Despite all efforts to reduce the global burden of malaria, the WHO estimates released in Decemberexhibit that 212 million cases of the disease occurred in resulting in about 429 thousand deaths. Furthermore, the highest mortality rates occurred in the sub-Saharan African countries where about of the global malaria cases occurred and led to about of the total world’s malaria deaths. Moreover, more than two third of the global malaria related deaths were children in the age group under the age five years old. In addition, despite the fact that malaria is curable and preventable, and despite all other advances to control and contain the disease, the world at large is still far from complete safety from the health and economic menace exhibited by the disease.
Indeed, WHO reports that in , nearly half of the world’s population was at risk of malaria, and the disease was actively and continuously transmitted in about countries in the world. Moreover, the most vulnerable populations include infants and children under the age of five, pregnant women, patients with HIV/AIDS and non-immune migrants, visitors and travellers to malaria endemic zonesWHO ; CDC . These facts and observations sound a loud call for understanding, cooperation, national solidarity, social and scientific investigation in the fight to eradicate or ameliorate the burdens of malaria.
Malaria is a vector-borne disease caused by protozoa (a micro-parasitic organism) of the genus Plasmodium. There are several different species of the parasite that cause disease in humans namely: P. falciparum, P. viviax, P. ovale and P. malariae. However, the species that causes the most severe and fatal disease is the P. falciparum. Malaria is transmitted between humans by the infectious bite of a female mosquito of the genus Anopheles. Similar to other mosquito-borne parasitic diseases such lymphatic filariasis, the complete life cycle of the malaria plasmodium entails two-hosts: (1) the female anopheles mosquito vector, and (2) the susceptible or infectious human being. The stage of the parasite infective to humans is called sporozoite, while the stage of the parasite infective to mosquitoes is called gametocyte.
Indeed, as an infected female anopheles mosquito persistently quests and successfully bites a human being to obtain a blood meal, she injects sporozoites through the salivary glands into the blood stream of the susceptible or infected person. Inside the exposed or infectious person, the sporoziotes are thought to develop in the liver into schizont which contain numerous merozoites222Schizonts and merozoites are intermediary developmental stages of the parasite.. The mature schizonts rupture releasing the meroziotes into the bloodstream. The meroziotes infect red blood cells, and within the red blood they either (1.) develop to form additional schizont in the blood stream that continue to infect the human body, or (2.) they develop to form a sexual stage of the plasmodium called gametocyte. The stages of maturation of the plasmodium from the sporozoite through the schizont stage within the human body is called the exo-erythrocytic cycle. Moreover, the total duration of the exo-erythrocytic cycle is estimated at between 7-30 days depending on the species of plasmodium, with the exceptions of the plasmodia- P. vivax and P. ovale that may be delayed for as long as 1 to 2 years.
The gametocyte stage of the plasmodium (also referred to as the sexual stage of the malaria parasite) which is infectious to susceptible or infectious female mosquitoes is ingested by the female mosquito when she successfully takes a blood meal from an infectious human being. Within the mosquito, the gametocyte develops into female and male gametes which undergo fertilization and develop into sporozoites which can infect humans. The stages of development from the gametocyte to infectious sporozoites within the mosquito is called the sporogonic cycle. It is estimated that the duration of the sporogonic cycle is over 2 to 3 weeks malaria ; WHO ; CDC . The delay between infection of the mosquito and maturation of the sporozoites suggests that the mosquito must survive a minimum of the 2 to 3 weeks to be able to transmit malaria. These facts are important in deriving a mathematical model to represent the dynamics of malaria.
In the general class of infectious diseases, vector-borne diseases such as malaria and dengue fever exhibit several unique biological characteristics. For instance, as observed in the description about the life cycle of the malaria parasite above, 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 sub-life cycles that are completed separately in the two hosts or directly involved in two separate and independent half-life 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 delayed incubation times namely:- (1) the incubation period of the infectious agent ( or the half-life cycle) in the vector, and (2) the incubation period of the infectious agent (or the other half-life cycle) in 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 8-12 days incubation period inside the female mosquito vector, which starts immediately after the ingestion of a dengue fever virus infected blood meal, that has been successfully taken from a dengue fever infectious human being via a mosquito bite, and (2) another delay incubation period of about 2-7 days in the human being when the hosting female infectious vector acquires another blood meal from a susceptible human being, whereby the virus is successfully transferred from the infectious mosquito to the susceptible personWHO ; CDC .
Malaria confers natural immunity after recovery from the disease. The strength and effectiveness of the natural immunity against the disease depends primarily on the frequency of exposure to the parasites and other biological factors such as age, pregnancy, and genetic nature of red blood cells of people with malaria. The naturally acquired immunity against malaria, especially in areas where malaria is highly endemic such as sub-sahara African, varies across age groups and people with various biological characteristics etc. For example, newborns, pregnant women and visitors from areas with little or no malaria history exhibit low immunity levels against the malaria parasites, while adults who have suffered repeated attacks tend to exhibit higher levels of protective natural immunity against the occurrence of severe disease. Other adults with history of malaria exposure are asymptomatic to subsequent malaria attacks. Furthermore, other biological characteristics related to the nature of red blood cells such as sickle cell trait, and Duffy blood group negativity etc. are also noted to confer long lasting protective resistance against certain species of the malaria parasite. For example, people with sickle cell trait are relatively more protected against p. falciparum malaria, while people who are Duffy negative show strong resistance against P. vivax malariaCDC ; lars ; denise .
Various types of compartmental mathematical epidemic dynamic models have been proposed and utilized to investigate the dynamics of infectious diseases. For instance, dengue fever and measles are studied in eric ; sya ; pang . Furthermore, several different authors have proposed various epidemic dynamic models for malaria beginning with Rossross who studied mosquito control, Macdonaldmacdonald who addressed superinfection, a combined dynamics of mosquitoes and humans investigated in ngwa-shu , the naturally acquired immunity by continuing exposure to malaria explored in hyun ; may and several other studies such as kazeem ; gungala ; anita which are based on the mosquito biting habit. There are also studies which have instead focused on the malaria parasite as the agent of disease transmission such as tabo .
In general, the compartmental mathematical epidemic dynamic models are largely classified as SIS, SIR, SIRS, SEIRS, and SEIR etc. epidemic dynamic models depending on the compartments of the disease classes directly involved in the general disease dynamicsqun ; qunliu ; nguyen ; joaq ; sena ; wanduku-fundamental ; Wanduku-2017 ; zhica . 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 . In addition, 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 dynamicsWanduku-2017 ; wanduku-delay ; kyrychko ; qun , other authors study one or more forms of the classes of delays represented as two separate delay timeszhica ; cooke-driessche ; 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 creates periodic solutions. In cooke ; baretta-takeuchi1 , the occurrence of a delay in the vector-borne 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 system such as the disease transmission, natural death, birth and disease related death rates etc. In Wanduku-2017 ; wanduku-fundamental ; wanduku-delay , 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 of the population. In Baretta-kolmanovskii
, 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 system generally exhibits more complex behavior in the disease dynamics. For instance, the presence of stochastic white noise process in the disease dynamics may destabilize a disease free steady state population by exhibiting high intensity values or high standard deviation values which generally displace the population from a disease free state. In some cases, the presence of white noise may lead to massive oscillations of the state of the system depending on the intensity value of the random fluctuations, which can decrease the population size over time and lead to the extinction of the population. For example, inqun ; Wanduku-2017 ; wanduku-fundamental ; wanduku-delay ; zhuhu ; yanli , the occurrence of stochastic noise in the system destabilizes the disease free steady population state. In qun , the disease free steady state fails to exist when the intensity value of the white noise process from the natural death process of the susceptible population is positive.
The interaction between susceptible, , and infectious individuals, , during the disease transmission process of an infectious disease sometimes exhibits more complex behavior than a simple representation by the frequently used bilinear incidence rate or force of infection given as for vector-borne diseases, or for infectious diseases that involve direct human-to-human disease transmission, where is the effective contact rate, and is the incubation period for the vector-borne disease. More complex behaviors such as the psychological or crowding effects stemming from behavioral change of susceptible individuals when the infectious population increases significantly over time 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, or exhibit other nonlinear behaviors for the disease transmission rate. For instance, in yakui ; xiao ; huo ; kyrychko ; qun ; muroya ; liu ; capasso-serio ; capasso ; hethcote ; koro several different functional forms for the force of infection or incidence rate are used to represent the nonlinear behavior that occurs during the disease transmission process. In yakui ; xiao ; capasso-serio ; 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 behavior of the incidence rate is represented by the general functional form, . In addition, the authors in yakui ; huo ; capasso-serio ; muroya ; capasso ; qun studied vector-borne diseases with several different functional forms for the nonlinear incidence rates of the disease.
Cookecooke presented a deterministic epidemic dynamic model for a vector-borne disease, 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 . The study above allows insight about the dynamics of the disease primarily in the human population while keeping tract of the influence of the vector on the dynamics via the disease transmission process. Whereas vector control aides in malaria preventionngwa-shu ; gungala , in various events of emergency malaria crisis such as when severe disease erupts, urgent medical interference on the involved human being requires direct intervention by medical experts through the use of anti-malaria medications. This observation necessitates a thorough continuous understanding of the dynamics of malaria with major emphasis based in the human population especially in a realistic framework where the system is constantly bombarded by random environmental fluctuations. Very little or nothing about the dynamics of malaria in the human population in a more realistic white noise driven mathematical dynamic system is known. This study bridges the gap by providing a comparative stochastic and deterministic study of a class of malaria models, in an attempt to elucidate the influence of underlying random perturbations on the dynamics of the disease, in particular, on disease eradication via studying the stability of equilibria, extinction and permanence of disease via studying the asymptotic properties of the solutions of the systems.
This paper employs similar reasoning in cooke
, to derive a general class of SEIRS stochastic epidemic dynamic models with three delays for vector-borne diseases such as malaria. The three delays are classified under the two general group types namely:- disease latency and immunity delay. Two of the delays represent the incubation period of the infectious agent (plasmodium for malaria) inside the vector and human hosts, and the third delay represents the period of effective naturally acquired immunity against the vector-borne disease, where the natural immunity is conferred after recovery from infection. Moreover, the delays are random variables. In addition, the general vector-borne 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 deterministic version of the epidemic dynamic model is a system of ordinary differential equations. The stochastic version of the epidemic dynamic model is a system of Ito-Doob type stochastic differential equations.
It should be noted that this study addresses some objectives of a sizeable ongoing project. To conserve space, a parallel detailed study about the qualitative behavior of the intensity of the random fluctuations in the disease dynamics ( which are represented by the white noise processes in the stochastic model) in relation to the stochastic asymptotic stability of the steady states of the system, and with critical examination of the effects of the intensities on disease eradication from the stochastic system appears elsewhere. In that parallel study, various novel mathematical techniques are utilized to diagnose and elucidate the finite properties of the white noise processes in the system, critically evaluate and describe their impact on the disease dynamics. In the current paper, the primary goal is to gain complete comparative insight about the general asymptotic properties of the deterministic and stochastic systems:- (2.8)-(2.11) and (2.15)-(2.18), and with attention given to show how the occurrence of noise and delays in the disease dynamics create several interesting features of the disease dynamics in relation to the qualitative behavior of (1) the equilibria of the systems, whenever the equilibria exist, and (2) the trajectories of the stochastic system near potential deterministic equilibria in the system. Moreover, the interconnection between the two different types of dynamical systems (stochastic and deterministic) with respect to the asymptotic stability of the equilibria, and asymptotic behavior of the solutions of the stochastic system near potential deterministic equilibria is established.
This work is presented as follows:- In section 2, the stochastic and deterministic epidemic dynamic models for malaria are derived. In section 3, the model validation results are presented for both the deterministic and stochastic systems. In section 4, the existence and asymptotic properties of the disease free equilibrium population in both systems are investigated. In Section 5, existence and the asymptotic properties of the endemic equilibria of both systems are also investigated. And in Section 6, numerical simulation results are given to justify the results of this paper.
2 Derivation of Model
A generalized class of stochastic SEIRS delayed epidemic dynamic models for vector-borne 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 incubation of the infectious agent for the vector-borne disease depends 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 connection 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. 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 a successful 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 of 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
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 cooke-driessche , the exposed population, , at time can be written as follows
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
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
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
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 vector-borne diseases in the absence of any random environmental fluctuations can be written as follows:
where the initial conditions are given in the following: Let and define
where is the space of continuous functions with the supremum norm
It is assumed 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 ). Indeed, the variability in the disease transmission and natural death rates are represented by the white noise processes as follows:
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 environmental white noise process due to the random fluctuations in the natural death rate in the state, and is the intensity value of the white noise process due to the random fluctuations in the disease transmission rate.
Indeed, the intensity values of the white noise processes: and representing the variability in the natural death rate, , and disease transmission rate, , at time , owing to the random fluctuations that occur during the disease transmission and natural death processes of the disease dynamics, measures the average deviation of the random variable disease transmission, , and natural death, , rates from their constant mean values - and , respectively, over the infinitesimally small time interval . This measure reflects the force of the random fluctuations that occur during the disease outbreak at anytime, which lead to oscillations in the natural death and disease transmission rates overtime, and consequently lead to oscillations of the sizes of the susceptible, exposed, infectious and removal classes of the total population over time during the disease outbreak.
Substituting (2.14) into the deterministic system (2.8)-(2.11) leads to the following generalized system of Ito-Doob stochastic differential equations describing the dynamics of vector-borne diseases in the human population.
where the initial conditions are given in the following: Let and define
where is the space of continuous functions with the supremum norm
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 the class of functions satisfying different general assumptions which characterize the nonlinear character of the incidence function of the disease dynamics. Some general properties of the incidence function assumed in this study include the following:
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 vector-borne disease scenario, where the disease dynamics is represented by the system (2.15)-(2.18), and the disease transmission rate between the vectors and the human beings initially increases or decreases for 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 .
Observe that (2.16) and (2.18), and the corresponding equations (2) and (2.11) all decouple from the other two equations in their respective systems: (2.15)-(2.18) and (2.8)-(2.11). Nevertheless, for convenience most of the results in this paper related to the systems (2.15)-(2.18) and (2.8)-(2.11) will be shown mostly for the vector . The following notations are utilized:
3 Model Validation Results
The analysis and results in this manuscript are exhibited for both the deterministic and stochastic systems (2.8)-(2.11) and (2.15)-(2.18). These necessitate the existence and uniqueness of the solutions of the stochastic and deterministic systems. The standard methods utilized in the earlier studieswanduku-determ ; Wanduku-2017 ; wanduku-delay ; divine5 are applied to establish the results. The following Lemma describes the behavior of the positive local solutions for the systems (2.8)-(2.11) and (2.15)-(2.18). This result will be useful in establishing the existence and uniqueness results for the global solutions of the deterministic and stochastic systems (2.8)-(2.11) and (2.15)-(2.18).
Suppose for some the systems (2.8)-(2.11) and (2.15)-(2.18) with initial conditions in (LABEL:ch1.sec0.eq06a)-(2.13) and (LABEL:ch1.sec0.eq12)-(2.20) respectively have unique positive solutions denoted , for all , then if , it follows that . In addition, the set denoted by
is locally self-invariant with respect to the systems (2.8)-(2.11) and (2.15)-(2.18), where is the closed ball in centered at the origin with radius containing the local positive solutions defined over .
The proof of the result for (2.8)-(2.11) and (2.15)-(2.18) are the same, hence without of loss of generality, the result will be shown only for the stochastic system (2.15)-(2.18). It follows directly from (2.15)-(2.18) that
The result then follows easily by observing that for , the equation (3.2) leads to . And under the assumption that , the result follows.
The following set of theorems presents the existence and uniqueness results for the global solutions of the deterministic and stochastic systems (2.8)-(2.11) and (2.15)-(2.18). First, the existence results for the deterministic system (2.8)-(2.11) is established. The standard technique applied in wanduku-determ is utilized to establish the results.
It is easy to see that the rate functions of the system (2.8)-(2.11) are nonlinear, continuous in their argument variables, and satisfy the local Lipschitz condition for the given initial data (LABEL:ch1.sec0.eq06a)-(2.13). Therefore, there exists a unique local solution on , where . The rest of the result such as showing that the local solution is positive and extending the local solution inductively to a global positive solution follow a standard technique wanduku-determ . Moreover, from Lemma 3.1, it follows that and (3.3) holds.
The next theorem presents the existence and uniqueness results for the global solutions of the stochastic system (2.15)-(2.18). The standard technique applied in Wanduku-2017 ; wanduku-delay is utilized to establish the results.
It is easy to see that the coefficients of (2.15)-(2.18) 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, where is defined in Lemma 3.1 (3.1). Define the following stopping time
and lets show that a.s. Suppose on the contrary that . Let , and . Define
It follows from (3.5) that
It follows from (3.6)-(LABEL:ch1.sec1.thm1.eq6) that for ,