Abstract

This work presents a stochastic model of malaria spread. We first calculated the basic reproduction number of the models and in order to show that the malaria-free equilibrium is asymptotically stable; then, we used a finite Markov chain model to describe the interactions between the different compartments of the model . We carried out numerical simulations of our results for two types of transmission zones: a zone with low malaria transmission and an endemic zone. Through these simulations, we first determined the invariant stationary distribution of the model, and then, we found that the use of the indoor residual spraying (IRS) method by regular application of insecticides is more effective for the elimination of malaria than the use of long-acting impregnated mosquito nets (LLINs).

1. Introduction

Mathematical models of the spread of malaria date back to the beginning of the 20th century by Ross [1]. Several mathematical models have been developed to study the dynamics of malaria spread. We can cite the models of Ross, the models in [2–4], and the models in [5, 6]. The mathematical theory of the spread of malaria was solidified by Reiner et al. [7] in the 1960s. In 2018, Mbogo et al. [2] use the Galton-Watson branching processes to propose an extension of the model formulated by [4]. There are also results in the following works [8–13] that could well explain the literature. In this work, we first present in Section 2 some preliminary results, necessary for the conduct of our work by successively studying the models , , and . Then, we present an extension of the Markovian SIS model in a hypoendemic and endemic area in Sections 3.2 and 3.3, respectively. And finally, Section 4 is devoted to the work of numerical simulations of our results.

2. Mathematical Preliminaries and Notations

2.1. Ross’s Model

In 1911, Sir Ronald Ross proposed a model which took into account both anopheline and human populations. This model is certainly the starting point for vector-host models. Ross divided hosts (humans) and vectors (anopheles) into two classes, susceptible and infected, respectively. Let be the population of susceptible humans and the population of infected humans. Ross assumes that there is no latency period and that, consequently, an infected person is automatically an infectious person. Similarly, he refers to the population of susceptible anopheles as and the population of infected anopheles as . In his model, Ross assumes that both the human and Anopheles populations are constant and that one mosquito bites “” humans per unit of time where “” is constant [14]. The malaria propagation graph in Ross’s model is shown in Figure 1.

Figure 1 

The final graph of disease transmission.

He obtained the differential system (1) governing the model of malaria.

Assuming that the human and Anopheles populations are constant, Ross used two ordinary differential equations (2) to model the evolution of the fraction of individuals in the classes of infected .

According to Ross, eradicating malaria requires reducing the number of infectious mosquitoes below a certain threshold. He determines the basic reproduction number per and formulates the following corollary [15, 16].

Corollary 1. (1)If , then the disease completely disappears from the population after a certain time(2)If , then the disease remains endemic in the population.

2.2. Model

Since malaria provides temporary immunity and is not lethal if treated, it is possible to use a SIRS (Susceptible-Infected-Recovered-Susceptible) model, since recovered individuals return to the class with probability or relapsed individuals become infected again with probability . So to Ross’s model, we add the recovered compartment. These types of model are also solved in [17, 18]. Figure 2 illustrates the scheme of disease progression.

Figure 2 

Malaria transmission diagram.

In this model, the differential equation system satisfies the following equation: where , , , , and represent the number of susceptible humans, infectious humans, recovered humans, susceptible mosquitoes, and infectious mosquitoes, respectively. These types of models are also studied by authors such as [2–4].

2.3. Model

According to the literature, the average delay between exposure to parasitised blood and the first clinical signs of infection is 4 to 17 days for P. falciparum. There is therefore an additional class of exposed individuals or latent individuals who are not yet infectious. It is therefore necessary to model the dynamics of the spread of malaria using the model, which would be an extension of the and models previously studied. The disease progression diagram in this type of model can be represented in Figure 3.

Figure 3 

Malaria progression graph.

The ordinary differential equations (ODEs) governing the deterministic SLIRS model are presented by the system of

Mosquito ODEs are neglected for simplicity. All parameters and their biological interpretation are recorded in Table 1.

Once we know the triplet , we can work with the reduced system (6) and then deduce because .

The disease-free equilibrium (DFE) of this model is therefore . Authors such as [17, 19, 20] have also studied these types of models.

3. Main Results

3.1. Basic Reproduction Rate

Theorem 2. The basic reproduction rates of the models in Section 2.2 and in Section 2.3 are given by

Proof. To determine the basic reproduction rate , we apply the Van den Driessche method [15]. The nonlinear ordinary differential equation system (4) integrating the compartments of the model in Figure 2 can be expressed as where represents new infections and represents the rate of individuals entering and exiting the compartment, respectively [15]. The Jacobian matrices of and at equilibrium without disease are The inverse matrix of is with The set of eigenvalues of is denoted and is equal The basic reproduction number is the spectral radius of the next generation matrix: The basic reproduction number of our model (4) in Section 2.2 is Similarly for the model (6) in Section 2.3, we have Hence, Therefore, Then, the basic reproduction rate is given by the spectral radius of .

Corollary 3. The disease-free equilibrium of the system (6) is locally and asymptotically stable if and unstable if .

Corollary 4. The malaria-free equilibrium of the system (4) is locally asymptotically stable if and unstable if .

3.2. Extension of the Markov SIS Model in a Hypoendemic Zone

In this section, we develop a stochastic model for malaria transmission by considering two types of host (“nonimmune” and “semi-immune”) in the human population. The “nonimmune” group comprises human individuals who have never acquired any immunity to malaria. It is assumed that these hosts are vulnerable because they can suffer and/or die from malaria. “Semi-immune” people are those who have acquired or lost at least some immunity to malaria in their lifetime. It is assumed that these hosts are nonvulnerable, so they cannot die from malaria but can only suffer from it. We envisage a model of the type for nonimmunes until they enter the semi-immune category and then follow a model of the type , where and denote the index for nonimmunes and semi-immunes, respectively. For the mosquito population, we use a model of the type .

3.2.1. Elaboration of the Model

We consider that it is the gametocyte form of Plasmodium in humans that transmits the infection to mosquitoes and the sporozoite form in the mosquito that transmits the infection to humans. In our model, we adopt a number of operational definitions.

(1) Terminology. (i)Susceptible : an individual bitten by infectious mosquitoes and capable of contracting malaria(ii)Latent : individual who has contracted malaria but does not transmit it initially(iii)Infectious : an individual who harbours a high level of parasites in gametocyte form. They can suffer and/or die from malaria(iv)Immune (): an asymptomatic carrier of the parasite. This is a state of protection against the disease, but not against the parasite; it is a healing stage

A human being is said to be ill with malaria if he or she is either in a latent or infectious state. In order to write the model, the nonimmune human host types are subdivided into four (4) compartments: susceptible , latent , infectious , and immune (). Semi-immune human host types are also subdivided into four (4) compartments: susceptible , latent , infectious , and immune (). The mosquito population is made up of susceptible and infectious mosquitoes. We are studying two heterogeneous populations made up of several pathological classes, in which individuals can move from one class to another. There are ten different disease states. The pattern of disease progression in this model is illustrated in Figure 4.

Figure 4 

Diagram of malaria progression.

The solid arrows indicate the direction of propagation of Plasmodium, thus creating states of health in the subjects.

The red dotted arrows indicating the direction of infection from an infectious mosquito () to a nonimmune susceptible human individual or a semi-immune susceptible human individual.

The black dotted arrows indicate the direction of infection of a susceptible mosquito on an infectious nonimmune human () or an infectious nonimmune human ().

3.2.2. Model Assumptions

A1: the probabilities , and are in the interval .

A2: the parameters , and are assumed to be strictly positive in the interval and represent the probabilities of moving from a subfund to a subfund .

A3: these passage probabilities and parameters designate the conditional probabilities satisfying the Markov property. These conditional probabilities linking two consecutive pathological states are between if the transition has a biological meaning and 0 otherwise.

3.2.3. Interaction between Humans and Mosquitoes

Initially, infectious mosquitoes () bite susceptible humans with a probability. The latter may remain susceptible with probability for some time or become latent with probability . Individuals can remain latent for a given time with probability or become infectious () with as the transition probability. In this infectious state (), humans can die with a probability or become immune with a probability equal to , or even remain infectious with a probability for an interval of time. Having completed the cycle, the immune () becomes susceptible to the immunity rate, and through the intermediary of the infectious mosquito (), the cycle resumes as shown in Figure 4. At stage (), having acquired a certain immunity, the individual re-enters the susceptible class at the rate . The susceptible mosquitoes inoculate the parasite from a blood meal on the infectious () and with probabilities and , respectively; then, they become infectious () with probability .

Tables 2–4 summarise the parameters that will be described in the model.

Proposition 5. Let be a probability space and the process of the mode of transmission of plasmodia with values in the discrete state space . Then, the process defined on with values in is an irreducible and aperiodic homogeneous Markov chain with initial distribution and transition matrix , where is the probability that the process is in state at time . And this Markov chain has a stationary distribution such that .

Proof. We consider that the ten compartments , , , , , , , , , and in Figure 4 represent the ten states of a Markov chain. These ten states are denoted , respectively. Let be the set of ten states of the Markov chain. Let be the process of the mode of transmission and evolution of Plasmodium. Let denote the probability that the process is in state at time knowing that it is in state at time . Furthermore, the probability is assumed to be independent of . For all , is called the transition probability from state to state . The probabilistic transition graph of the Markov chain model is given in Figure 5, where the correspond to the different parameters of the model and represent the transition probabilities of the chain. The chain is irreducible because its representative graph (Figure 5) is strongly connected.

Figure 5 

Transition graph of irreducible Markov chain.

Under hypothesis A3, the are grouped together in a matrix defined by the relation

The matrix thus constructed is stochastic by extension of the SIS model.

Furthermore, during the clinical manifestations of malaria, the future state of the patient does not depend on the previous state, but rather on the current state (the system has no “memory”). All the information needed to predict the future is contained in the current state of the process. So we can say that the spread and evolution of malaria satisfy the weak Markov property (discrete time, discrete space): for any , for any sequence of states , as soon as . This property expresses the fact that the law of depends on only through the value of : the “present” gives as much information about the “future” as if we knew all the “past” . By hypothesis, the transition mechanism does not change over time. The weak Markov property then takes the following form:

i.e., which expresses the homogeneity of the process in time.

3.3. Extending the SIS Markov Model to an Endemic Area

Children under 5 years old are the most vulnerable to malaria in endemic areas [21–24]. They have not yet developed their own immunity and are therefore considered nonimmune. We do not take into account the age of the individual but rather his or her immunological status, because according to studies [25, 26], children and adults have the same probability of being infected by malaria. It is assumed that there has been one case of death in the infectious nonimmune human compartments. Then, the Plasmodium propagation diagram takes the form of Figure 6.

Figure 6 

Diagram showing the progression of malaria in an endemic area.

The solid arrows indicate the direction of propagation of Plasmodium, thus creating states of health in the subjects.

The blue dotted arrows indicate the direction of infection from an infectious mosquito () to a nonimmune susceptible human individual or a semi-immune susceptible human individual.

The black dotted arrows indicate the direction of infection of a susceptible mosquito on an infectious nonimmune human () or an infectious nonimmune human ().

The red dotted arrows indicate a case of death: (i)of infectious nonimmune human () thus entering compartment , where represents the probability of dying from malaria knowing that the nonimmune human is infectious(ii)mosquitoes by intradomiciliary spraying

The green dotted arrow indicates the use of long-lasting impregnated mosquito nets.

3.3.1. Model Assumptions

We supplement the list of assumptions in the previous Section 3.2 with two more:

A4: infectious mosquitoes are eliminated by intradomiciliary insecticide spraying with a probability .

A5: humans use long-acting impregnated mosquito nets with a probability of , and for reasons of model simplification, environmental factors that could have more or less significant effects are neglected.

Proposition 6. Let be a probability space and the process of the mode of transmission of plasmodia with values in the discrete state space . Then, the process defined on with values in is a homogeneous absorbing Markov chain with transition matrix .

Proof. The graph representing the Markov chain model is given in Figure 7, where the correspond to the different parameters of the model and represent the transition probabilities of the chain.
There are three absorbing states on the graph. As a result, the chain is absorbent. And under the assumptions A1,...,A5 and under the same interactions between humans and mosquitoes described in the previous section, we group the in a matrix whose canonical form is represented by the relation The matrix of transient states and that of absorbing states are given by the relations The fundamental matrix will be determined by application.

Proposition 7. The matrix elements give us the expectation of the total number of stays in before passing beyond the transient states, and the line of gives us the “expected life balance,” with a given initial state .

Proof. Let be the indicator of being present in the transient state at time . We pose We have . Considering also the breakdown into indicators where is the indicator of being in position at time . As a result, By changing the order of summation, we obtain where is the total expected time spent in starting from state .

Figure 7 

Transition graph of the absorbing Markov chain.

4. Numerical Simulations

4.1. Values Assigned to Transition Probabilities

In this section, we numerically analyse the stochastic matrices (18) and (22). We consider two malaria transmission zones: zone A corresponding to a low transmission zone and zone B corresponding to a high transmission zone. We begin by determining the values of the biological parameters equivalent to the transition probabilities in each zone. Most of these data are obtained from the literature, in particular from models developed by [27–31].

4.1.1. The Transition Probabilities between the Three Types of Host

The probability of effective transition between the three host types is the average number of contacts per unit of time (in this case the day) that could lead to infection of a specific host type.

: we estimate that the average number of mosquito bites a human being can suffer per day is 0.28 for the low transmission zone and 0.38 for the high transmission zone. This average number is a function of the exposed surface area of the human and any vector control interventions used by humans to reduce exposure to mosquitoes [27, 30].

: we estimate that the probability of transmission of an infection from an infectious mosquito to a susceptible nonimmune human being , knowing that there has been contact between the two, is 0.021 for the low transmission zone and 0.07 for the high transmission zone [30].

: we estimate that the probability of transmission of an infection from an infectious mosquito to a susceptible semi-immune human being , knowing that there has been contact between the two, is 0.012 for the low transmission zone and 0.022 for the high transmission zone [28, 31].

: we assume that the probability of transmission of infection from an infectious nonimmune human being to a susceptible mosquito , given that there has been contact between the two, is 0.11 for a low transmission zone and 0.45 for a high transmission zone [31].

: we assume that the probability of transmission of infection from an infectious semi-immune human being to a susceptible mosquito , given that there has been contact between the two, is 0.08 for an area of low transmission and 0.35 for an area of high transmission.

4.1.2. Transitions in Nonimmune Host Compartments

: the probability of transition from the susceptible state to the latent state . This parameter results from the infection force which is defined by where is the human population size and is the mosquito population size [28–30].