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 .
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.
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 .
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 .
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.
Under hypothesis A3, the are grouped together in a matrix defined by the relation
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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 .
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
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oxgFjN7N5S/j5OO2LBOrIUPNcTAVVB04N8gZM+RT2ZMelQ95EmbA30P+9t+UCKGJ94gsbog7/LXz
iwXxlhZv7x3o2K+hq5b7V4VpG5IQLakutjjQAQUMcYv7XreoA/miFRm1eU3IXc3vpaSjZWDo2f2O
UkOfIlDDMSWF0UkHHASrDIPALZ9UMUJny4KIiFN7PPbFaaff7k81Zjs4ZBTrt16X1Q2mudFISBtf
V6uj5fjP8CflsQVvN64i6JpTRX91KydnqpOFQCR/XqeI1ueHma/+npcXR4wUJv8HP8MmG2ybG0cx
wPPtapuqGc+TSUApsKpIJcF4JiFR29p+ER3EB64orrFeoAnT34erA7PHYFu4Z6wj
The matrix of transient states and that of absorbing states are given by the relations
KLUv/QBYpL4DnpXEBgousBLNNG1KVWEkB3oaqsQq1nREAkzTRd9va/Mbandvrtz20TOFPSH8KX1d
7acEF+ILCgr2CFZOFMaiJIgmHkeSI+8ojGSPXPKQVMkkwQSrMO58xkTtVRXVclwWmQgQ9YdI64h0
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BMktjAIUBiFGCqPsMHULI0EsjAMPROrwUBKFcUhhJMZNLB51o8LATN09jiRRFEqSpDDMytaSiVQS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ZIimR4pnO7qGXqi4NkPT4Y11RUhmm51o99ITkjieh/ezqzNRYn3YgYceCZIkllhHKS5PKItSGImC
KMgqmjfFnEgSyZIoyaGJXpGCJIsoIkeOHGJFyG1hIHIr+zp80JHYoxYtDEUTRfTQVb2e2oJkYr4S
uKCCiVZhnJ4ojEVJkEsQeaEELbQsSZIgyKGkemZyEZHrlAcltFCC7LL5TheG+nDixVQuqxcmRsQS
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cbrkpJ7gGTKFEYGNBBOBgcKCg8MDvs48m8YI9dgMml/5eOzuTt5sJR00xhgK2A5v08c16hlNDxWG
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1Crcpfwt5VXtMI/q3uqQKU26Q2ar/PVqKh2undXK8Fk+PWS0ymV2a03LQ8YKX3mU/qop71ojXfpl
1SFjuzpkNAnXtPdK8x4yqGlH6pMRER0yZv2hrxBtesi8bqj25zdpD5mbyaRrdruipDAUcpnP5/uo
Vq/Xhw8jTlwYiRUrXqioXFVVVoeVW0dSSi2VKDKbrczs7IaWprZ2GGKpSWHoHkhCVmyKFQYiCqIo
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lr6sMKoogXidVyYvpJBCCB20pKQ32e6YpKSbsMUUSwyxIw3tdmZWVhg2skQtpVRSSB21pZWFhfW6
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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 .
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 ].
: the probability of transition from the latent state to the infectious state . We assume that for both types of zone.
: the probability of transition from the infectious state to the immune state . We have assumed that this rate is 0.005 for the zone of low transmission and 0.001 for the zone of high transmission zone.
: the probability of transition from the immune state to the susceptible state . This is a phase of change of status from nonimmune to semi-immune hosts. We have assumed that this probability is for the low transmission zone and for the high transmission zone [27 , 29 , 31 ].
4.1.3. Transitions in the Compartments of Semi-immune Hosts : the probability of transition from the susceptible state to the latent state . This parameter results from the strength of infection of the semi-immunes, which is defined by
where is the size of the human population and is the size of the mosquito population [28 –30 ].
: the probability of transition from the latent state to the infectious state . We assume that for both types of zone.
: the probability of transition from the infectious state to the immune state . We have assumed that this rate is 0.01 for both types of zone.
: the probability of transition from the immune state to the susceptible state . Here, the immune cells heal and return to the susceptible compartment. We assume that for the low transmission zone and for the high transmission zone [30 ].
For , the are obtained by stochastic effect. The other transition probabilities not quoted here are obtained by using the Markov property.
All these estimated probabilities are summarised in Table 5 .
4.2. Study of the Matrix in a Zone A In this section, we numerically analyse the matrices derived from the relationship (18 ) using the parameter values obtained in Table 5 (zone A), which correspond to a stable transmission zone. We obtain the matrix of the relationship (31 ) with values in .
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We consider a heterogeneous population of humans and mosquitoes made up of eighty people and twenty mosquitoes assumed to be under normal conditions of temperature and reproduction. The human population is made up of fifty-seven nonimmune host types and twenty-three semi-immune host types. This gives the initial distribution given by the relationship
that is, 20% of susceptible nonimmune, 15% of latent nonimmune, 10% of infectious nonimmune, 12% of recovered nonimmune, 9% of susceptible mosquitoes, 11% of infectious mosquitoes, 5% of susceptible semi-immune, 8% of latent semi-immune, 6% of infectious semi-immune, and 4% of recovered semi-immune. We use R version 4.2.1 (2022-06-23 ucrt)-“Funny-Looking Kid” for the simulation results. Tables 6 and 7 give the probabilities for . They illustrate the evolution of the different stages of malaria over a period of twenty-one malaria episodes, using the initial distribution of the relationship (32 ).
The stationary distribution obtained is and is given by the relation
Graphical representations of the results of Tables 6 and 7 are listed in Figure 8 .
(a) Graph of susceptible hosts (b) Graph of infectious hosts (c) Graph of latent hosts (d) Graph of recovered hosts (a) Graph of susceptible hosts (b) Graph of infectious hosts (c) Graph of latent hosts (d) Graph of recovered hosts For a given initial distribution, the population of susceptible Anopheles mosquitoes increases and stabilises at around 25%. There has also been an increase in the number of infectious mosquitoes. The rapid decline in the susceptible nonimmune curve led to rapid growth in the susceptible semi-immune curve, creating a peak of probability 0.22 which stabilised after the twentieth episode. After several episodes of malaria, nonimmune individuals gradually migrate to the semi-immune class to acquire a certain level of immunity, leading to a high level of semi-immune individuals, which stabilises from the twentieth episode onwards.
4.3. Study of the Matrix in an Endemic Area Using the parameters obtained in Table 5 (zone B), we obtain the numerical values of the absorbing matrix of the relation (22 ) described by the relation
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fqNAAkBk2B7iVcbFfVMhmdAYYxPVKSr8mWRwZyUrtLo3VhJNMUFE9wlLq7s5moR9qAoetroDjuiD
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XAG0GiqxJV0EkCEFTCBAqKdOnglCkE8IQbjMkIyNDCl5iUBOMooiDBxIKgNwbdGPDDiRZki5AZnj
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qxmXamG45mIzbJZjKBAk8cXSTLqXRDHCSyaCAjagWL2skQRLXQ0JwHCeySWUqiEh66VrjQUSBczU
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p1B6ZFJR7oabAbWa4qDwEy11wYa0M9zMy6soQ48kT/rVdFehgaDye0i7gqAYTGRGN6/9M8Up4nNS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==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The matrix of transient states and that of absorbing states of the matrix are given in the following the relations.
KLUv/QBYVL4DfpTEBQousBLNNG1KVWEkB3oaqsQq1nREAkzTRd9va/Mbandvrtz20TOFPSH8KX1d
7acEF+ILCArzCOUknZZQKycKY1ESRBOPI8mRdxRGskcueUiqZJJgglUYdz5jovaqimo5LotMBIj6
Q6R1RDpqmm0V6xQTj0x0PrXnIZOJSGenu2hEwnNobLXHdjtk9ih3U79r+tHQdAs3dwnp7L25zpah
6iGDlXerGtFPinSm+uuHhkaXTYd0M7NjZtC4MBJVkmWpqx90hYHPZ7IkXIeHi1gYTva+p1UYiiV2
HNlSih4mVoWx3JIURqKWLV2sGHGi6ClJLAmEksKgJLEgCaIgSHIkojekMAwxUomDkOVIHMqhHMeR
V3LzSCyC5BZGAQqDECOFUXaYuoWRIBbGgQcidXgoicI4pDAS4yYWj7pRYWCm7h5HkigKJUlSGGZl
a8lEKokkkpWkoKJYGLp8Oeh+Fsbv7pxQ8oZmVYmkMJxcSRJKFmTBwthE90ASZSFXYRxyuyOFceSY
R2EoiLLcoSwJvzCehXEsyIIoibrcIyGX6b0wkETRJWE7LQxbYRzJoSRKscI4ljuUA0GQLQzVTbyZ
CmNrsfB1vLNKOKFsYeTLLowFIUTvhKKkguSRx6Zm5lnRoiCH4vEMK6EkL4zkyBPvZuW4xcMrDLqE
LVkYalk2Edz5kKk02qLbrZWs1MZSXNP6khmt/kMm7bWktvOeZISUd8jk4e3QiOoqWpcWrtnebpJt
XXaZFanuTJfMmpmn1Ki+8E5lwtq0TUv0jXvI2J1h5h1q4tHIjol3tUumwqCAbmI/u4/VzcSChVEA
JUEOxbPTRRJKYlkSeeyBLMli5cjCuIQlag9TqjCSwsjCOGHiy5Uni/8Lg5LFKozFWg4rDGWrMDCB
FEYlR3IoXvmgJSmMY1c3F79X64SCrGoq3hZG4s8Kw4nCQI7VLIyflakOsg/FS2EkKlFrh2JLnX3v
QK5cUcKEtDASSutDcVIYmiwJKVZMuUzoCzpRiUs8cXRKoS5WfK1dDAtDj4QlfHmKJIvthWG/H6/I
k8SXk6EhpuImCVyQnJWVl5apqblKLi41vV5R9UBU0YQuSkvrNZnKfkJM3cJIFIbLrjDu8/GHlwjt
kHkdMvvfS4Q3vohSy5Z3XlPrdnJaMzeP7SwsJZv6eEv+FqbRf6ZWaHeiTcUlqj2F5g6Zq0MGfURN
W09n6FIyRNMjxbMdXUMvVFyboenwxroiJLPNTrR76QlJHM/D+9nVmSixPuzAQ48ESRJLrKMUlyeU
RSmMREEUZBXNm2JOJIlkSZTk0ESvSEGSRRSRI0cOsSLktjAQuZV9HT7oSOxRixaGookieuiqXk9t
QTIxXwlcUMFEqzBOTxTGoiTIJYi8UIIWWpYkSRDkUFI9M7mIyHXKgxJaKEF22XynC0N9OPFiKpfV
CxMjYoktRuczvcO33/H644VBWHi7IKUzN5HW0jsU8dp+Mdll6ahjv19c3DyQwlASS4f9QD32OJJD
IcSIkcI4XXJST/AMmcKIwEaCicBAYcHB4QFfZ55NY4R6bAbNr3w8dncnb7aSDhpjDAVsh7fp4xr1
jKaHCsO/k6XaRXh4qDAUXbmmami0eD688r2slIfM2BRGBAwREQwOEhogYCIu8GCBgWC4AAMLjJYS
rsu3ujuvjqe4dWpoZkZbpaNFh0weMqh4125ioRWSGaImGt7rbvF8o10tIsSktetBrB8yqntKqLdp
p7rT0ydqFe5S/pbyqnaYR3VvdciUJt0hs1X+ejWVDtfOamX4LJ8eMlrlMru1puUhY4WvPEp/1ZR3
rZEu/bLqkLFdHTKahGvae6V5DxnUtCP1yYiIDhmz/tBXiDY9ZF43VPvzm7SHzM1k0jW7XVFSGAq5
zOfzfVSr1+vDhxEnLozEihUvVFSuqiqrw8qtIymllkoUmc1WZnZ2Q0tTWzsMsdSkMHQPJCErNsUK
AxEFUZSFFi9e6pYvuOQux4IsyaIkFko4UQlLFEZhbGI3NZF4iUtQYlmQ5dAvtlSiHIoqeSuMn5Ul
FqWQYytLX1YYVZRAvM4rkxdSSCGEDlpS0ptsd0xS0k3YYoolhtiRhnY7MysrDBtZopZSKimkjtrS
ysLCel1ZVbmoEl6sODHiw2t1Gn3fZ3PZJidoIYUSKowC0HprbW+/uLm6u8cim6yyyxZGYunwUYcd
d/zAIw899jgO5FCO5UgchBgpxJBDjiCSiCKLHAiCJIiCLEgCocRJJZZc8gSTTOwKA1P3QOhIFEVJ
KL4Fl1xywaUwctkFWZRlWShBCSfsiScwkYlNIMiSUBIlkSSQBJJIElpqJU2kkkkeuamJeWnppCyM
BCqI4IGLiYiHhoXGxJUjP67P/3d73U/ccsohd5xe3u/Ozke3mGKIramlFUaZVaIsiosHIpigggsm
kNJZaXn5xMzU3DwSSSWXTCSptdT09Iqaqrp6KKKJKrpoQmmVJbGQyxKppCp6KMcmZioJgiAHokgi
iBxiSCGEJI7lSA7kOA498sDjDjvoWFbZZJHd1cX93tp6aROqKKKrqainvysyMjoqjAIcdlhhHEhi
iyWsMOh8dXZ3P7w8vb3jkEtOueUSR/fr9x+f1++PI0+ufHnihMZCw8MjYqLiLrsUxh5Ioix0yimF
cWV3VFLRJZfCwOXLLbbUUosXWhLKoihKoiDKKqqkgsqVU0qxQgopS6IkSYIkR6JJJpg8saQSJ5Qk
kAVRkAQpjAIUhgEKAwmpMJCSkPDojqfCKMANmcKIQAAMEhwWIBoQwWFRELsC8qBBYViAPGhgiGzo
Bg2ZMsYYCORBA3OFcVs/VBi3Uj3lYdrnQ4WRV3mm292/dqgwqnKVqNaHunmoMChQGBEQU4ZMYUTg
AQ1YYCQ4NERYSEBgJIhwYGiIsABxAAfEUDlwcAoMERU8YAHDBAgNERALFFCo0DDhwQIUPGABQwYN
CqgQwYFhAgQLIiCOCIgDAxELFDg0wAITAcMFCxiwwERwwODhAgoPDBENGFhgIBgiODRcAAMRCxRM
eHhwwISHBwcYDFTAAAIxEBowcMEChISGCRMWEglEMDgsTKAgwqFhQQEQCw4LEypYoEEBGIAQscAB
CyAeIhoiLMYBgiEiYsFk0BABgeGCChAMFFzwMIEhA4eFAsTDBRgUCEwEhgFiYUIiggsaFg4EJgIJ
Dg0QFiAcYCAwEURAYMiABRpgIjB8oAHiIQICAwQgKIyHCzhMcAEGDxAHcGiQgMEECh4oAwpDgkND
hEMDBAUODwz4BRhEMFgg4WHCAwQFDg0SGCpEODBQoPCQRHBYGjQoDAsWcIgWkDwoB4aIiIUJiTrr
jblbS7U26VFTGaEB4gAPctOBIQkcGiQ0QDSAQcICBQ2SAwwNEHgEhgcRHBYOLEPEggMEQ8SE40Ha
ZZN5anO1htY2P16rIYN4yqzLgTE1EjA0QODBAQ4MFzTgAfnMlQQREh5k8OACDQ8iHBgiHB6SBw4N
EhiQB+kKEAweWFAABQwYoBEYTHiowFBBBIMEiAUHPLgXJiQmMER4WMDaSMAwYQGCBQwSIBYU8OCq
jAQVIhocHgDBMAGDCojuLIiAOB6cweGBAUYCDjA0ZMBA68AUpfkpSgeG1sGEBgf84JoD84NkDBEL
Ey6gwOCABzbBUUEDEAsULniAiKACDg8NDRIYKqAABGQkQA5QgZGgggSHhogIBgsYJCxQmOBBGgkY
JCxUeMADsyojAQSDBShgwIAIExowQB6UJzhsIhYoQDwkGCY0JB58gqNBYWgAwsIExHDgYcIEhgmY
hQgHBgkLCzAOCx5IjAQVIB4mPCQVKjBAeICgIMKBgQIGEB6UBg0K42VNOhXhnqft9c9TdfNU197m
Gp3Nc2BeSOe7OhUYKmhAgM2BsbohPQdm1UjQgQdEgkNDBAcMHpAGDQrD1NBGAoYPNFSosJCBw8IC
KDB3EthcUQ3gHoAZCRgiFhSHh4YIJGRQ4STRYXBomODQ4GADDKeO/icERgKGCRENDmjQoDAigMBE
AIEJzEMGFg5gKCxgYAHCAgcERgKIh8QPlpHA4aGhAgQDBB7oBEeDBoVB58pEUAEEEA8RCxARFTCY
MGECYySAeIhYYIBIGjQoDEPHHDBY4CHC4cACCgDQIGpFMtNENJZ3B9MkXtl2FNHcFR1D47s0ze66
xzdbh9ZjNLprLdOy5jGi2UU8m1iKx9asU1N/7mjusg6twf9rO6JRk51Ts6VoWOcS09g9UX008dCa
pPJt6TnrPNFYWWnl0uZPkk3pHc30mHf0eIqkzr0iPLYNmpd8SoWKMd4BiIAETAQQGKwIEjB+wDDB
AxE7wfGAAsTDBRwwwZFAYB40KIwONEBELCxgwgJDBQOIwBAB4gECQwURDwwYgBkAghjSS3vu0epJ
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hedmW3v227YO+WUHpq7m24iOK9EcSMCYpet2rT2WmOYo18bv7dm0dGDypId16zswt3ZHu4R6PGvz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EA==
4.3.1. Fundamental Matrix The fundamental matrix of an absorbing Markov chain can be used to extract many properties of this chain. In particular, it can be used to determine the average number of visits to a given state before absorption, the time expectancy until absorption starting from a given state, and the probabilities of being absorbed in a given state , starting from a state . The fundamental matrix resulting from the matrix is given by the relation
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NRkULpYsNAFioUtYtaKGEgchTQZcMtVw41aj9VjDtbXG2Li7EGX9AL6Gs8c4Ucp3EG37UmbXJ6W8
XOepjlcw5gCCluoAYwjk+LjzNQnNSL5M75KgHDy3SWai58fFhduuOQ9M7AG/s1TBkQ==
oJk+LKPTcQl4pBlKKAmuYGB1PSo3Xt0mji/SrrRzVv8iwDDj6nM6HV+Kmt+FXwr/QXD3BxU85tHL
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a7OHrW4aps/2UekxtVI5iWnZI8vEHf4sJ07/QNsc0tru5JlwlEs7Z/4+9PktEs0U2M2hSIqOhGMb
qRvmuxZT3xQPlG+PGx6dqLwxURnHEX8/8iNmknEyBaQ4DmlloWQ6
Thus, for an individual in the state , the average number of months before absorption by or is days in , days in , days in , days in , and in . For infectious mosquitoes , the average number of months before absorption by (eliminated by insecticides) or (eliminated by LLINs) is days in , days in , and days in . With absorbing Markov chains, all equilibrium distributions will be limited to state absorbents, here , , and . Also, we can determine the probabilities of absorption of states to by states , , and . To do this, let us consider the matrix defined by the relation
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According to these results, the probability of an individual , , , , , , , and dying from malaria is equal to 0.006, 0.006, 0.046, 0.000, 0.005, 0.002, and 0.001, respectively. It can be seen that the probability of an infectious individual dying from malaria is consistent with the clinical results. Susceptible mosquitoes (infectious , respectively) are absorbed either by insecticides at 95.3% (at 82.9%, respectively) or by LLINs at 04.6% (at 16.6%, respectively). The results show that to eliminate malaria, it is preferable to use indoor residual spraying (IRS) by regular application of insecticides rather than long-acting impregnated mosquito nets (LLINs). Susceptible individuals and are absorbed by IRS at 62.1% and 84.5% and by LLIN at 37.4% and 14.8%, respectively. This means that susceptible humans prefer IRS rather than LLINs to protect themselves against parasite infection.
5. Conclusion In this study, we proposed a Markovian stochastic approach to a compartment model regarding malaria transmission. This Markovian model generalizes the host-vector compartment models of types , , and . The study of this model made it possible to identify a technique suitable for combating malaria. Our model constitutes a valuable tool for the stochastic modeling of epidemics: it is used to predict the evolution of the dynamics of malaria in a human and anopheline population. In the forthcoming paper, we will study the geometric V-ergodicity of the model by including the effect of abiotic factors such as temperature.
Data Availability All the data is in the article.
Conflicts of Interest The authors declare that they have no conflicts of interest.