Analyzing the qualitative properties of white noise on a family of infectious disease models in a highly random environment
Abstract
A class of stochastic vectorborne infectious disease models is derived and studied. The class type is determined by a general nonlinear incidence rate of the disease. The disease spreads in a highly random environment with variability from the disease transmission and natural death rates. Other sources of variability include the random delays of disease incubation inside the vector and the human being, and also the random delay due to the period of effective acquired immunity against the disease. The basic reproduction number and other threshold conditions for disease eradication are computed. The qualitative behaviors of the disease dynamics are examined under the different sources of variability in the system. A technique to classify the different levels of the intensities of the noises in the system is presented, and used to investigate the qualitative behaviors of the disease dynamics in the infectionfree steady state population under the different intensity levels of the white noises in the system. Moreover, the possibility of population extinction, whenever the intensities of the white noises in the system are high is examined. Numerical simulation results are presented to elucidate the theoretical results.
keywords:
Diseasefree steady state, Stochastic stability, Basic reproduction number, Lyapunov functional technique, Intensity of white noise1 Introduction
In the general class of infectious diseases, vectorborne diseases exhibit several unique biological characteristics. For instance, the incubation of the disease requires two hosts  the vector and human hosts, which may be either directly involved in a full life cycle of the infectious agent consisting of two separate and independent segments of sublife cycles which are completed separately inside the two hosts, or directly involved in two separate and independent halflife cycles of the infectious agent in the hosts. Therefore, there exists a total latent time lapse of disease incubation which extends over the two segments of delay incubation times namely: (1) the incubation period of the infectious agent ( or the halflife cycle) inside the vector, and (2) the incubation period of the infectious agent (or the other halflife cycle) inside the human being. For example, the dengue fever virus transmitted primarily by the Aedes aegypti and Aedes albopictus mosquitos undergoes two delay incubation periods: (1) about 812 days incubation period inside the female mosquito vector, which starts immediately after the ingestion of a dengue fever virus infected blood meal, which is successfully taken from a dengue fever infectious human being via a mosquito bite, and (2) another delay incubation period of about 27 days in the human being when the hosting female infectious vector bites a susceptible human being, whereby the virus is successfully transmitted from the infectious mosquito to the susceptible person. See WHO (); CDC ().
Indeed, for dengue fever transmission, a susceptible vector acquires infected blood meal from a dengue fever infectious person via a mosquito bite. The virus incubates in the mosquito for about 812 days, and at the end of the first incubation period the exposed mosquito becomes infectious. The virus is transferred to a susceptible human being after another successful mosquito bite, and it subsequently undergoes a second incubation phase in the exposed human being. The second incubation phase is mostly a viremia phase that involves the complete circulation of the virus in the human blood stream, and at the end of the phase the exposed person develops full blown fever. See WHO (); CDC ().
While the infectious dengue fever vector is known to stay infectious for the rest of the life span, it is important to note in the modelling of the vectorborne disease dynamics that no relationship between vector survival and viral invasion of the mosquito has been determinedWHO (); CDC ().
For the vectorborne disease, malaria, the parasite undergoes a first halflife cycle called the sporogonic cycle in the female Anopheles mosquito lasting approximately days after the first successful mosquito bite from a malaria infectious person. The parasite further completes the remaining halflife cycle called the exoerythrocytic cycle lasting about 730 days inside the exposed human beingWHO (); CDC (), whenever the parasite is transferred to a susceptible person after another successful infectious mosquito bite by the hosting female mosquito.
Several vectorborne diseases induce or confer natural immunity against the disease after infection and recovery. The effectiveness and duration of the natural protective immunity varies depending on the type of disease and also on other biological factors. For example, the exposure and successful recovery from one dengue fever viral strain confers lifelong immunity against the particular viral serotypeWHO ().
Also, the exposure and successful recovery from a malaria parasite, for example, falciparum vivae induces natural immunity against the disease which can protect against subsequent severe outbreaks of the disease. Moreover, the effectiveness and duration of the naturally acquired immunity against malaria is determined by several factors such as the species and the frequency of exposure to the parasites. Furthermore, it has been determined that the naturally acquired immunity against malaria has bearings on other biological factors such as the genetic characteristics of the human beingCDC (); lars (); denise ().
Many infectious diseases have been investigated utilizing compartmental mathematical epidemic dynamic models. For instance, malaria and dengue fever are studied in eric (); sya (), and measles is studied in pang (). In general, these models are largely classified as SIS, SIR, SIRS, SEIRS, and SEIR etc.qun (); qunliu (); nguyen (); joaq (); sena (); wandukufundamental (); Wanduku2017 (); zhica () epidemic dynamic models depending on the compartments of the disease classes directly involved in the general disease dynamics.
Several studies devote interest to SEIRS and SEIR modelsjoaq (); sena (); cesar (); sen (); zhica () which allow the inclusion of the compartment of individuals who are exposed to the disease, , that is, infected but noninfectious individuals. This natural inclusion of the exposed class of individuals allows for more insight about the disease dynamics during the incubation stage of the disease. For example, the existence of periodic solutions are investigated in the SEIRS epidemic studyjoaq (); zhica (). And the effects of seasonal changes on the disease dynamics are investigated in the SEIRS epidemic study in zheng ().
Many epidemic dynamic models are modified and improved in reality by including the time delays that occur in the disease dynamics. Generally, two distinct classes of delays are studied namely:disease latency and immunity delay. The disease latency has been represented as the infected but noninfectious period of disease incubation, and also as the period of infectiousness which nonetheless is studied as a delay in the dynamics of the disease. The immunity delay represents the period of effective naturally acquired immunity against the disease after exposure and successful recovery from infection. Whereas, some authors study diseases and disease scenarios under the realistic assumption of one form of these two classes of delays in the disease dynamicsWanduku2017 (); wandukudelay (); kyrychko (); qun (), other authors study one or more forms of the classes of delays represented as two separate delay timeszhica (); cookedriessche (); shuj (); Sampath (). The occurrence of delays in the disease dynamics may influence the dynamics of the disease in many important ways. For instance, in zhica (), the presence of delays in the epidemic dynamic system leads to the existence of periodic solutions. In cooke (); barettatakeuchi1 (), the occurrence of a delay in the vectorborne disease dynamics destabilizes the equilibrium population state of the system.
Stochastic epidemic dynamic models more realistically represent epidemic dynamic processes because they include the randomness which naturally occurs during a disease outbreak, owing to the presence of constant random environmental fluctuations in the disease dynamics. The presence of stochastic white noise process in the epidemic dynamic system may directly impact the density of the system or indirectly influence other driving parameters of the infectious system such as the disease transmission, natural death, birth and disease related death rates etc. In Wanduku2017 (); wandukufundamental (); wandukudelay (), the stochastic white noise process represents the random fluctuations in the disease transmission process. In qun (), the white noise process represents the variability in the natural death rate of the population. In Barettakolmanovskii (), the white noise process represents the random fluctuations in the system which deviate the state of the system from the equilibrium state, that is, the white noise process is proportional to the difference between the state and equilibrium of the system.
A stochastic white noise process driven infectious system generally exhibits more complex behavior in the disease dynamics, than would be observed in the corresponding deterministic system. For instance, the presence of stochastic white noise process in the disease dynamics may destabilize a disease free steady state population and drive the system into an endemic state. In other cases, the presence of white noise with high intensity in the disease dynamics may continuously decrease the population size over time, and eventually lead to the extinction of the population. For example, in qun (); Wanduku2017 (); wandukufundamental (); wandukudelay (); zhuhu (); yanli (), the occurrence of stochastic noise in the system destabilizes the disease free steady population state. In qun (), it is observed that the disease free steady state fails to exist when the intensity of the noise in the system from the natural death rate of the susceptible population is high.
The interaction between susceptible, , and infectious individuals, , during the disease transmission process can sometimes generate complex nonlinear responses from the susceptible population as the infectious population increases. Such complex nonlinear responses can no longer be represented by the the frequently used bilinear incidence rate (or force of infection) given as for vectorborne diseases, or , for infectious diseases that involve direct humantohuman disease transmission, where is the effective contact rate, and is the incubation period for the vectorborne disease. Some examples of nonlinear complex responses from the susceptible population include psychological or crowding effects stemming from behavioral change of susceptible individuals when the infectious population increases significantly over time. These nonlinear response behaviors exist for certain types of infectious diseases and disease scenarios, where the contact between the susceptible and infectious classes are regulated, and consequently prevent unboundedness in the disease transmission rate.
For instance, in yakui (); xiao (); huo (); kyrychko (); qun (); muroya (); liu (); capassoserio (); capasso (); hethcote (); koro () several different functional forms for the force of infection are used to represent the nonlinear behavior of the disease transmission rate. In yakui (); xiao (); capassoserio (); huo () the authors consider a Holling Type II functional form, , that saturates for large values of . In muroya (); xiao (); capasso (), a bounded Holling Type II function, , is used to represent the force of infection of the disease. In hethcote (); koro (), the nonlinear incidence rate is represented by the general functional form, . And the authors in yakui (); huo (); capassoserio (); muroya (); capasso (); qun () studied vectorborne diseases with several different functional forms for the nonlinear incidence rates of the disease.
Cookecooke () presented a deterministic epidemic dynamic model for vectorborne diseases, where the bilinear incidence rate defined as represents the number of new infections occurring per unit time during the disease transmission process. It is assumed in the formulation of this incidence rate that the number of infectious vectors at time interacting and effectively transmitting infection to susceptible individuals, , after number of effective contacts per unit time per infective is proportional to the infectious human population, , at earlier time .
This paper employs similar reasoning in cooke (); shuj (), to derive a class of SEIRS stochastic epidemic dynamic models with three delays for vectorborne diseases. The three delays are classified under the two general formsdisease latency and immunity delay. Two of the delays represent the incubation period of the infectious agent inside the vector and human hosts, and the third delay represents the period of effective naturally acquired immunity against the vectorborne disease conferred after recovery from infection. Furthermore, the delays are random variables. In addition, the general vectorborne disease dynamics is driven by stochastic white noise processes originating from the random environmental fluctuations in the natural death and disease transmission rates in the population. The epidemic dynamic model is represented as a system of ItoDoob type stochastic differential equations.
It is important to note that this study is part of the broader project investigating vectorborne diseases in the human population. As part of this project, a deterministic study of malaria will appear in Wandukuwandukubiomath (). Some specialized stochastic extensions of this project addressing the impacts of noise on the persistence of malaria in the endemic equilibrium population will appear in Wandukuwandukucomparative (). Moreover, the stochastic permanence of malaria and existence of stationary distribution will appear in Wandukuwandukupermanence (). The primary focus of the current study is to develop and study the fundamental properties of the class of stochastic models for vectorborne diseases in a very noisy environment comprising of variability from the disease transmission and natural death rates. In this study, the intensities of the noises in the infectious system are classified, and their qualitative impacts on the eradication of the vectorborne disease in the steady state population are characterized.
This work is presented as follows: In section 2, the epidemic dynamic model is derived. In section 3, the model validation results are presented. In section 4, the existence and asymptotic stochastic stability of the disease free equilibrium population is investigated. In Section 5, the asymptotic behavior of the stochastic system under the influence of the various intensity levels of the white noise processes in the system is characterized. In section 6, numerical simulation results are given.
2 Derivation of Model
A generalized class of stochastic SEIRS delayed epidemic dynamic models for vectorborne diseases is presented. The delays represent the incubation period of the infectious agents in the vector , and in the human host . The third delay represents the naturally acquired immunity period of the disease , where the delays are random variables with density functions , and and . Furthermore, the joint density of and is given by . Moreover, it is assumed that the random variables and are independent (i.e. ). Indeed, the independence between and is justified from the understanding that the duration of incubation of the infectious agent for the vectorborne disease depends only on the suitable biological environmental requirements for incubation inside the vector and the human body which are unrelated. Furthermore, the independence between and follows from the lack of any real biological evidence to justify the interconnection between the incubation of the infectious agent inside the vector and the acquired natural immunity conferred to the human being. But and may be dependent as biological evidence suggests that the naturally acquired immunity is induced by exposure to the infectious agent.
By employing similar reasoning in cooke (); qun (); capasso (); huo (), the expected incidence rate of the disease or force of infection of the disease at time due to the disease transmission process between the infectious vectors and susceptible humans, , is given by the expression , where is the natural death rate of individuals in the population, and it is assumed for simplicity that the natural death rate for the vectors and human beings are the same. Assuming exponential lifetime for the random incubation period , the probability rate, , represents the survival probability rate of exposed vectors over the incubation period, , of the infectious agent inside the vectors with the length of the period given as , where the vectors acquired infection at the earlier time from an infectious human via for instance, biting and collecting an infected blood meal, and become infectious at time . Furthermore, it is assumed that the survival of the vectors over the incubation period of length is independent of the age of the vectors. In addition, , is the infectious human population at earlier time , is a nonlinear incidence function for the disease dynamics, and is the average number of effective contacts per infectious individual per unit time. Indeed, the force of infection, signifies the expected rate of new infections at time between the infectious vectors and the susceptible human population at time , where the infectious agent is transmitted per infectious vector per unit time at the rate . Furthermore, it is assumed that the number of infectious vectors at time is proportional to the infectious human population at earlier time . Moreover, it is further assumed that the interaction between the infectious vectors and susceptible humans exhibits nonlinear behavior, for instance, psychological and overcrowding effects, which is characterized by the nonlinear incidence function . Therefore, the force of infection given by
(2.1) 
represents the expected rate at which infected individuals leave the susceptible state and become exposed at time .
The susceptible individuals who have acquired infection from infectious vectors but are non infectious form the exposed class . The population of exposed individuals at time is denoted . After the incubation period, , of the infectious agent in the exposed human host, the individual becomes infectious, , at time . Applying similar reasoning in cookedriessche (), the exposed population, , at time can be written as follows
(2.2) 
where
(2.3) 
represents the probability that an individual remains exposed over the time interval . It is easy to see from (2.2) that under the assumption that the disease has been in the population for at least a time , in fact, , so that all initial perturbations have died out, the expected number of exposed individuals at time is given by
(2.4) 
Similarly, for the removal population, , at time , individuals recover from the infectious state at the per capita rate and acquire natural immunity. The natural immunity wanes after the varying immunity period , and removed individuals become susceptible again to the disease. Therefore, at time , individuals leave the infectious state at the rate and become part of the removal population . Thus, at time the removed population is given by the following equation
(2.5) 
where
(2.6) 
represents the probability that an individual remains naturally immune to the disease over the time interval . But it follows from (2.5) that under the assumption that the disease has been in the population for at least a time , in fact, the disease has been in the population for sufficiently large amount of time so that all initial perturbations have died out, then the expected number of removal individuals at time can be written as
(2.7) 
There is also constant birth rate of susceptible individuals in the population. Furthermore, individuals die additionally due to disease related causes at the rate . A compartmental framework illustrating the transition rates between the different states in the system and also showing the delays in the disease dynamics is given in Figure 1.
It follows from (2.1), (2.4), (2.7) and the transition rates illustrated in the compartmental framework in Figure 1 above that the family of SEIRS epidemic dynamic models for a vectorborne diseases in the absence of any random environmental fluctuations can be written as follows:
(2.8)  
(2.10)  
(2.11) 
It should be noted that the deterministic model (2.8)(2.11) with the initial conditions in (LABEL:ch1.sec0.eq12)(2.22) has been studied in the special case of malaria in Wandukuwandukubiomath ().
It is assumed in the current study that the effects of random environmental fluctuations lead to variability in the disease transmission and natural death rates. For , let be a complete probability space, and be a filtration (that is, sub  algebra that satisfies the following: given and ). The variability in the disease transmission and natural death rates are represented by independent white noise processes, and the rates are expressed as follows:
(2.12) 
where and represent the standard white noise and normalized wiener processes for the state at time , with the following properties: . Furthermore, , represents the intensity value of the white noise process due to the natural death rate in the state, and is the intensity value of the white noise process due to the disease transmission rate.
The ideas behind the formulation of the expressions in (2.12) are given in the following. The constant parameters and represent the natural death and disease transmission rates per unit time, respectively. In reality, random environmental fluctuations impact these rates turning them into random variables and . Thus, the natural death and disease transmission rates over an infinitesimally small interval of time with length is given by the expressions and , respectively. It is assumed that there are independent and identical random impacts acting upon these rates at times over subintervals of length , where , and . Furthermore, it is assumed that is constant or deterministic, and is also a constant. It follows that by letting the independent identically distributed random variables represent the random effects acting on the natural death rate, then it follows further that the rate at time , that is,
(2.13) 
where ,and . Note that can similarly be expressed as (2.13). And for sufficient large value of , the summation in (2.13) converges in distribution by the central limit theorem to a random variable which is identically distributed as the wiener process , with mean and variance . It follows easily from (2.13) that
(2.14) 
Similarly, it can be easily seen that
(2.15) 
Note that the intensities of the independent white noise processes in the expressions and that represent the natural death rate, , and disease transmission rate, , at time , measure the average deviation of the random variable disease transmission rate, , and natural death rate, , about their constant mean values and , respectively, over the infinitesimally small time interval . These measures reflect the force of the random fluctuations that occur during the disease outbreak at anytime, and which lead to oscillations in the natural death and disease transmission rates overtime, and consequently lead to oscillations of the susceptible, exposed, infectious and removal states of the system over time during the disease outbreak. Thus, in this study the words ”strength” and ”intensity” of the white noise are used synonymously. Also, the constructions ”strong noise” and ”weak noise” are used to refer to white noise with high and low intensities, respectively.
Under the assumptions in the formulation of the natural death rate per unit time as a brownian motion process above, it can also be seen easily that under further assumption that the number of natural deaths over an interval of length follows a poisson process with intensity of the process , and mean , then the lifetime is exponentially distributed with mean and survival function
(2.16) 
Substituting (2.12)(2.16) into the deterministic system (2.8)(2.11) leads to the following generalized system of ItoDoob stochastic differential equations describing the dynamics of vectorborne diseases in the human population.
(2.17)  
(2.18)  
(2.19)  
(2.20) 
where the initial conditions are given in the following: where ever necessary, we let and define
where is the space of continuous functions with the supremum norm
(2.22) 
Furthermore, the random continuous functions are , or independent of for all .
Several epidemiological studies gumel (); zhica (); joaq (); kyrychko (); qun () have been conducted involving families of SIR, SEIRS, SIS etc. epidemic dynamic models, where the family type is determined by a set of general assumptions which characterize the nonlinear behavior of the incidence function of the disease. Some general properties of the incidence function assumed in this study include the following:
Assumption 2.1

.

is strictly monotonic on .

is differentiable concave on .

has a horizontal asymptote .

is at most as large as the identity function over the positive all .
An incidence function that satisfies Assumption 2.1  can be used to describe the disease transmission process of a vectorborne disease scenario, where the disease dynamics is represented by the system (2.17)(2.20), and the disease transmission rate between the vectors and the human beings initially increases or decreases for relatively small values of the infectious population size, and is bounded or steady for sufficiently large size of the infectious individuals in the population. It is noted that Assumption 2.1 is a generalization of some subcases of the assumptions  investigated in gumel (); zhica (); kyrychko (); qun (). Some examples of frequently used incidence functions in the literature that satisfy Assumption 2.1 include: , , and .
It can be observed that (2.18) and (2.20) decouple from the other equations for and in the system (2.17)(2.20). It is customary to show the results for this kind of decoupled system using the simplified system containing only the nondecoupled system equations for and , and then infer the results to the states and , since these states depend exclusively on and . Nevertheless, for convenience, the existence and stability results of the system (2.17)(2.20) will be shown for the vector . The following notations will be used throughout this study:
(2.23) 
3 Model Validation Results
In this section, the existence and uniqueness results for the solutions of the stochastic system (2.17)(2.20) are presented. The standard method utilized in the earlier studiesWanduku2017 (); wandukudelay (); divine5 () is applied to establish the results. It should be noted that the existence and qualitative behavior of the positive solutions of the system (2.17)(2.20) depend on the sources (natural death or disease transmission rates) of variability in the system. As it is shown below, certain sources of variability lead to very complex uncontrolled behavior of the solutions of the system. The following Lemma describes the behavior of the positive local solutions for the system (2.17)(2.20). This result will be useful in establishing the existence and uniqueness results for the global solutions of the stochastic system (2.17)(2.20).
Lemma 3.1
Suppose for some the system (2.17)(2.20) with initial condition in (LABEL:ch1.sec0.eq12) has a unique positive solution , for all , then if , it follows that
if the intensities of the independent white noise processes in the system satisfy , and , then , and in addition, the set denoted by
(3.1) 
is locally selfinvariant with respect to the system (2.17)(2.20), where is the closed ball in centered at the origin with radius containing the local positive solutions defined over .
If the intensities of the independent white noise processes in the system satisfy , and , then , for all .
Proof:
It follows directly from (2.17)(2.20) that when , and , then
(3.2) 
The result in (a.) follows easily by observing that for , the equation (3.2) leads to . And under the assumption that , the result follows immediately. The result in (b.) follows directly from Theorem 3.1. The following theorem presents the existence and uniqueness results for the global solutions of the stochastic system (2.17)(2.20).
Theorem 3.1
Given the initial conditions (LABEL:ch1.sec0.eq12) and (2.22), there exists a unique solution process satisfying (2.17)(2.20), for all . Moreover,
the solution process is positive for all a.s. and lies in , whenever the intensities of the independent white noise processes in the system satisfy , and . That is, a.s. and , where is defined in Lemma 3.1, (3.1).
Also, the solution process is positive for all a.s. and lies in , whenever the intensities of the independent white noise processes in the system satisfy , and . That is, a.s. and .
Proof:
It is easy to see that the coefficients of (2.17)(2.20) satisfy the local Lipschitz condition for the given initial data (LABEL:ch1.sec0.eq12). Therefore there exist a unique maximal local solution on , where is the first hitting time or the explosion timemao (). The following shows that almost surely, whenever , and , where is defined in Lemma 3.1 (3.1), and also that , whenever , and .
Define the following stopping time
(3.3) 
and lets show that a.s. Suppose on the contrary that . Let , and . Define
(3.4) 
It follows from (3.4) that
(3.5) 
where
(3.6)  
and
It follows from (3.5)(LABEL:ch1.sec1.thm1.eq6) that for ,