key: cord-322862-dcb237an
authors: Bekiros, Stelios; Kouloumpou, Dimitra
title: SBDiEM: A new Mathematical model of Infectious Disease Dynamics
date: 2020-04-23
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.109828
sha: 
doc_id: 322862
cord_uid: dcb237an

A worldwide multi-scale interplay among a plethora of factors, ranging from micro-pathogens and individual or population interactions to macro-scale environmental, socio-economic and demographic conditions, entails the development of highly sophisticated mathematical models for robust representation of the contagious disease dynamics that would lead to the improvement of current outbreak control strategies and vaccination and prevention policies. Due to the complexity of the underlying interactions, both deterministic and stochastic epidemiological models are built upon incomplete information regarding the infectious network. Hence, rigorous mathematical epidemiology models can be utilized to combat epidemic outbreaks. We introduce a new spatiotemporal approach (SBDiEM) for modeling, forecasting and nowcasting infectious dynamics, particularly in light of recent efforts to establish a global surveillance network for combating pandemics with the use of artificial intelligence. This model can be adjusted to describe past outbreaks as well as COVID-19. Our novel methodology may have important implications for national health systems, international stakeholders and policy makers.

The World Health Organization (WHO) reported on December 31, 2019 cases of pneumonia of undetected etiology in the city of Wuhan, Hubei Province in China. A novel coronavirus (CoViD- 19) was identi…ed as the source of the disease by the Chinese authorities on January 7, 2020. Eventually, the International Committee on Taxonomy of Viruses on 11 February, 2020 named the Severe Acute Respiratory Syndrome Coronavirus as SARS-CoV-2 [1] . Concerns on public health were dispersed on a global scale about potentially infected countries. The virus might have been generated by animal populations and transmitted via the Huanan wholesale market [2] [3] [4] albeit not proven, while clinical …ndings demonstrated that international spread was caused mainly by commercial air travel [4] [5] [6] [7] . The WHO declared SARS-CoV-2 a pandemic on March 11, 2020 . Throughout the globe, huge e¤orts were in progress to limit the spread of the virus and …nd medications and vaccines. However, the scienti…c community could not fully comprehend the dynamics of the spread [8] [9] [10] .

Several outbreaks of infectious diseases have occurred in the past with immense impact on public health. For instance, the Severe Acute Respiratory Syndrome (SARS) occurred in 2003, the swine ‡u in 2009 and the Middle East Respiratory Syndrome Coronavirus (MERS) in Saudi Arabia in 2012, which still survives at a sub-critical level causing some peaks [11] [12] [13] [14] . Additionally, the Ebola epidemic emerged between 2014 and 2016 and caused over 28,000 cases in West Africa [15] . Its temporal decline coincided with the outbreak of Zika virus in Brazil [16] . Consequently, the outbreak of severe pathogens such as the aforementioned, require global interdisciplinary e¤orts in order to decode key epidemiological features and their transmission dynamics, and develop possible control policies.

Insights from mathematical modelling can be extremely bene…cial. Indeed, dealing with infectious diseases from a mathematical angle could reveal inherent patterns and underlying structures that govern outbreaks. Stakeholders utilize available data from current and previous outbreaks in order to forecast infection rates, identify how to restrict the spread of diseases, and eventually introduce vaccination policies that will be most e¤ective. Epidemiology is essentially a biology discipline concerned with public health and as such, it can be heavily in ‡uenced by mathematical theory. Most phenomena observed at population level are often very complex and di¢ cult to decode just by observing the characteristics of isolated individuals [17] . Statistical analyses of epidemiological data help to characterize, quantify and summarize the way diseases spread in host populations. Interestingly, mathematical models appear as e¢ cient ways to explore and test various epidemiological hypotheses, mostly due to the existence of ethical and practical limitations when deducting experiments on living populations. Models provide conceptual results on e.g., the basic reproduction number, threshold e¤ects or herd immunity. One additional element of epidemiological modeling is the link with data via statistical methods. Although simple epidemiological models are often used, viral and bacterial infections commonly require increased complexity. There are many models in the literature on single epidemics, endemic diseases and spatiotemporal disease dynamics. The aim is to develop robust public health policies in de…ning optimal vaccination strategies.

Our study presents for the …rst time a new stochastic mathematical model for describing infectious dynamics and tracking virus temporal transmissibility on 3-dimensional space (earth). This model can be adjusted to describe all past outbreaks as well as CoViD- 19 . As a matter of fact, it introduces a novel approach to mathematical modelling of infectious dynamics of any disease, and sets a starting point for conducting simulations, forecasting and nowcasting investigations based on real-world stereographic and spherical tracking on earth.

In short, a single epidemic outbreak as opposed to disease endemicity occurs in a time span short enough not to have the demographic changes perturbing the dynamics of contacts among individuals. The most popular mathematical model in this category is the Susceptible-Infected-Recovered (SIR) epidemic model, in which all individuals of a …nite population interact in the same manner. Individuals at time t are susceptible (S), infected (I) or recovered (R). The …nal size of the epidemic will strongly depend upon the initial conditions of the number of susceptible and infected individuals as well as the infection parameter. The …nal size distribution of the simple SIR model in most cases is bimodal presenting two local maxima. This bimodal feature is caused by two likely scenarios; either the epidemic dies out quickly infecting few individuals, or it becomes long-lasting and substantial. However, stochasticity in the form of random walk transmission mechanisms related to spreading processes has never been explored in epidemiology widely [18] [19] [20] . For example, in computer science, some arti…cially created viruses propagate randomly by a plethora of online communication channels. To the best of our knowledge, we are the …rst to scrutinize extensively the role of random walks in epidemic spreading and provide the proper mathematical arsenal to model it robustly. Interestingly, random walk paths converge in distribution to Brownian motions [21] . In this work, we assume that a biological carrier of virus Y is at position X(t) at any given time t. We call this the inaugural contamination focal point on earth.

The path de…ned by its motion is considered infectious. X t ; t 0 is supposed to follow a Brownian motion on a 2-dimensional sphere S 2 of radius a, i.e the sphere in R of dimension 3. We consider this a proxy for earth, spreading via spherical and stereographic coordinates. Next, using the Laplace-Beltrami operator we construct the Brownian motion infectious process on the 2-dimensional sphere, using spherical and stereographic coordinates as local coordinates. We evaluate explicitly certain quantities related to generated di¤usion processes. In what follows, we compute the transition and transmission density for the X t ; t 0, and we derive the stochastic di¤erential equations that govern the infectious disease dynamics for X t ; t 0 in those local coordinates. We continue with the calculation of expectations of outbreak exit times in time and space of speci…c domains, possessing certain symmetries. Moreover, the moment generating functions are produced. In mathematical terms, we derive the stochastic re ‡ection principle on S 2 for the infectious disease transmission process. Re ‡ection points can be extremely useful to calculate the distribution functions of certain temporal quantities related to the dynamics. Additionally, we evaluate boundary local times of …rst hitting of the outbreak for an epidemic or a hybrid endemicepidemic model. Hence, biological carrier(s) of a virus (infectious individuals) are tracked at any given time on earth coordinates, and the path(s) de…ned by each infectious dynamical motion. In the following two chapters we present a thorough literature review and a state-of-the-art analysis in order to pose clearly our novel approach optimally among the various methodologies followed thus far.

The rest of paper is organized as follows: section 2 provides a brief literature review, past and recent, of mathematical epidemiology. Section 3 presents the state-of-the-art, and focal concepts and term de…nitions required to introduce our novel model. It also states which category the new model falls into, according to the o¢ cial taxonomy of the various methodologies already utilized so far in the relevant literature. Next, section 4 exposes in detail the mathematical formulation of the model. Lastly, section 5 discusses proposed policies and future paths of research, and concludes.

The beginning of mathematical modeling in epidemiology dates back to 1766, when Bernoulli developed a mathematical model to analyze the mortality of smallpox in England [22] . Bernoulli used his model to show that inoculation against the virus would increase the life expectancy at birth by about three years. A revision of the main …ndings and a presentation of the criticism by D'Alembert, appears recently in Dietz and Heesterbeek [23] . Lambert in 1772 as well as Laplace in 1812 extended the Bernoulli model by incorporating age-dependent parameters [24, 25] . However, further systematic research was absent until the beginning of the twentieth century with the pioneering work of Ross in 1911, which is considered the inaugural study of modern mathematical epidemiology [26] . Ross used a set of equations to approximate the discrete-time dynamics of malaria via a mosquito-based pathogen transmission [27] . Importantly, the past century has witnessed the rapid emergence and development of substantial theories in epidemics. In 1927, Kermack and McKendrick [28] derived the celebrated threshold theorem, which is one of the key results in epidemiology. It predicts -depending on the transmission potential of the infectionthe critical fraction of susceptibles in the population that must be exceeded if an epidemic is to occur. Kermack and McKendrick published three seminal papers, establishing what is called the deterministic compartmental epidemic modelling [29] [30] [31] , wherein they addressed the mass-action incident in disease transmission cycles, assuming that the probability of infection of a susceptible is analogous to the number of its contacts with infected individuals. This deterministic representation was in line with the Law of Mass Action [32] introduced by Guldberg and Waage in 1864 and renders the basic most commonly used SIR model, which assumes homogeneous mixing of the contacts and conservation of the total population and low rates of interaction. MacDonald extended Ross's model to explain in depth the transmission process of malaria. Utilizing modern computer power, the mathematical model for the dynamics and the control of mosquito-transmitted pathogens provided robust results in real-word applications. Overall, the family of models they introduced is known by now as Ross-MacDonald models [33] . Moreover, the classic work of Bartlett [34] examined models and data to explore the factors that determine disease persistence in large populations. Arguably, a landmark book on mathematical modelling of epidemiological systems was published by Bailey [35] and highlighted the importance of public health decision making [36] . Given the diversity of infectious diseases studied since the middle of the 1950s, an impressive variety of epidemiological models have been developed. In addition, we should highlight the 19th century works by Enko [37] [38] [39] , who …rst published a probabilistic model for describing the epidemic of measles, yet in discrete time. This model is the precursor of the popular Reed-Frost chain binomial model introduced by Frost in 1928 in biostatistics'lectures at Johns Hopkins University [40] . It assumes that the infection spreads from an infected to a susceptible individual via a discrete time Markov chain, and set the basis of contemporary stochastic epidemic modelling, on which we will also focus in our present work.

Moving to the 21st century, we mention some interesting works; Xing et al., [41] introduced a mathematical model on H7N9 in ‡uenza among migrant and resident birds, domestic poultry and humans in China. In this study they concluded that temperature seasonality might be a source of the disease, yet they suggested for the …rst time that controlling markets could help controlling outbreaks. Lee and Pietz [42] developed a mathematical model for Zika virus using logistic growth in human populations. Sun et al., [43] proposed a transmission model for cholera in China and observed that reducing the spread requires extensive immunization coverage of the population. Nishiura et al. [44] developed a Zika mathematical model which exhibited the same dynamics as dengue fever, and Khan et al. [45] introduced a model whereby a saturation function describes well the typhoid fever dynamics. Gui and Zhang [46] , developed a modi…ed SIR model demonstrating nonlinearities in recovery rates. Their model exhibited a backward bifurcation phenomenon, which in turn implied that a plain reduction of the reproduction number less than one, was not rendered su¢ cient to stop the disease spread. Li et al. [47] constructed a multi-group brucellosis model and found out that the best way to contain the disease is to avoid cross infection of animal populations. Moreover, Yu and Lin [48] identi…ed complex dynamical behaviour in epidemiological models and particularly the existence of multiple limit cycle bifurcations using a predictor-prey model. Shi et al. [49] proposed an HIV model with a saturated reverse function to describe the dynamics of infected cells. Additionally, Bonyah et al. [50] developed a SIR model to study the dynamics of buruli ulcer and suggested policy measures to control the disease. Lastly, Zhang et al. [51] developed a model with a latent period of the disease wherein the person is not infectious with saturated incidence rates and treatment functions, called SEIR epidemic model.

The SIR model is the basic one used for modelling epidemics. Kermack and McKendrick created the model in 1927 [29] in which they considered a …xed population with only three compartments, susceptible (S), infected (I) and recovered (R). There are a large number of modi…cations of the SIR model, including those that include births and deaths, the SIR without or with vital dynamics, a model where upon recovery there is no immunity called SIS and where immunity lasts for a short period of time, called SIRS model. Furthermore, a model where there is a latent period of the disease and where the person is not infectious is indenti…ed as SEIS and SEIR respectively, or where infants can be born with immunity is named MSIR. Also, we mention the herd immunity model [52, 53] .

Overall, the transmission mechanism from infective populations to susceptibles is not well-comprehended for many infectious diseases. Interactions in a population are very complex, hence it is extremely di¢ cult to capture the large scale dynamics of disease spread without formal mathematical modeling. An epidemiological model uses microscopic e¤ects -the role of an infectious individual -to forecast the macroscopic behavior of disease spread via a population.

Deterministic models do not incorporate any form of uncertainty and as such, they can be thought to account for the mean trend of a process, alone. On the other hand, stochastic models describe the mean trend as well as the variance structure of the underlying processes. Two basic types of stochasticity are commonly used: demographic and environmental. Within the context of demographic stochasticity, all individuals are subject to the same potential events with the exact same probabilities but di¤erences in the fates of population individuals. Disease propagation in large populations obeys to the weak law of large numbers, thus e¤ects of demographic stochasticity can be decreased signi…cantly, and many times a deterministic model becomes more suitable. However, random events cannot be neglected and a stochastic model can be equally appropriate. Environmental stochasticity involves variations in the probability associated with an exogenous event. Model parameters of stochastic models are characterized by probability distributions, whilst for …xed parameter values deterministic models will always produce the same results, except when chaotic behaviour emerges.

In the classic SIR model it is assumed that the individuals leave the infectious class at a constant rate and even if this assumption seems most intuitive, it is not always the most realistic, regarding the duration individuals stay infective [54] [55] [56] . Usually, random variables describe the time of recovery since infection. For discrete random variables (e.g., number of individuals) it is easy to de…ne a probability distribution, whilst for continuous variables the time of recovery since infection is modelled. Often, in this last category it is not possible to …x a probability as there is in…nity of such times. Hence, we …rst de…ne a cumulative distribution and then express a probability density function from this cumulative distribution. Infectious periods are exponentially distributed with a mean infectious duration, however as frequently real data does not back up this assumption, we rather use constant duration. To account for such more realistic distributions, the assumption that the probability of recovery does not depend on the time since infection, is often relaxed. Then, a common method of stages can be used to replace the infective compartment by a series of successive ones, each with an exponential distribution of the same parameter, leading to a total duration of the infectious period modelled by a gamma distribution [17] .

Epidemic models presented above describe rapid outbreaks during which normally the host population is assumed to be in a constant state. For longer periods, deaths and births feed the population with new susceptibles, possibly allowing the disease to persist at a constant prevalence. This state renders an endemic state in the population [17] . In this case, we account for birth and death rate of the host population, whereby a good approximation is that the population size N=S+I+R is constant. When deterministic dynamics prevail a threshold on the value of the basic reproduction number exists. Conditions regarding this number guarantee the disease persistence, but in epidemic models such persistence can be dependent upon the magnitude of the stochastic ‡uctuations around the steady-state equilibrium. Furthermore, many times diseases are in an endemo-epidemic state. As endemic models exhibit damped oscillations which converge toward an endemic equilibrium, this equilibrium can be weakly stable with perturbations (intrinsic or extrinsic), which excite and sustain the inherent oscillation behaviour [57] . This behaviour is due to heterogeneity that is added temporally to the coe¢ cient of transmission, spatially in the context of meta-populations, or by cohorts for age-structured models. Lastly, heterogeneity can be added statistically in case of stochastic versions. For example, a stochastic version of the endemic SIR model can utilize a Markov process, in which the future is independent of the past given the present, with a state space de…ned by the number of individuals in each of the three classes, and changes in the state space characterized by probabilistic transition events. And as future events are independent on past events, the time to the next event follows a negative exponential distribution.

Over the years, a vast number of mathematical modeling approaches has been proposed, tackling the problem from di¤erent perspectives. The prevailing taxonomy proposed by Siettos and Russo (2013) [58] encompasses three general categories: (1) statistical methods of outbreaks and their identi…cation of spatial patterns in real epidemics, (2) state-space models of the evolution of a "hypothetical"or on-going epidemic spread, and (3) machine learning methods, all utilized also for predictability purposes vis-à-vis an ongoing epidemic. In particular, the …rst category includes i) regression methods [59] [60] [61] [62] [63] [64] , ii) times series analysis, namely ARIMA and seasonal ARIMA approaches [65] [66] [67] [68] , iii) process control methods including cumulative sum (CUSUM) charts [69] [70] [71] [72] [73] [74] and exponentially weighted moving average (EWMA) methods [75, 76] , as well as iv) Hidden Markov models (HMM) [77, 78] . The second category incorporates i) "continuum"models in the form of di¤erential and/or (integro)-partial di¤erential equations [79] [80] [81] [82] , ii) discrete and continuous-time Markov-chain models [83] [84] [85] , iii) complex network models which relax the hypotheses of the previous stochastic models that interactions among individuals are instantaneous and homogeneous [86] [87] [88] [89] [90] [91] , and iv) Agent-based models [92] [93] [94] [95] . Lastly, the third category includes well-known machine learning approaches widely used in computer science, such as i) arti…cial neural networks [96] , ii) web-based data mining [97, 98] and iii) surveillance networks [99] , to name a few.

For the …rst time in the relevant literature, we introduce a new stochastic model laying in the intersection of categories (1) and (2), called "Stereographic Brownian Di¤usion Epidemiology Model (SBDiEM)". Figure 1 presents a graphical overview of the models utilized so far, and the "positioning" of our novel approach for modelling infectious diseases.

Insert Figure 1 here Let n 2 N = f1; 2; 3; : : :g. The n-dimensional sphere S n with center (c 1 ; :::; c n+1 ) and radius a > 0 is (de…ned to be) the set of all points x = (x 1 ; x 2 ; ::

De…nition 4.2. We consider R n R n+1 to be the hyperplane given by x n+1 = 0. For convenience, we will let (x 1 ; x 2 ; :::; x n ; x n+1 ) be coordinates on R n+1 and ( 1 ; 2 ; :::

The stereographic projection coordinates of S n is the map : S n f0; 0; : : : ; 2ag ! R n given by

This map de…nes coordinates ( 1 ; 2 ; :::; n ) on S n so that the point (x 1 ; x 2 ; :::; x n ; x n+1 ) of S n has coordinates ( 1 ; 2 ; :::; n ) ; where

The inverse map is given by

The points of the 2-sphere with center at the origin and radius a may also be described in spherical coordinates in the following way:

x 1 = a cos sin '

where 0 < 2 and 0 ' : 

;

3. The family fU ; x g is maximal relative to conditions 1 and 2.

Each pair (x ; U ) is called a coordinate chart on M: (For more details see [101] )

Let g = [g ij ] be the Riemmanian metric tensor on a Riemmanian manifold M . This means that, in any coordinate chart (x 1 ; x 2 ; :::; x n ) on M , the length element can be computed by

Given local coordinates (x 1 ; : : : ; x n ); we can easily compute the matrix g = [g ij ] by the inner product

(see [101] ). We denote by g ij the elements of the inverse matrix g 1 .

De…nition 4.5. The Laplace-Beltrami operator M associated with the metric g is de…ned by

where f is a C r function on M:

In this work we are interested in the case where M = S 2 , i.e., the 2-dimensional sphere. We will denote the corresponding Laplace-Beltrami operator of S 2 by 2 or just using the spherical coordinates.

we have x = @x @ = ( a sin sin '; a cos sin '; 0)

x ' = @x @' = (a cos cos '; a sin cos '; a sin ')

Hence the Laplace-Beltrami operator of a smooth function f on S 2 is

where x 1 = and x 2 = ': Thus

In case where the function f is independent of the Laplace-Beltrami operator of f is

Generally the Laplace-Beltrami operator of a smooth function f on S n is we have x k = @x @ k = = 8a 2 1 k ( 2 1 + + 2 n + 4a 2 ) 2 ; : : : ;

; g ij = 0; if i 6 = j; and p det(g) = (4a 2 ) n 2 1 + + 2 n + 4a 2 n : Therefore, the Laplace Beltrami operator of a smooth function f on S 2 , using Stereographic projection coordinates is

De…nition 4.6. Let M be a Riemannian manifold (see de…nition 1.5) and its corresponding Laplace-Beltrami operator. Any function P (t; x; y) on (0; 1) M M satisfying the di¤ erential equation @P @t

where x is acting on the x-variables and the initial condition P (t; x; y) ! x (y) as; t ! 0 + (4.10)

(where x (y) is the delta mass at x 2 M ) is called a fundamental solution of the heat equation (4.9) on M .

The smallest positive fundamental solution of the heat equation (4.9) and (4.10) is the heat kernel on M . It has been proved by J. Dodziak [102] , that the heat kernel always exists, and is smooth in (t; x; y). Moreover the heat kernel possesses the following properties. De…nition 4.7. A process X t ; t 0 is a Markov process if for any t; s 0, the conditional distribution of X t+s , given the information about the process up to time t, is the same as the conditional distribution of X t+s , given X t .

De…nition 4.8. The Brownian motion X t , t 0, on a Riemannian manifold M is a Markov process with transition density function P (t; x; y) the heat kernel associated with the Laplace-Beltrami operator.

Remark 4.1. In the case where M = S n , n 2, the transition density function P (t; x; y) of the Brownian motion X t depends only on t and d(x; y), the distance between x and y. Thus in spherical coordinates it depends on t and the angle ' between x and y. Hence, the transition density function of the Brownian motion can be written as P (t; x; y) = p(t; ');

(4.13)

where p(t; ') is the solution of

and lim t!0 + aA n 1 p(t; ') sin n 1 (') = ('):

Here ( ) is the Dirac delta function on R and A n denotes the area of the n-dimensional sphere S n with radius a. It is well known that [103] A n = 2 n+1 2 a n ( n+1 2 )

;

where ( ) is the Gamma function. More precisely A n = 2 n+1 2 a n ( n 1 2 )! for n odd (4.17)

A n = 2 n ( n 2 1)! n 2 a n (n 1)! for n even (4.18)

Remark 4.2. The fact that S n is a compact and smooth manifold implies that (4.14) -(4.15) has a unique positive solution which also satis…es Z S n P (t; x; y)d (y) = 1:

Furthermore, as t ! 1, P (t; x; y) approaches the uniform density on S n , i.e. P (t; x; y) ! c; where c = 1 A n :

In the sequel for typographical convenience we will write X t instead of fX t g t 0.

In this section we shall represent the transition density function p(t; ') of the position X(t) of a biological carrier (infected individual) of virus Y at any given time t. For the next sections we suppose that the infected individual is at position X(t) at any given time t, namely the path de…ned by its motion is considered infectious. X t ; t 0 describes a Brownian motion on a 2-dimensional sphere S 2 of radius a. The solution of the di¤usion equation

is given by the function

see [104] . Here P 0 n ; n = 0; 1; 2; : : : is the associated Legendre polynomials of order zero, i.e. Proof. First we prove that p(t; ') satis…es the di¤erential equation

We have that

where K(t; ') is given by the (4.24), therefore

However from the (4.22) @K @t

i.e. @p(t; ') @t = 1 2a 2 sin ' cos ' @p(t; ') @' + sin ' @ 2 p(t; ') @' 2 : 

We recall the following well-known fact be such that a(x) = (x) T (x) is positive de…nite. If X t is the Ito di¤ usion process

(4.28)

then, its generator A is given by the formula

Conversely, the operator A given above is the generator of di¤usion (4.28). For the proof see [105] .

The generator of Brownian motion on S 2 in spherical coordinates is

Therefore, the Brownian motion on S 2 in spherical coordinates is the solution of the stochastic di¤erential equation

where X t = ( (t); '(t)) :

Case of Sterographic Projection Coordinates Expressed in stereographic projection coordinates, the generator of Brownian motion on S 2 is

Hence, the Brownian motion on S 2 in stereographic projection coordinates is the solution of the stochastic di¤erential equation

where X t = (x 1 (t); x 2 (t)) :

We recall some basic de…nitions.

De…nition 4.9. A measurable space f ; Fg is said to be equipped with a …ltration {F t }, t 2 [0; +1), if for every t 0 {F t } is a -algebra of subsets of such that F t F and for every t 1 ; t 2 2 [0; +1) such that t 1 < t 2 , we have that F t1 F t2 . (i.e. {F t } is an increasing family of sub -algebras of F). Let X t be the Brownian motion in S n and D S n a domain. Then

is a stopping time with respect to F t = f X s j 0 s tg, called the exit time on @D . and

then the expectation of T is given by

Proof. Based on [106] ,

we have the unique solution of the di¤erential equation Thus

Therefore,

However (see [105] )

Consequently,

Finally, and

then the expectation of T is given by

Proof. According to [106] , E ' [t] satis…es the Poisson equation on D with Dirichlet boundary data. By uniqueness

is the unique solution of the di¤erential equation (4.31), i.e., Hence from (4.34)

However u( ; ' 1 ) = u( ; ' 2 ) = 0;

i.e.

Thus

1 sin x dx and c 2 = 0:

Consequently,

namely,

Proposition 4.4. We consider the 2-dimensional sphere S 2 of radius a: Let two circles pass through the North pole, such that in stereographic coordinates are represented by the parallel lines 2 = b and 2 = c; where b; c 2 R; say b < c: We consider the set D in S 2 ; whose stereographic projection is

If X t is the position of the carrier of virus Y at a given time t starting at the point A; where the stereogrpaphic projection coordinates of A are

then,

and g( ; t) = 2a 2 ln (c b) 2 ln 2 j j 2 + t 2 + 4a 2 (4.41)

As we have seen the function

satis…es the di¤erential equation

Here, 2 is the Laplace-Beltrami operator on S 2 expressed in stereographic projection coordinates. Hence, the di¤erential equation takes the form

(4.42)

However the function

satis…es the di¤erential equation (4.42) .

with boundary conditions

If we take the transformation of variables x = 1 and y = 2 b and set the function (x; y) = f ( 1 ; 2 ); then (x; y) we satisfy @ 2 @x 2 + @ 2 @y 2 = 0;

with boundary conditions (x; 0) = 2a 2 ln(

and

where = c b: Now let z = x + yi and w = exp z ; i.e. z = ln w : Thus, if w = u + vi; u; v 2 R then u = exp

x cos y and v = exp x sin y : (4.43)

Introducing the function (u; v) = (x; y): It follows that (u; v) satis…es

with boundary conditions This is the standard Dirichlet boundary value problem for the half line, and it is well known that (see e.g. [107] ) its solution is given by the Poisson integral formula for the half-plane:

where g( ; t) = 2a 2 ln 2 ln 2 j j 2 + t 2 + 4a 2 :

Notice that g( ; t) = g( ; t): Hence,

where u; v are given in (4.43). Therefore

exp 2 x sin 2 y + exp x cos y 2 d ;

i.e. and in case T 1 = inf ft 0 j X t = 2 D 1 g ;

then the probabilities P r A fT = T 1 g and P r A fT = T 2 g are given by

and

:

Proof. It is known that (see [21] ), u( ; ') = P r A fT = T 1 g is the unique solution of the di¤erential equation In we set f (') = du d' ;

hence from (4.47)

However, u(' 1 ) = 1 and u(' 2 ) = 0;

hence

: Of course, @D 1 = f ( 1 ; 2 ) j 1 2 R and 2 = bg and @D 2 = f ( 1 ; 2 ) j 1 2 R and 2 = cg :

Let X t be the position of the carrier of virus Y at a given time t starting at the point A; and the stereographic projection coordinates of A are

Proof It is known that (see [21] ) the function u( 1 ; 2 ) = P r A fT = T 1 g is the unique solution of the di¤erential equation Here, 2 ; is the Laplace-Beltrami operator on S 2 expressed in the stereographic projection coordinates. Hence from (4.8) the di¤erential equation (4.53) takes the form and

then the expectation of exp( T ) is given by

where is such that ( + 1) = 2a 2 and P ( ) is the Legendre function

where the multiple-valued function (z + p z 2 1 cos ') is to be determined in such a way that for ' = 2 it is equal to (the principal value of ) z (which is, in particular, real for positive z and real ).

If > 1 2 ; where 1 is the …rst Dirichlet eigenvalue of D S 2 ; then

it satis…es the di¤erential equation

with boundary condition u(' 0 ) = 1: (4.58)

Here 2 is the Laplace-Beltrami operator on S 2 . By the symmetry of D, it follows that the expectation of exp[ T ] is independent of . Hence u is independent of . From (4.4) the di¤erential equation (4.57) takes the form

If we set z = cos ';

and (4.59) transforms to

This is Legendre's di¤erential equation. However, u(') is bounded for all ' 2 [0; ] and u(' 0 ) = 1. Therefore (see [107] ), the solution of (4.59) is

where is such that ( + 1) = 2a 2 : where

then P r A fT < tg = 2P r A fX t = 2 Dg : (4.60)

Proof.

P r A fT < tg = P r A fT < t; X t = 2 Dg + P r A fT < t; X t 2 Dg :

However, if X t = 2 D then of course T < t:

On the other hand, if we setX

then by the strong Markov property of X t

Therefore from (4.61), (4.62) and (4.63) we obtain that P r A fT < tg = 2P r A fX t = 2 Dg :

The re ‡ection principle can help to calculate the distribution functions of certain exit times.

Let i.e.

where p(t; ') is the transition density function of the Brownian motion on S 2 of radius a: Hence from (4.26) It is known that (see [107] )

However, P 0 n (1) = 1 for every n 2 R:

It is also known that for every n 2 N P 0 2n (0) = ( 1) n (2n)! 2 2n (n!) 2 and P 0 2n+1 (0) = 0:

Thus, if n is even then I n = 0: If n is odd, i.e. n = 2k + 1; then

i.e. I n = ( 1) n (2k)! (k + 1)(k!) 2 ( 1) n exp (2n + 1)(2n + 2) p t a (2n)!(4n + 3) 2 2n+1 (n!) 2 (n + 1)

:

Furthermore, if S(0; ) namely the South Pole of S 2 , then P r S fX t = 2 Dg = P r N fX t = 2 Dg = P r N fX t 2 Dg = 1 P r N fX t = 2 Dg: Therefore P r S fX t = 2 Dg = 1 2

( 1) n exp (2n + 1)(2n + 2) p t a (2n)!(4n + 3) 2 2n+1 (n!) 2 (n + 1)

:

( 1) n exp (2n + 1)(2n + 2) p t a (2n)!(2n + 3) 2 2n+1 (n!) 2 (n + 1)

: is a subset of S 2 : The re ‡ected Brownian motion in D 1 is the di¤ usion Y t whose generator is n in D 1 with Neuman boundary condition at @D 1 :

Roughly speaking Y t behaves like X t inside D 1 but when it reaches the boundary, it is re ‡ected back in D 1 : we de…ne the boundary local time L t of Y t ; as

It can be shown that the limit exist in the L 2 sense.

Proposition 4.8. Let ' 0 ; ' 1 2 (0; ), such that ' 0 < ' 1 , both …xed. We consider the sets D; 0 in S 2 ; such that D = f ( ; ')j 2 [0; 2 ) and ' 2 (' 0 ; ' 1 )g :

and

Let Y t be the re ‡ected Brownian motion in 0 starting at the point

and L t is the boundary local time of Y t ; then , as long as the function z is positive (see [109] ). Here 2 is the Laplace-Beltrami operator on S 2 . By the symmetry of D it follows that E A [ exp ( L T ) ] is independent of : From (4.2) the di¤erential equation takes the form

We have shown that the solution of (4.71) is

However, z( ; ' 0 ) = 1 and @z @' ( ; ' 1 ) + z( ; ' 1 ) = 0:

Hence

and c 2 = 1:

Thus

However,

Therefore,

:

A worldwide multilevel interplay among a plethora of factors ranging from micro-pathogens and individual interactions to macro-scale environmental, socio-economic and demographic conditions, necessitate the development of highly sophisticated mathematical models for robust representation of contagious dynamics of infectious diseases that would lead to the establishment of e¤ective control strategies and prevention policies. Ethical and practical reasons defer from conducting enormous experiments in public health systems, hence mathematical models appear to be an e¢ cient way to explore contagion dynamics. A key aspect of epidemiological models is their link to real data, which is of particular utility toward the design of vaccination policies. Two major vaccination strategies exist currently, i.e., the mass vaccination, which is most applied, and the recently developed pulse vaccination which is used in an increasing number of countries. However, most vaccination strategies are imperfect in the sense that they decrease the number of cases, without however eradicating the disease.

Public-health organizations in the world use the epidemiological models that fall in the three categories already presented in this work, to evaluate disease outbreak policies for epidemics. As we pointed out, many shortcomings exist for those models. All the models already used in the literature assume that the host population has constant size. However, this excludes diseases in exponentially growing populations as in most developing countries, or disease-induced mortality as childhood diseases in developing countries e.g., malaria. Modeling infectious dynamics in non-stationary host populations requires explicit modeling of the host population as well as of the disease per se. Models sometimes can be highly complicated in order to improve best …t to real data. Nonetheless, very complex models do not always perform optimally in real-world applications or in simulations. Real-world models allow for swift decision making, and suitable quanti…cation of the spatiotemporal dynamics of an outbreak. Multidisciplinary research e¤orts are speeding up, integrating the advances in epidemiology, molecular biology, computational science and applied mathematics. Mathematical modeling allows better understanding of the transmission process of infectious diseases in space and time, by setting forth rigorously the proper assumptions, the variables, the equations and their parameters.

Due to the complexity of the underlying complex interactions, either deterministic or stochastic epidemiological models are built upon incomplete information about e.g., the basic reproduction number, threshold e¤ects, intensity of spread, precise data of infected versus susceptible individuals, and other inaccuracies regarding the entire infectious network. Simulations or brute-force computational techniques have been implemented in that direction to provide approximate solutions with encouraging results. Nevertheless, some of the underlying generating processes of the outbreaks, such as the virus pathogenicity or variant social network topologies, ethnological characteristics and other quantities, may in ‡uence the spread of an outbreak. Simulations often prove to be ine¢ cient for the systematic analysis of an emergent epidemic. New rigorous mathematical modeling methodologies, such as the one presented in this work for the …rst time, can be used to address inherent incomplete data structure and hidden nonlinear complex dynamics, with an aim to enhance forecastability in combating epidemic outbreaks.

In the present study we introduced a novel approach for surveillance and modeling of infectious disease dynamics, called SBDiEM. We explicitly described the mathematical framework underpinning the implementation and conceptualization of our new-age epidemiological model. Our goal is to contribute to the arsenal of models already developed so far. It can be of particular interest, in light of a recent intensive worldwide e¤ort to speed up the establishment of a global surveillance network for combating pandemics of emergent and re-emergent infectious diseases. Toward this aim, mathematical modeling will play a major role in assessing, controlling and forecasting potential outbreaks. We have to better understand and model the impact of numerous variables on contagious dynamics, ranging from the microscopic hostpathogen level, to individual and population interactions, as well as macroscopic environmental, social, economic and demographic factors all over the world.

As a path for future research, we intend to conduct simulations, and empirical analyses based on real-time spatiotemporal datasets, in case of past outbreaks of infectious diseases as well as for COVID-19. Furthermore, we plan to convey an extensive comparative evaluation investigation of SBDiEM vis-à-vis the three major categories set forth by the taxonomy of Siettos and Russo (2013) , and more speci…cally versus (1) statistical methods for epidemic surveillance, (2) state-space models of epidemic spread and (3) machine learning methods. In this way, the forecasting and nowcasting capabilities of the new model will be thoroughly explored. We also intend to investigate embedding the proposed analytical model into integrated arti…cial intelligence systems in the near future. Our novel methodology apart from o¤ering a much better understanding of the complex and heterogeneous infectious disease dynamics could enhance predictability of epidemic outbreaks as well as have potentially important implications for national health systems, stakeholders and international policy makers.

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