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International Journal of Mathematics and Mathematical Sciences
Volume 2017, Article ID 5058085, 25 pages
https://doi.org/10.1155/2017/5058085
Research Article

The Effect of Seasonal Weather Variation on the Dynamics of the Plague Disease

1Nelson Mandela African Institution of Science and Technology (NM-AIST), Arusha, Tanzania
2Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda

Correspondence should be addressed to Rigobert C. Ngeleja; moc.oohay@ajelegnr

Received 21 February 2017; Accepted 28 June 2017; Published 10 August 2017

Academic Editor: Ram N. Mohapatra

Copyright © 2017 Rigobert C. Ngeleja et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Plague is a historic disease which is also known to be the most devastating disease that ever occurred in human history, caused by gram-negative bacteria known as Yersinia pestis. The disease is mostly affected by variations of weather conditions as it disturbs the normal behavior of main plague disease transmission agents, namely, human beings, rodents, fleas, and pathogens, in the environment. This in turn changes the way they interact with each other and ultimately leads to a periodic transmission of plague disease. In this paper, we formulate a periodic epidemic model system by incorporating seasonal transmission rate in order to study the effect of seasonal weather variation on the dynamics of plague disease. We compute the basic reproduction number of a proposed model. We then use numerical simulation to illustrate the effect of different weather dependent parameters on the basic reproduction number. We are able to deduce that infection rate, progression rates from primary forms of plague disease to more severe forms of plague disease, and the infectious flea abundance affect, to a large extent, the number of bubonic, septicemic, and pneumonic plague infective agents. We recommend that it is more reasonable to consider these factors that have been shown to have a significant effect on for effective control strategies.

1. Introduction

Plague is the ancient disease caused by the bacterium Yersinia pestis and has had significant effects on human societies throughout the history [1]. Dynamics of plague disease are the result of complex interactions between human beings, rodent population, flea population, and pathogens in the environment. Seasonal variation particularly temperature, humidity, rainfall, and precipitation greatly affects the normal transmission capacity of plague disease by either lowering it or raising it. It affects pathogen in the environment, fleas, rodents, and even human behavior by altering their normal immigration rate, death rate, survival rate, and infectious capability [2].

1.1. Seasonality in Flea Development Stages and Behavior

Flea’s survival is greatly affected by temperature and relative humidity [3]. The ectothermic characteristics of fleas make them very sensitive to temperature fluctuations. Xenopsylla cheopis is the primary vector flea for Yersinia pestis. It is significantly affected by seasonal weather variation as most of its life stages depend on temperature, humidity, and precipitation. The rate of metamorphosis of this kind of flea from egg to adult is also regulated by temperature.

Flea larvae feed on almost any organic debris but mostly they feed on adult excreta which consist of relatively undigested blood [4]. This adult fecal matter when dried falls from the host to serve as food for the larvae. Thus the availability of food (dried flea dirt) for larvae to feed depends on the weather condition particularly temperature and humidity. The larvae develop well in areas where the relative humidity is greater than 75 percent and the temperature is between 21°C and 32°C [5, 6]. At constant temperature fleas become most sensitive to air saturation and are massively killed when the air saturation is insufficiency [7]. Considering the fact that all immature flea stages occur outside the host, development rates of flea increase with temperature until it reaches a critical value which makes flea most vulnerable. High temperature combined with low humidity hinders flea’s survival at immature stages [8].

The condition where relative humidity is below is unfavorable for flea growth. It is at this condition that the biting rate of flea onto the infected human and rodent or of the infected flea onto the susceptible human and rodent is significantly low. But when the relative humidity is the flea becomes very active and as a result the biting rate and infection increase significantly. Moreover when temperature is above 27.5°C the rapid disappearance of plague bacilli from the flea stomach occurs, resulting in reduced rates of plague disease transmission. This in turn reduces the flea’s efficiency in its ability to transmit the plague bacillus to human beings and rodents [9, 10].

When fleas are in rodent burrows, their survival of immature stages is affected by soil moisture that is partly controlled by outside precipitation [11]. As a way of getting rid of detrimental moisture losses and temperature swings, rodents normally shift to start living underground [12]. On the other hand, when they are attached with a high organic load, excessively wet conditions in rodent burrows (e.g., relative humidity ) can stimulate the growth of destructive fungi that diminish flea’s larval and egg survival [13].

Different studies justify the negative correlation between rainfall and plague epidemics. For example, Cavanaugh and Marshall Jr. [3] reported that, in areas where drains are absent, or where drainage is insufficient as a result of soil composition or impoundments of water, flooding unquestionably causes a drop in the flea population. In areas with improved drainage, such as those with sandy soils, the lessening of the flea population is minimal. Precipitation also influences plague infection for it influences the concentration of rodents, fleas, and humans in the same shelter.

1.2. Seasonality in Rodents

The direct effect posed on rodent population due to temperature change is minor. This is due to the fact that rodents are homoeothermic and hence do not respond immediately to changes in ambient temperatures [14]. Temperature indirectly affects the spread of plague in rodent population in different ways as follows: at a low mean temperature of 10°C the bacteria within host (rodent) become very active as a result a large number of infected rodents dying before even the plague bacilli appear in their blood. At this particular temperature rodents also lose the ability to infect other susceptible individuals.

Rainfall may pose positive or negative effect on the increase of rodent population depending on its intensity [11]. A season of moderate rainfall may be considered to affect positively the increase of rodent abundance but when the amount of rainfall is extremely heavy it results in a tremendous rodent population decline [15]. When it is moderate and upon a proper timing, rainfall may foster the increase of rodent population [3]. This is due to the fact that rodent’s reproduction period normally follows wet seasons [1618]. That is to say, the increase of rodent population during wet period is expected to be higher than that during the dry seasons. This clearly concurs with the result in the study by Leirs et al. [19], which narrates that, in Tanzania, rodent population densities show clear association with the annual rainfall and its seasonal distribution. However when rainfall is of high intensity, it causes flooding of rodent burrows. Large number of rodents population dies and the remaining ones normally move from forest to the households where they can protect themselves [3, 8, 20]. In other cases, increased precipitation or drought stalwartly disturbs rodent population dynamics, as it deters food availability.

1.3. Seasonality in Pathogens in the Environment

When the bacteria are in lungs, the transmission of Yersinia pestis is possible through various ways: contact transmission, in which one may be infected through physical contact with respiratory particles on the infected surface; airborne transmission, which occurs through inhaling the bacteria causing the disease through successive contact with the nose or mouth of an infected individual; respiratory particles, which occurs through respiratory droplets which is through shedding of respiratory particles (i.e., droplets or aerosols) from an infected human or rodent into the environment [21].

Extreme temperatures regularly are ruinous to the survival of pathogens causing plague. The changes in temperature may lead to varying effects on the pathogens in the environment and vectors that live in an environment. When the mean temperature approaches the maximum limit that can be endured by the pathogens, a small increase in temperature may be very dangerous to the pathogen survival. Conversely when pathogens are in the environment characterized by low mean temperature, a small increase in temperature may result in increased development, incubation, and replication of the pathogen in the environment [22, 23].

Davis [24] compared the seasonal incidence of plague with usual atmospheric conditions in particular temperature and rainfall. It was depicted that human plague is more frequent in warm moist weather between 15°C and 27°C than in hot dry (over 27°C) or cold weather (under 15°C). Mitscherlich and Marth [25] narrate that the solar exerts a detrimental effect on bacterial aerosol and the decay rate of Yersinia pestis is proportional to the increase of UV light.

The reports by Ayyadurai et al. [26] and Mollaret [27] justify the ability of the Yersinia pestis to culture the organism from deep within contaminated soil. Eisen et al. [28] were able to show the great potential durability of Yersinia pestis in the soil substrate. The long duration of their survival in the soil supports indirectly the virulence maintenance.

Yersinia pestis exhibit a very slow growth at the temperature between 35°C and 37°C but they grow very fast at the temperature 28°C. They die very rapid if exposed to a UV light or temperature exceeding 40°C or when exposed to intensive desiccation [2931]. Bacteria decrease their sensitivity when the level of humidity drops below 76% [25].

When an infected individual coughs or sneezes, thousands of the bacteria are released in air [32]. The released respiratory particles may be large and heavy that they cannot remain suspended in the air. When respiratory particles are large the transmission can only occur when these particles are expelled directly onto another close susceptible individual. In some cases the release of smaller respiratory particles may occur; this is when the airborne transmission is possible. The smaller released particles are easily suspended in the air respired (i.e., passed to the lower respiratory tract) [33].

Relative humidity and temperature affect the transmission of Yersinia pestis from one individual to the other. Humidity affects the size of the respiratory particle [34]. When humidity is low the large drops partially evaporate to create smaller, lighter drops that are more likely to remain airborne for extended periods of time [35]. That is to say, when the air is sufficiently dry the large sized particles shrink to a size that favors long-range transport which in turn leads to increased infection.

1.4. Seasonality in Human Behavior

Human activities and behavior in plague-infected areas are also to be considered as important determinants of plague transmission to and by humans [42]. When occurrences of plague are due to human intrusions in natural plague areas, it is thus important to consider season variation as a second-order variable that influences disease incidence through human behavior. In Tanzania drought and famine which are the result of lack of rainfall and temperature fluctuation have a great impact on the farmers and pastoralists as they force them to move from one area to another searching for food for themselves and their cattle. These human intrusions from one place to another may lead to the increase of plague disease transmission in rodents, fleas, human population, and pathogens in the environment.

2. Model Formulation

We describe the complex interaction that leads to plague disease transmission and use it to formulate a model for the dynamics of the plague disease coupled with the effect of seasonal weather variation in its transmission. The model includes four populations, namely, human beings, rodents, fleas, and pathogens, in the environment. We generally assume that all individuals from each population are susceptible to the disease, the recovered individuals confer temporary immunity and return to be susceptible again, and the infectious are all individuals with either bubonic plague or pneumonic or septicemic plague.

2.1. Variables and Parameters Used in the Model

In Notations and Table 1 we present variables and parameters, their description, and their values as used in the model. We have obtained the parameter values from the literature that relate to this study and the present information on plague disease and through estimation.

Table 1: Parameters and their description.
2.2. Model Description

The human population is divided into six subgroups: the subgroup of people who have not contracted the disease, to be referred to as susceptible and denoted by , but may get it if they come into contact with , , , , , or ; people who have the disease but have not shown any symptom and are incapable of transmitting the disease to be referred to as exposed and denoted by ; those who are infected and capable of transmitting the disease are divided into three subgroups: there are those who have bubonic plague denoted by , those with septicemic plague denoted by , and those who have pneumonic plague disease denoted by . The fraction of population in if treated or through strong body immunity may recover and move to subgroup ; otherwise they progress either to a septicemic disease infective agent or to pneumonic plague disease infective agent or else they die. The population in the subgroup through strong body immunity or if treated recover and progress to the subgroup and if not treated they progress and join subgroup ; otherwise they die. The population of the subgroup is considered as a very dangerous stage of plague disease; it is a very fatal stage of plague disease with the fatality rate of about ; however if treated they recover and join subgroup ; otherwise they die. So the total human population is as given by

Fleas are divided into two subgroups, those who have not contracted the disease but may get it if they get in contact with infectious agent (rodent or human) referred to as susceptible flea and denoted by and those who are infected and are capable of transmitting the disease referred to as infective agents and denoted by . The total flea population is as given by

The rodents are divided into five subgroups; those who have not contracted the disease but may get it if they get in contact with , , , , , or , referred to as susceptible rodents and denoted by ; those who have the disease but have not shown any symptom and are incapable of transmitting the disease referred to as exposed and denoted by ; those who are infected and capable of transmitting the disease are divided into three subgroups: those who have bubonic plague denoted by , those with septicemic plague denoted by , and those who have pneumonic plague . The fraction of population in may progress either to a septicemic plague disease infective agent or to pneumonic plague disease infective agent . The rodent population in the subgroup may either progress to pneumonic plague disease infective agent ; otherwise they die. The population in the subgroup is considered as a very dangerous stage of plague disease and very fatal so the mortality due to disease in this subgroup is approximated to be 100%. Then the total rodent population is as given by

The individuals with pneumonic plague may release pathogens causing plague disease to the environment denoted by through coughing or sneezing. When the condition in soil/environment is favorable, pathogens may remain infectious in the environment for a long time. When a susceptible individual adequately interacts with the environment infested with Yersinia pestis, he/she gets the disease even in the absence of any vector.

2.3. Description of Interactions

The susceptible fleas in subgroup get Yersinia pestis bacteria through biting the infected rodent or who are the primary reservoir for the bacteria and become infected at the rates and , respectively. Fleas may also get the disease when they bite the infected human being with bubonic plague or septicemic plague at the rates and , respectively. Thus the flea population gets plague infection with the force of infection given in

The human population may get the disease in one of the following ways: when the infected flea bites and infects the susceptible human being at a rate ; when they interact with one another; this can be with either a person with pneumonic plague through airborne transmission or septicemic plague through physical or sexual contact at the rates and , respectively. Another infection is through airborne transmission through interaction with rodent infected with pneumonic plague or through touching or eating the infected rodent with septicemic plague at rates of and , respectively. Human beings may also get the infection from the environment when they breath in the bacteria or physically contact the infected material at the rate of . That is to say, human population acquire plague disease following effective contact with infected human, rodent, flea, and the environment with force of infection given by

The subgroup , after the infection, progresses and becomes latent to the disease at a rate . After to days the subgroups become infected into one of the three infectious classes, , , or (depending on the mode of transmission an individual is exposed to), and are capable of transmitting the disease. The proportion of progresses and becomes infected by bubonic plague , septicemic plague , or pneumonic plague at the rate and proportion to , , or , respectively. The compartment either through strong body immunity or if they get treatment they recover and move to subgroup at a rate ; otherwise they either progress to subgroup or at a rate or die either naturally at a rate or due to the disease at a rate . The fraction of humans with septicemic plague either through strong body immunity or if treated recover at a rate and join ; otherwise they either progress to subgroup at a rate or die due to the disease at a rate or naturally at a rate . The compartments if treated recover at a rate ; otherwise they die either naturally at a rate or due to the disease at a rate . The subgroup attain temporary immunity and then return and become susceptible at a rate .

The rodent population may get a disease in one of the following ways: when the infected flea bites and infects the susceptible rodent at a rate , through interaction between rodents themselves, which may be with rodent infected by pneumonic plague or septicemic plague at the rates and , respectively. The other infection may be through interaction with human infected with either pneumonic plague or septicemic plague at rates of and , respectively. When the susceptible rodent sufficiently interacts with the pathogens in environment through breathing in the bacteria or physically touches the infected material, it gets the infections at the rate of . Rodent also gets the disease through adequate interaction with rodent, human, flea, and pathogens in the environment with force of infection given by

The subgroup , after the infection, progress and become latent to the disease at a rate . After to days the subgroup become infected and capable of transmitting the disease; the fraction of it progresses and becomes infected by bubonic plague , septicemic plague , or pneumonic plague at the rate and proportional to , , or respectively. The rodent in subgroup may either progress to subgroup or at a rate or die either naturally at a rate or due to the disease at a rate . The compartment may either progress to at a rate or die due to a disease at a rate or naturally at a rate and the compartments die either naturally at a rate or due to the disease at a rate .

With regard to the pathogens in the environment, we assume that the adequate interaction with and has a negligible effect on the dynamics of pathogens population size in the environment. The pathogens in the environment are populated at a constant rate . The infected human with pneumonic plague and rodent with pneumonic plague also populate the environment with the bacteria at the rates and , respectively. Thus the environment is populated with pathogens causing plague disease with the force of infection given by

The pathogens within the environment suffer natural mortality at a rate . Human population in subgroups and , flea population in subgroup , and rodent population in subgroups and suffer natural mortality at rates , and , respectively. The compartments , , , , , , and suffer both natural death at the rates , , and and disease induced mortality at rates , , , , , , and respectively. Human, flea, and rodent are recruited at the rates , , and , respectively.

2.4. Model Equations for Plague Disease

Now we assume that the variation of infection capability from one individual to the other, migration of individuals from one place to another, and recruitment and death rates of individuals in different stages due to seasonal weather variation affect only the rate at which the disease is transmitted from one infected individual to the other. We now use the variables and parameters and their description given in Notations and Table 1 and the description of interactions to drive the system of differential equations given as follows.

Human Beings

Rodents

Fleas

Pathogens

3. Basic Properties of the Model

In this section we discuss the feasible region and positivity of the plague disease model. For convenience purpose and easy presentation of the result we let denote all continuous functions on the real line. If is a periodic function in then we use for the average value of on time interval defined byfor a continuous -periodic function .

3.1. Invariant Region

Plague disease affects human, rodent, flea, and pathogens in the environment populations. For the possible modeling process all state variables and parameters of the model must be nonnegative for . We thus need to verify whether the solutions of the model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) are in suitable feasible region where all state variables are positive. Inspired by Dumont et al. [43] and Mpeshe et al. [44] we first write system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) in the following compact form:where , , is a matrix, and is a column vector.

We then havewherewhere , and .

Now from submatrices , , , and we can deduce that matrix is a Metzler matrix such that all of its off-diagonal elements are nonnegative, , and is Lipschitz continuous. Thus the feasible region for the model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is the set

This means that any trajectory of the system starting from an initial state in the positive orthant of remains forever in .

3.2. Positivity of the Solution

We need to show that all variables and parameters of the model are nonnegative, . We now solve the equations of the system in their patches for testing the positivity. We found that, by letting the initial values of the systems ((8a), (8b), (8c), (8d), (8e), (8f)), ((9a), (9b), (9c), (9d), (9e)), ((10a), (10b)), and (11) be , , and , , , , , , , , , and , in the solution set , , , , , , , , , , , , , and are nonnegative, .

4. Model Analysis

4.1. Disease-Free Equilibrium Solution

The periodic model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) with nonnegative, continuous periodic functions has disease-free equilibrium solution in which we consider the following equations:

Now given initial conditions , , and for (17), (18), and (19), respectively, we will have

As , (17), (18), and (19) admit unique solution , , and , respectively, which is globally attractive in .

To find the disease-free equilibrium point we set the derivatives of system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) equal to zero. Then the model system has disease-free solution which is obtained by setting , , , and for human, rodent, flea, and pathogen system, respectively. Hence system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) has a disease-free equilibrium point

5. Basic Reproduction Number

Let () be the standard ordered -dimensional Euclidean space with a norm . For we write provided , provided , and if .

Now let be the continuous, cooperative, irreducible, and -periodic matrix function with period , be the fundamental solution matrix of the linear ordinary differential systemand be the spectral radius of . By Aronsson and Kellogg [45] it follows that is a matrix with all elements positive for each . By the Perron Frobenius theorem, is the principal eigenvalue of in the sense that it is simple and admits an eigenvector . The following result is important for our subsequent comparison argument.

Proposition 1. let , and then there exists a positive, -periodic function such that is a solution of .

Proof. Let be the eigenvector associated with the spectral radius .
By the change of variablesystem (22) becomeswhere is an identity matrix.
Thus is a positive solution of (24). We can easily see thatMoreoverThus, is a positive -periodic solution of (24) and hence, is a solution of (22).

The plague disease model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) has unique disease-free equilibrium point given in (21).

We consider a heterogeneous population whose individuals are distinguishable by stage of the disease and hence identifiable and put into epidemiological compartments which are , , , , , , , , , , , , and . We sort the compartments so that the first compartments correspond to infected individuals.

We now let be the rate of appearance of new infections in the th compartments; be the rate of transfer of individuals into compartment by all other means, other than the epidemic ones; be the rate of transfer of individuals out of compartment .

Then the plague disease transmission model in ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is governed by a periodic ordinary differential system given inwhere .

We rearrange the system by sorting the infectious classes (, , , , , , , , , ) coming first. We then have

Then we havewith .

Now using (30) the matrices and are as given below:where

Following the setting by Wang and Zhao [46] and van den Driessche and Watmough [47] for epidemic models we check conditions (A1)–(A7) for plague disease epidemic model. System ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is equivalent to periodic ordinary differential system (27). Now considering this system we can easily see that conditions (A1)–(A5) stated below are satisfied.(A1)Since each function represents a directed transfer of individuals (human, rodent, flea, and pathogens in the environment), they are all nonnegative. Thus, for each the functions , , and are nonnegative and continuous on and continuously differentiable with respect to .(A2)There is a real number such that for each the functions , , and are -periodic in .(A3)If a compartment is empty, there will be no transfer of individuals out of the compartment by any means. That is to say, if then . In particular if , then for .(A4)The incidence of infection for uninfected compartments is zero. That is to say, for .(A5)If the population is disease-free then the population will remain free of disease. Thus if , then and for .We know that system (27) has disease-free periodic solution given in (21). Now we define and , , where and are the th components of and , respectively. Now from (28) and (29) we obtain a matrix given inWe then let be the monodromy matrix of the linear -periodic system . Then which implies that is linearly asymptotically stable in the disease-free subspace , where are the infected compartments. Thus condition (A6) stated below holds.(A6)The disease-free periodic solution is asymptotically stable in a disease-free subspace ; that is, , where is the principal eigenvalue of .Next we set and as two matrices defined by (30); then using (28) and (29) we get matrices and given in (31) and (32), respectively. We can further see that matrix is nonnegative, and is cooperative in the sense that the off-diagonal elements are nonnegative. Let , be the evolution operator of our -periodic systemThat is, for each the matrix satisfieswhere is a identity matrix. Thus the monodromy matrix of (35) equals . Therefore condition (A7) stated below holds.(A7)The evolution of individuals in the infectious compartments decays exponentially due to natural and disease induced mortalities. Thus .

Now using the standard theory of linear periodic system by Hale [48], there exist and such that

We then have

Considering the periodic environment we suppose that , -periodic in , is the distribution of the new infection at a rate produced by the infected individuals who were introduced at time . Given then yields the distribution of those infected individuals who were newly infected at time and remain in the infected class at . We then havewhich is the distribution of accumulative new infections at time produced by all those infected individual introduced at previous time to .

Let be the ordered Banach space of all -periodic function from to , which is equipped with the maximum norm and the positive cone . Define a linear operator by

Now by Wang and Zhao [46], Diekmann et al. [49], and van den Driessche and Watmough [47] we name as the next infection operator; then the basic reproduction number of the periodic system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is given:where is the spectral radius of .

5.1. Characterization of

In this subsection we investigate whether the basic reproduction number in our periodic system can characterize the threshold of the disease invasion. To do this we consider the following linear -periodic equation:with parameter . Let , be the evolution operator of system (42) on . We can clearly see that , . Considering matrices (31) and (32) we note that, for each , all off-diagonal elements of matrix are nonnegative (cooperative matrix). It follows that the linear operator is positive in for each , . Now using Perron-Frobenius theorem by Smith and Waltman [50] it entails that is an eigenvalue of with a nonnegative eigenvector. Also using matrix similarity concept by Shores [51] we can easily verify that matrix is similar to the matrix and hence for any , where is a spectrum of the matrix .

Proposition 2 (see [46]). We let (A1)–(A7) hold for system ((8a), (8b), (8c), (8d), (8e), (8f))–(11); then (i)if has a positive solution , then is an eigenvalue of , and hence ;(ii)if , then is the unique solution of ;(iii) if and only if .

This result shows that, in order to find the basic reproduction number, we need to find the monodromy matrix of system (42) and evaluate it. We then find the spectral radius of and solve the equation for which is the basic reproduction number.

5.2. Computation of the Basic Reproduction Number

We compute a time-averaged basic reproduction number using the next-generation matrix as outlined by Wesley and Allen [52], Heesterbeek [53], and Diekmann et al. [49]. The method has the advantage over the usual next-generation method in that the steps to reach an estimate of and the matrix elements of the next-generation matrix have a clear biological basis. It is easy to handle complex diseases like plague disease which has multiple transmission roots from different infectious agents.

To do this we first categorize individuals by their state at the moment they become infected (type at infection). These types at infection refer specifically to the birth of the infection in the individual. These categories (types at infection) differ in the way they transmit plague disease which in turn differentiates their ability to produce secondary cases.

In our case we categorize the individuals into eight states and label them as follows: human infected with bubonic plague (type 1), human infected with septicemic plague (type 2), human infected with pneumonic plague (type 3), rodent infected with bubonic plague (type 4), rodent infected with septicemic plague (type 5), rodent infected with pneumonic plague (type 6), flea infested with pathogens (type 7), and the pathogens in the environment (type 8).

We assume and label individual with bubonic plague as stage one of the disease, septicemic plague as stage two, and pneumonic plague as stage three. We also assume that when an individual in stage one graduates to stage two we only consider the current stage and ignore the latter. We assume that the infection only goes in ascending direction that is from stages one to two, or two to three, but not in the reverse direction.

Since the system has eight types at infection, the next-generation matrix, , will be an matrix with elements ’s. Each of the elements ’s stands for expected number of new cases of caused by one infected individual of . For example, is the expected number of new cases of humans infected with bubonic plague caused by one infected human with bubonic plague.

We now define a matrix whose entries are . The resulting next-generation matrix is as given in

Then, , where is spectral radius of .

Some elements equal because not all types of infections cause all other types of infection. For example, humans with bubonic plague (type at infection 1) do not produce type at infections 1 (human infected with bubonic plague), 4 (rodent infected with bubonic plague), 5 (rodent infected with septicemic plague), 6 (rodent infected with pneumonic plague), and 8 (pathogens in the environment). This means that , , , , and are 0. The type at infection 2 (human infected with septicemic plague) also does not produce type at infections 1 (human infected with bubonic plague), 4 (rodent infected with bubonic plague), 6 (rodent infected with pneumonic plague), and 8 (pathogens in the environment). This also means that , , , and are zero (0). The type at infection 3 does not produce type at infections 1 (human infected with bubonic plague), 2 (human infected with septicemic plague), 4 (rodent infected with bubonic plague), 5, and 7 which means that , , , , and are zero. Type at infection 4 does not produce type at infection 1, 2, 3, 4, or 8 which means that , , , , and are zero. Type at infection 5 does not produce type at infections 1, 3, 4, and 8; then , , , and are zero. The type at infection 6 does not produce type at infections 1, 2, 4, 5, and 7; thus , , , , and are zero. Type at infection 7 also does not produce type at infections 3, 6, 7, and 8; thus , , , and are zero. And the type at infection 8 does not produce type at infections 1, 2, 4, 5, 7, and 8 which means that , , , , , and are zero. Incorporating these, we modify the matrix as shown in the following matrix:

We will now explain the derivation of each matrix element in detail. We employ the derivation steps by Gail and Benichou [54] to drive the expressions for . We mainly base our derivation on the adequate contact rate between the infected individual type and the susceptible individual type , the expected duration of infection of individual type , and the probability that the individual type survives the duration between the latent stage and the time an individual experiences the onset of clinical disease as in

The production of depends on the probability that the total number of fleas that become infectious at the rate of and the infected immigrants survive the incubation period. We also consider the rate at which adequately bites the susceptible human and the bite results in a human infected with bubonic plague at the average value of transmission rate . The total number of humans infected with bubonic plague caused by one flea infested with pathogens is as given in

Septicemic plague in human may be produced in various ways: progression of untreated human with bubonic plague to human with septicemic plague, adequate contact (including sexual contact) between humans with septicemic plague, adequate contact between rodent and human with septicemic plague, and being acquired from the flea infested with pathogens. We consider the progression rate of infected human with bubonic to septicemic , the adequate contact (it may be sexual contact) rate between humans with septicemic plague, rodent infected with septicemic plague, and the infected flea to human with septicemic plague at the average rates , , and . Then the number of humans infected with septicemic plague from all the mentioned infectious agents is as given in

The proportions and of untreated and may progress and become at the progression rates and , respectively. We multiply the average period remain infected by the rate at which they progress to . may also result from the airborne transmission from the human or rodent with pneumonic plague at the average rate or , respectively, and through the direct interaction with the environment at the average rate . Then the total number of humans infected with pneumonic plague from the stated five sources is given in

Production of number of rodents with bubonic plague depends only on the flea infested with pathogens. The infection depends on the infection period of the flea that survives the incubation period and the proportion at which the adequate contact between infected flea and susceptible rodent causes bubonic plague at the average rate as given in

The septicemic plague in rodent is produced in three ways; the first way is when infected rodent with bubonic plague progresses and becomes septicemic plague infective agent at the rate . The second way is after adequate contact (it may also be a rodent eating or biting an infected individual) between the susceptible rodent and a rodent infected with septicemic plague or human at the average rate or , respectively. The third way is from the flea infested with pathogens with the proportion that the adequate contact between and the susceptible rodent results in . The total number of infected from these infectious agents is as given in

may be the result of airborne transmission between the susceptible rodent and the human and rodent with pneumonic plague at the average rates and , respectively. It may also occur from the progression of untreated and at the rates and , respectively. The pathogens in environment may also cause after the adequate interaction at the average rate . Now the total number of resulting from these interactions is in

Fleas are infested with pathogens from humans and rodents infected with bubonic and septicemic plague at the average rates , , , and . The infection is dictated by the probability that humans and rodents with bubonic and septicemic plague survive the incubation period and the adequate rates of contact. From these interactions, we get the total number of infectious fleas, given in

The pathogens are released into the environment at the average rates and from and , respectively. The released number of pathogens at a given time depends on the infectious period of the rodent and human infected with pneumonic plague and the probability that and survive the incubation period. The total pathogens in soil/environment is as given in

Each element in the matrix represents the expected number of secondary cases produced by infected individual during the entire infectious period of that particular individual into a completely susceptible population [55].

5.2.1. Basic Reproduction Number

We obtain the average basic reproduction number by computing the maximum modulus of the eigenvalues of the next-generation matrix [49, 53]. Now using Maple computing software package, the basic reproduction number is