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Global incidence of dengue, a vector-borne tropical disease, has seen a dramatic increase with several major outbreaks in the past few decades. We formulate and analyze a stochastic epidemic model for the transmission dynamics of a single strain of dengue virus. The stochastic model is constructed using a continuous time Markov chain (CTMC) and is based on an existing deterministic model that suggests the existence of a backward bifurcation for some values of the model parameters. The dynamics of the stochastic model are explored through numerical simulations in this region of bistability. The mean of each random variable is numerically estimated and these are compared to the dynamics of the deterministic model. It is observed that the stochastic model also predicts the co-existence of a locally asymptotically stable disease-free equilibrium along with a locally stable endemic equilibrium. This co-existence of equilibria is important from a public health perspective because it implies that dengue can persist in populations even if the value of the basic reproduction number is less than unity.

Dengue, a vector transmitted disease, has seen a dramatic increase in global incidence over the past decades [1,2]. Originally restricted to a handful of countries, dengue is now endemic in more than a hundred tropical and subtropical countries worldwide [1-3]. With an estimated 50 - 100 million cases and nearly 10,000 - 20,000 deaths annually, dengue ranks second to Malaria amongst deadly mosquito-born diseases [1,2,4-6]. The disease is caused by one of four virus serotypes (strains) of the genus Flavivirus [2,3,7]. Most infected individuals suffer from dengue fever, a severe flu-like illness characterized by high fever, which poses only a limited threat to mortality [2,8]. The symptoms usually last for one to two weeks, after an initial incubation period of about 4 - 7 days [

Dengue is transmitted to humans through mosquito bites. Female mosquitos of the Aedas genus, primarily Aedas aegypti, acquire the dengue virus through a blood meal from infected humans [2,11]. The dengue virus has an incubation period of about 7 - 10 days in the vector, and is then spread to susceptible humans who are bitten by the infected mosquito [

Historical records indicate the occurrence of dengue epidemics in North America, Asia and Africa in the late 18th century [

The dengue virus (DENV), which causes dengue fever in humans, is a single positive-stranded RNA virus of the family Flaviviridae and genus Flavivirus [2,3,7]. The virus has four distinct serotypes, DENV1, DENV2, DENV3 and DENV4, each of which can cause the full spectrum of the disease [2,3,7]. Owing to the difficulty of developing an immunization against all four serotypes, there is currently no vaccine against the disease [6,8], [

The clinical symptoms and effects of dengue disease vary greatly. Nearly 80% of individuals suffering from a primary infection with DENV are asymptomatic or display only a mild, uncomplicated fever [2,8]. A minority of infected individuals suffer from DHF and DSS, the more severe forms of dengue disease [3,5,9]. As mentioned previously however, risk of DHF and DSS is associated primarily with secondary infection with a heterologous serotype of DENV [4,12,13,15]. In general, dengue disease is marked by three separate phases: febrile, critical and recovery. The characteristic symptoms of dengue in the febrile phase are the sudden onset of high fever, rash, headaches and muscle and joint pains, which lead to the alternative name “breakbone fever” for dengue disease [

Over the past several years, a number of deterministic mathematical models have been proposed to analyze the transmission dynamics of dengue in urban communities [5,11-17]. L. Esteva and C. Vargas [

In addition, over the past few decades, several stochastic epidemic models for the spread of infectious diseases have also been proposed and analyzed [19-27]. An important qualitative difference between deterministic and stochastic epidemic models in general is the asymptotic dynamics [

The purpose of this study is to formulate and analyze a stochastic model for the transmission dynamics of a single strain of dengue virus using a continuous time Markov chain (CTMC). The stochastic model is based on an existing deterministic model proposed by Garba et al. [

This paper is organized as follows. The second section contains a description of the deterministic model formulated in [

The deterministic model we have considered is a deterministic vector-host ODE model that assumes a homogenous mixing of the host (human) and vector (mosquito) populations. The total human population at time t, denoted by, is divided into four mutually exclusive classes comprising of susceptible humans, exposed humans, infected humans and recovered humans. It is assumed that individuals who recover from infection with a particular serotype of Dengue gain lifelong immunity to it [

The model assumes that the susceptible human population has a constant recruitment rate and natural death rate. Susceptible individuals are infected with Dengue virus (due to contact with infected vectors) at a rate and thus enter the exposed class E_{H}. The exposed population is depleted at the natural death rate. Additionally, exposed individuals develop symptoms and move into the infected class I_{H} at a rate. The infected population is depleted via the natural death rate mu, the disease-induced death rate and the recovery rate of infected individuals. Finally, the recovered population decreases due to the natural death rate mu.

Similarly, the susceptible vector population has a constant recruitment rate and a natural death rate. Susceptible vectors are infected with Dengue virus (due to effective contact with infected humans) at a rate and thus move to the exposed vector class. The exposed vector class is depleted at the natural death rate. In addition exposed vectors develop symptoms and move to the infected vector class at a rate. Infected vectors, in addition to the natural death rate die at a disease induced death rate.

Mathematically, the modified determinstic model is as follows:

where,

The basic reproduction number for the model (2.1) can be calculated using the next generation operator method given in [

where, , and.

It follows from [

Lemma 1. The system (2.1) has a locally asymptotically stable disease-free equilibrium (DFE) given by

, whenever. The DFE given by is locally unstable whenever.

It should be pointed out however, that we have not proven that the DFE is globally asymptotically stable for values of the basic reproduction number. Indeed, as shown in the next section, this is not true.

As mentioned in the introduction, Garba et al. proved that the deterministic model presented in [

Theorem 1. Define the constants and as follows:

Then, the system (2.1) has:

• A unique endemic equilibrium if.

• A unique endemic equilibrium if, and or.

• Two endemic equilibria if and .

• No endemic equilibrium otherwise.

Furthermore, if case 3) from above holds and we define as follows

then a backward bifurcation of the system (2.1) occurs for values of the basic reproduction such that

.

Proof. By setting, the result follows readily from the proof of Theorem 1 in [

As a consequence of Theorem 1, the model (2.1) does not possess a globally asymptotically stable DFE for. Indeed, it is clear that for certain values of, there is a simultaneous co-existence of a locally asymptotically stable DFE alongside a locally asymptotically stable endemic equilibrium. Thus, lowering the value of the basic reproduction number to less than unity is no longer a sufficient condition to ensure disease elimination. From an epidemiological perspective, this possibility is extremely important because it implies that the dengue virus can persist in a population even when.

We now formulate an analogous stochastic model using a continuous time Markov chain. Our main purpose in this study will be to explore the dynamics of the stochastic model using numerical simulations in the region of bistability predicted by Theorem 1 and compare them to the corresponding results from the deterministic model (2.1). It has been observed [

Following [

with

The continous-time stochastic dynamical system

is a multivariate process with a joint probability function with

where

and.

We further assume that the stochastic process is time homogeneous. Thus, using the notation of [

and therefore, the probability of transition from the state to the state

is defined by

where

In order for the probabilities to be meaningful (non-negative and bounded by 1), the following is assumed:

.

Since all the parameters are positive, a small value of ensures that the above condition is satisfied.

Although, we now have the infinitesimal transition probabilities in hand, it is difficult to express the transition matrix and the generator matrix in easily expressible forms. One possible solution to this problem is given in [

We now assume that the stochastic process satisfies the Markov property [

Using the Markov property and the infinitesimal transition probabilities, we can express the state probabilities at time in terms of the state probabilities at time t. For the sake of simplicity, we assume that the total population at time t is less than the upper bound K and furthermore that none of the random variables is zero. The purpose of these two assumptions is to ignore numerous tedious sub-cases.

Proposition 1. Assume. Further assume that none of the variables s_{h}, e_{h}, i_{h}, r, s_{v}, e_{v}, i_{v} is zero. Then, the state probabilities satisfy the following difference equations

where

The remaining cases, which result from removing the previous assumptions, are straightforward to calculate.

Some properties of the Markov chain can now be discussed. The Markov chain is reducible, with two important communication classes [

and

Furthermore, E is not closed since, for example,. On the other hand, D is closed since

Since E is an open class, all states in E are transient. Hence and in view of D being the only closed communication class, all sample paths will eventually be absorbed into the communication class D. However, while D is indeed a closed class, it does not contain an absorbing state since ,

In view of the above discussion, the following theorem is established.

Theorem 2. The CTMC is a reducible chain with no absorbing states. However, all sample paths are eventually absorbed in the closed class

.

The two communication classes D and E correspond to the disease-free and endemic stages of the disease respectively. Therefore, the above result indicates that every sample path is eventually absorbed into the disease-free class and therefore, irrespective of parameter values, the stochastic model will always converge to a disease-free state. This is in direct contrast to the deterministic case in which a unique endemic steady state solution always exists for. Thus, the deterministic case allows for the possibility of the disease having an endemic equilibrium, depending on the value of a threshold parameter, while the stochastic model always predicts an eventual disease-free equilibrium. Furthermore, disease extinction takes place in finite time in the stochastic case while in the deterministic case a disease-free equilibrium is only approached asymptotically. Depending on the value of however, the sample paths for the stochastic model might remain in the endemic class E for a very long time. As a consequence, for most practical purposes, the stochastic model follows closely the behavior of the corresponding deterministic model for the majority of time. This can be demonstrated using numerical simulations.

Our purpose in this section is to explore the dynamical behavior of the stochastic model for values of the model parameter, which result in a backward bifurcation and produce a region of bistability. The behavior of the stochastic model in this region is important due to the public health implications of the backward bifurcation. As mentioned previously, the existence of a backward bifurcation in model (2.1) results in the possibility of a locally asymptotically stable endemic equilibrium even for values of. Therefore, lowering the value of the basic reproduction number below unity is no longer a sufficient condition for disease elimination. Moreover, as mentioned in [

For the purpose of numerical simulations, we will formulate our stochastic model as a discrete time Markov chain (DTMC) and employ a constant time step. This simplifies many of the numerical simulations and allows us to calculate the numerical mean, which is not possible when using a CTMC with random, exponentially distributed interevent times. As mentioned in [

For the numerical simulations in this section, the following parameter values were chosen [

This results in a value of R_{0} = 0.855 and R_{c} = 0.571 and thus we have. Therefore, in view of Lemma 1 and Theorem 1, the deterministic model (2.1) now has a locally asymptotically stable disease-free equilibrium (DFE) as well as a locally asymptotically stable endemic equilibrium. We point out however, that these parameter values are not necessarily realistic from a biological and epidemiological point of view.

We numerically simulate 500 sample paths to estimate the mean of each discrete random variable,. The following initial conditions were chosen:

Initial Population:

For these initial conditions, the behavior of the stochastic model closely follows that of the deterministic model (2.1) as shown in _{0} being less than unity, we see that the stochastic model predicts that the disease will remain endemic in the host population. As a consequence, the behavior of the stochastic model suggests that lowering the value of R_{0} to less than unity is no longer a sufficient condition for disease elimination.

On the other hand, the following initial conditions result in the stochastic model as well as the deterministic model (2.1) tending towards the DFE:

Initial Population:

In this case, both the deterministic model (2.1) and the stochastic model predict that the disease will be eliminated from the host population in time, as shown in

The behavior of both the stochastic and the deterministic model is therefore highly dependent on the initial conditions. We therefore conclude that due to the existence of the backward bifurcation in model (2.1), disease elimination or persistence in the region of bistability is dependent on the initial conditions of the system. If a sufficient number of infectives are present in the population, then both the stochastic as well as the deterministic model predict that there is a possibility that the disease will persist, despite R_{0} being less than unity. Figures 3 and 4 display the forces of infection for the two different initial conditions.

The values of R_{0} and R_{c} depend on the variables, and. While, deterministic models implicitly assume that the model parameters are not stochastic in nature, an element of uncertainty is always associated with estimates of these parameters due to factors such as natural variation, errors in measurements and lack of measuring techniques. In general, uncertainty analysis quantifies the degree of confidence in the parameter estimates by producing 95% confidence intervals (CI) which can be interpreted as intervals containing 95% of future estimates when the same assumptions are made and the only source of noise is observation error. This enables us to estimate the probability of, while taking into account the uncertainty associated with estimates of the model parameters. Our purpose in this section is to estimate the likelihood of a backward bifurcation in model (2.1) for our chosen parameter values.

We use the Latin Hypercube Sampling (LHS) to quantify the uncertainty in R_{0} and R_{c} as a function of the 7 model parameters and.

It is assumed that the recruitment rates and are constants. The assumed distributions of the model parameters used in the two analyses are mentioned in

We formulate a stochastic model, based on an existing deterministic ODE model, for the transmission dynamics of dengue virus using a continuous time Markov chain. The deterministic model is known to undergo a backward bifurcation for certain values of the model parameters and consequently presents the possibility of the co-existence of both a locally asymptotically stable DFE along with a locally asymptotically stable endemic equilibrium for certain values of the basic reproduction number less than unity. It is important to note that the deterministic model (2.1) undergoes a backward bifurcation as a consequence of both the use of standard incidence, as opposed to mass action, as well as inclusion of the vector dynamics. Thus, removing the vector dynamics from model (2.1) and using a direct transmission model as suggested in [

The numerical mean of each discrete random variable is calculated using Monte Carlo simulations. It is found that the numerically calculated means are in excellent agreement with the corresponding results from the deterministic case for parameter values, which result in the backward bifurcation. Therefore, both the stochastic model as well as the deterministic model in [

Parameter

Mean, variance (dist)

The current study focuses on a model for the spread of single strain dengue virus. However, as demonstrated by Garba et al. in [