.
Denoteσ=c, k12=m2a2, and k22=mS−g2b2. Using Maple routines, we find t+e=extremamt+:(i)
extrema−c2+sqrtc4+4∗m2∗a2+S∗m−g2∗b2+k32∗A/2∗m2∗a2+S∗m−g2∗b2+k32,,m
Answer:(68)12−c2+S2b2c4+4AS2b2k32+4Aa2b2g2+a2c4+4Aa2k32/S2b2+a2S2b2+a2S2b2k32+a2b2g2+a2k32.
For this optimal solutiont+e, we can write(69)Ect+e−Q2≤ɛr2⇔1−1r2Ect+e−Q2≥1−ɛ,that is, ℙct+e∈DQ,r≥1−ɛ.The previous theorem confirms that almost sure any two points in the “world” are joined by a stochastic curve. So, contamination can get anywhere from anywhere.


## 7. Conclusions

The most difficult thing for an epidemiological model is to predict human behavior. Infected population size estimation is a common problem in any epidemic analysis and it is the most important aspect for planning appropriate care and prevention policies. We underline that a model, even complex, is a simplification of reality, since a lot of other variables are still not completely known. An interactive approach with Maple can clarify some of these issues.The flow models and even differential inclusion models are varied, often confusing to interpret, and are not crystal balls, especially because the ideal data are not yet available. But they are a large part of what government leaders use to make decisions, influencing how resources are allocated to health care facilities and how social distancing orders are issued to the public.The three points of view in this paper, infectious disease flow, infectious disease wind, and stochastic connectivity, can input the biology of the virus and give positive answers to the following questions: how does it spread, how quickly does it infect, how quickly does it lead to symptoms, and how quickly does it replicate to a level where it can jump from person to person?The infectious disease wind is determined by the flow and by the geometry of space (created by nature or by human activities, conscious or not). All (the flow, the geometry, and the least squares technique) generate gyroscopic forces that lead to spiraling (uncontrolled) evolutions. The Riemannian structure can generate also chaos in geometric dynamics (sensitive dependence on initial conditions, and three or more dimensions). The experts in the fields of surveillance, epidemiology, and prevention of communicable disease spread, infection preventionists, must understand that infectious diseases modeled as wind and stochastic modeling of geometric structures are responsible for reading the infectious disease disasters.



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*Source: 6395410-2021-06-30.xml*