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

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 RT for effective control strategies.


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].

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, 2 International Journal of Mathematics and Mathematical Sciences 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 50% 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 80% 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 95%) 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.

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 [16][17][18].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 [29][30][31].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.

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.

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.

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.

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 100%; however if treated they recover and join subgroup   ; otherwise they die.So the total human population  1 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  2 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  3 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 Recruitment rate of rodents 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.

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 Γ  and Γ  , respectively.Fleas may also get the disease when they bite the infected human being with bubonic plague   or septicemic plague   at the rates Γ ℎ and Γ ℎ , 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 Γ ℎ ; 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 Γ ℎℎ and Γ ℎℎ , 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 Γ ℎ and Γ ℎ , 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  1 .That is to say, human population acquire plague disease following effective contact with infected human, rodent, flea, and the environment with force of infection  1 given by The subgroup   , after the infection, progresses and becomes latent to the disease at a rate  1 .After 2 to 7 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  2 and proportion to ] 1 , ] 2 , or ] 3 , respectively.The compartment   either through strong body immunity or if they get treatment they recover and move to subgroup   at a rate  3 ; otherwise they either progress to subgroup   or   at a rate  3 or die either naturally at a rate  1 or due to the disease at a rate  1 .The fraction of humans with septicemic plague   either through strong body immunity or if treated recover at a rate  4 and join   ; otherwise they either progress to subgroup   at a rate  4 or die due to the disease at a rate  1 or naturally at a rate  1 .The compartments   if treated recover at a rate  5 ; otherwise they die either naturally at a rate  1 or due to the disease at a rate  1 .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 Γ  , through interaction between rodents themselves, which may be with rodent infected by pneumonic plague   or septicemic plague   at the rates Γ  and Γ  , respectively.The other infection may be through interaction with human infected with either pneumonic plague   or septicemic plague   at rates of Γ ℎ and Γ ℎ , 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  2 .Rodent also gets the disease through adequate interaction with rodent, human, flea, and pathogens in the environment with force of infection  2 given by The subgroup   , after the infection, progress and become latent to the disease at a rate  1 .After 2 to 7 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  2 and proportional to  1 ,  2 , or  3 respectively.The rodent in subgroup   may either progress to subgroup   or   at a rate  3 or die either naturally at a rate  3 or due to the disease at a rate  3 .The compartment   may either progress to   at a rate  4 or die due to a disease at a rate  3 or naturally at a rate  3 and the compartments   die either naturally at a rate  3 or due to the disease at a rate  3 .
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  4 .The infected human with pneumonic plague   and rodent with pneumonic plague   also populate the environment  with the bacteria at the rates  1 and  2 , respectively.Thus the environment is populated with pathogens causing plague disease with the force of infection  4 given by The pathogens within the environment suffer natural mortality at a rate

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 Pathogens

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 [0, ] defined by for a continuous -periodic function ().

Basic Reproduction Number
Let (R  , R  ) be the standard ordered -dimensional Euclidean space with a norm ‖ ⋅ ‖.
).Now let () be the continuous, cooperative, irreducible, and -periodic  ×  matrix function with period  > 0, Φ (⋅) () be the fundamental solution matrix of the linear ordinary differential system and (Φ (⋅) ()) be the spectral radius of Φ (⋅) ().By Aronsson and Kellogg [45] it follows that Φ (⋅) () is a matrix with all elements positive for each  > 0. By the Perron Frobenius theorem, (Φ (⋅) ()) is the principal eigenvalue of Φ (⋅) () in the sense that it is simple and admits an eigenvector V * ≫ 0. The following result is important for our subsequent comparison argument.
By the change of variable system (22) becomes where  is an identity matrix.Thus V() = Φ ((⋅)−) ()V * is a positive solution of (24).We can easily see that Moreover Thus, V() is a positive -periodic solution of (24) and hence, () =   V() is a solution of (22).
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 F  () be the rate of appearance of new infections in the th compartments; V +  () be the rate of transfer of individuals into compartment  by all other means, other than the epidemic ones; V −  () be the rate of transfer of individuals out of compartment .
(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 1 ≤  ≤  the functions F  (, ), V +  (, ), and V −  (, ) are nonnegative and continuous on R × R  + and continuously differentiable with respect to .
(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   = 0 then V −  = 0.In particular if  ∈   , then V −  = 0 for  = 1, . . ., . (A4) The incidence of infection for uninfected compartments is zero.That is to say, F  = 0 for  > .
(A7) The evolution of individuals in the infectious compartments decays exponentially due to natural and disease induced mortalities.Thus Φ  () < 1.
Now using the standard theory of linear periodic system by Hale [48], there exist  > 0 and ℘ > 0 such that We then have Considering the periodic environment we suppose that Φ(), -periodic in , is the distribution of the new infection at a rate ()Φ() produced by the infected individuals who were introduced at time .Given  ≥  then (, )()Φ() yields the distribution of those infected individuals who were newly infected at time  and remain in the infected class at .We then have which is the distribution of accumulative new infections at time  produced by all those infected individual Φ() introduced at previous time  to  ( ≤ ).

Characterization of 𝑅 𝑇 .
In this subsection we investigate whether the basic reproduction number in our periodic system can characterize the threshold of the disease invasion.
This result shows that, in order to find the basic reproduction number, we need to find the monodromy matrix Φ − () of system (42) and evaluate it.We then find the spectral radius of Φ − () and solve the equation (Φ − ()) = 1 for  which is the basic reproduction number.

Computation of the Basic Reproduction Number.
We compute a time-averaged basic reproduction number  0 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  0 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 International Journal of Mathematics and Mathematical Sciences 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 nextgeneration matrix, K, will be an 8 × 8 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,  11 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 K whose entries are   .The resulting next-generation matrix is as given in ) . (43) Then,  0 = (), where () is spectral radius of .Some elements equal 0 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  11 ,  14 ,  15 ,  16 , and  18 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  21 ,  24 ,  26 , and  28 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 ) . (44) 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 × (Probability that the individual survive the incubation period) .

(45)
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 Γ ℎ .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  3  3 , 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  ℎℎ ,  ℎ , and  ℎ .Then the number of humans infected with septicemic plague from all the mentioned infectious agents is as given in The proportions  1 and  of untreated   and   may progress and become   at the progression rates  3 and  4 , 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  ℎℎ or  ℎ , respectively, and through the direct interaction with the environment at the average rate  1 .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  RB 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  3 Γ  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  3 (1 − ).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 Γ  or Γ ℎ , 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 Γ ℎ and Γ  , respectively.It may also occur from the progression of untreated   and   at the rates  3 and  4 , respectively.The pathogens in environment may also cause   after the adequate interaction at the average rate  2 .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  ℎ ,  ℎ ,   , 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  1 and  1 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 K 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].

Basic
Reproduction Number  0 .We obtain the average basic reproduction number  0 by computing the maximum modulus of the eigenvalues of the next-generation matrix K [49,53].Now using Maple computing software package, the basic reproduction number is in which where Since the system has multiple infectious types from multiple hosts, then the next-generation matrix produces the average value of the geometric mean of the number of infections per generation and the basic reproduction number International Journal of Mathematics and Mathematical Sciences  is the average number of secondary infections [56] ).The result may also be interpreted as follows: among all elements of the matrix K,   , which appear in   , gives more significant involvement in the dynamics and spread of plague disease.

Numerical Results and Discussion
Here we use the parameters values of model system ((8a), (8b), (8c), (8d), (8e), (8f))- (11) given in Table 1 to study the transmission trend of plague disease.Simulation results are given to show the effect of different parameters on the periodic reproduction number.We have also chosen temperature data obtained from Tanga region from January to December 2013 to show the seasonal distribution in the number of secondary cases of plague infections.It is observed that simulation results from time-averaged seasonal parameters and those seasonal parameters treated mathematically as sinusoidal functions match the real seasonal fluctuation data (temperature).
Results shows that the increase in number of individuals infected with bubonic plague, to a large extent, affects the increases in number of individuals with septicemic and pneumonic plague disease.This is due to the fact that individuals with bubonic plague progress and become either septicemic or pneumonic plague infective agents.This in turn leads to the significant increase of plague disease transmission rate and the average number of secondary infections.Figures 2 and 1 show how the progression rates from individuals with bubonic plague to individuals with septicemic plague affect the average number of secondary infections in human beings and rodents, respectively.It is illustrated that the increase of human beings and rodents progressing to become septicemic plague infective agents affects the disease dynamics by increasing the average number of secondary infections.We see similar result when we evaluate the periodic reproduction number based on the temperature data from Tanga region (Figures 1(c) and 2(c)) and time-averaged parameters (Figures 1(b) and 2(b)) for human beings and rodents, respectively.These findings necessitate the need for early treatment of plague disease infective agents especially the primary forms (bubonic and septicemic plague) before they progress to highly fatal and fast spreading plague forms like pneumonic plague disease.It is thus important for the government and other health stakeholders to ensure the availability of effective plague disease treatment especially in high risk areas.
Increase of plague disease transmission through flea bite in human beings and rodents populations alters the whole dynamics of plague disease.Results in Figures 3 and 4 show the effect of infection from infectious flea to human beings with bubonic and septicemic plague on the average number of secondary infections.The infection from flea to rodents with bubonic and septicemic plague disease also shows the significant effect on   as illustrated in Figures 5 and 6, respectively.The results generally show that when the periodic infection rates from flea increase, those of the infectious human beings and rodent increase as well; this in turn affects the general plague disease periodic transmission and spread.These results are in conjuncture with what is observed when   is evaluated using the temperature data and time-averaged seasonal parameter as in Figures 3(c The results call attention for the need to control the number of infectious fleas and flea population in general as the way of controlling the plague disease.The study recommends that for the appropriate and most effective way to control flea population we first need to study the flea's ecology and its local patterns of disease transmission.One of the most important and cost-effective strategies of controlling the vector flea is environmental management strategies that can reduce or eliminate vector breeding grounds.For example, in residential areas people must be educated to improve their surrounding environment in a way that does not favor the survival and growth of vector flea.This may be through improving the design of water systems, improving waste disposal and water storage, discouraging deforestation and loss of biodiversity, and living in well ventilated housing that is not close to animals.
There are also biological control tools like bacterial larvicides and larvivorous fish that may be used to control flea population [57].These control methods aim at killing vector larvae without generating the ecological impacts as they do not use chemicals.Another strategy is using chemical methods, which mainly shorten the lifespan of vectors.These tools include indoor residual sprays, space spraying, and use of chemical larvicides and adulticides.Since most of these methods have side effects to the environmental ecology they are recommended to be used when other safe strategies fail.Moreover, even though these chemicals are not environmentally friendly, we advise the environmental stakeholders to recommend the use of chemical methods of vector control that reinforces linkages between health and environment.
Reducing the number of flea population will reduce the infection rates to human beings and rodents and ultimately reduce the number of secondary infections.Figure 7(a) shows how reducing the number of infectious fleas reduces the number of secondary infections.This is also true when  the parameters that are affected by seasonal weather variation are evaluated using the temperature data in Tanga (Figure 7(c)) and using the time-averaged seasonal parameters (Figure 7(b)).This result is in light of the fact that the reduction of flea population will reduce the number of individuals with bubonic and septicemic plague and as a result reduce the number of pneumonic plague infective agents that result from the progression of individual with bubonic and septicemic plague.The reduction of flea population will reduce not only the infection from flea to other individuals but also the rate at which flea gets the disease from other individuals (human International Journal of Mathematics and Mathematical Sciences  beings and rodents).When the flea population is reduced it will as a result reduce the interaction between susceptible fleas and other infectious individuals and vice versa.The number of fleas getting the disease increases with the increase of the rate at which fleas acquire infection from infectious human beings with bubonic plague (see Figure 8(a)) and those with septicemic plague (see Figure 9(a)).We further observe that the increase of infectious fleas may be contributed by the infectious rodents with bubonic plague (see Figure 10(a)) and those with septicemic plague (see Figure 11(a)).We can also observe the similar results when the parameters are evaluated based on the temperature data in Tanga region and when the parameters are timely averaged as in Figures 8(c), 8(b), 10(c), and 10(b) for human beings and rodents, respectively.Therefore, using these results, we settle to the point that increasing transmission rate in flea International Journal of Mathematics and Mathematical Sciences population from human beings and rodents with bubonic and septicemic plague raises the average number of secondary plague disease infections.Physical contact that includes sexual contact between two infectious individuals (human beings and rodents) may lead to septicemic plague.The increase in the number of individuals with septicemic plague affects the general dynamics of plague disease, particularly the average number of secondary infections.It is illustrated that increasing infection rate from a human being with septicemic plague to the other susceptible sex using protective gears and taking necessary precaution when handling people or animals with septicemic plague.This also tells us that there is a necessity to quarantine people and animals that immigrate from areas that are infected by septicemic plague so that they do not affect other human beings or animals and thus increasing the endemicity of the disease.
The distribution of the basic reproduction number is based on the seasonal weather condition of a particular area at a particular time.This is what causes the unpredictability of the number of secondary cases of plague disease infection (bubonic, septicemic, and pneumonic plague) as it will change whenever the weather conditions change.We evaluate the distribution of the basic reproduction number based on the data we obtained on daily temperature ( ∘ C) and humidity (%) from five regions in Tanzania from January to December 2013.Figures 14,15,16,17,and 18 show the seasonal distribution of basic reproduction number when evaluated based on the temperature and humidity data from Kigoma, Mbeya, Mtwara, Singida, and Tanga regions, respectively.
The features displayed in these results clearly show how seasonal weather fluctuation can have significant effects on the number of secondary cases of plague disease.It can be noted that there is a seasonal pattern in new plague disease infection cases.We therefore vindicate the fact that the number of plague disease infective agents peaks whenever the weather condition is favorable for plague disease transmission and it drops when the weather condition does not favor plague disease transmission.

Conclusion
The transmission of plague disease occurs in several pathways which makes the modeling of the disease challenging and International Journal of Mathematics and Mathematical Sciences very complex.Moreover all ways that lead to plague disease transmission are directly or indirectly affected by seasonal weather variation which causes seasonality in plague disease epidemic.The effect of seasonal weather variation has been a glowing concern in different epidemiological studies.This in turn dictates that in order to study the dynamics and propose the effective control strategies of the plague disease we must incorporate the effect of seasonal weather variation.In this study we have analysed the plague disease model incorporated with the factors that are affected by the seasonal weather variation in order to study its effects on the dynamics of the plagues disease.We have computed basic reproduction number and depicted how it can be affected by seasonal weather variation through numerical simulation.We were able to deduce that progression rates from one primary form to secondary form of plague infection, flea's infection rate, and the vector flea abundance pose the significant effect on the increase of the average number of secondary cases of plague infection.Therefore the effective control strategies must take into account these factors as they have been shown to have a significant contribution to the increase of the average number of secondary cases of plague infection.

Figure 1 :
Figure 1: Effect of progression rates from   to   on the periodic reproduction number.

TFigure 2 :
Figure 2: Effect of progression rates from   to   on the periodic reproduction number.

Figure 3 :
Figure 3: The effect of infection from   to human beings with bubonic plague on the periodic reproduction number.

Figure 4 :
Figure 4: The effect of infection from   to human beings with septicemic plague on the periodic reproduction number.

Figure 5 :
Figure 5: The effect of infection from   to rodents with bubonic plague on the periodic reproduction number.

Figure 6 :
Figure 6: The effect of infection from   to rodents with septicemic plague on the periodic reproduction number.

Figure 7 :Figure 8 :
Figure 7: Effect of increased number of fleas' death rate on the periodic reproduction number.

Figure 9 :Figure 10 :
Figure 9: The effect of increased infection rate to fleas from the infectious human beings (  ) on the periodic reproduction number.

Figure 11 :Figure 12 :Figure 13 :Figure 14 :
Figure 11:  The effect of increased infection rate to fleas from the infectious rodents (  ) on the periodic reproduction number.

Figure 15 :Figure 16 :Figure 17 :Figure 18 :
Figure 15: Distribution of  0 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya.

Table 1 :
Parameters and their description.Progression rate out of   to    2  3 Progression rate out of   to    3  3 Progression rate out of   to International Journal of Mathematics and Mathematical Sciences and  3 , respectively.The compartments   ,   ,   ,   ,   ,   , and   suffer both natural death at the rates  1 ,  2 , and  3 and disease induced mortality at rates  1 ,  1 ,  1 ,  2 ,  3 ,  3 , and  3 respectively.Human, flea, and rodent are recruited at the rates  1 ,  2 , and  3 , respectively.
. It is shown that the basic reproduction number of plague disease depends on the expected number of new cases of humans infected with bubonic plague caused by one infected flea ( 17 ), the expected number of new cases of humans infected with septicemic plague caused by one infected human with bubonic plague ( 21 ), the expected number of new cases of humans infected with septicemic plague caused by one infected human with septicemic plague ( 22 ), the expected number of new cases of rodents infected with bubonic plague caused by one infected flea ( 47 ), the expected number of new cases of rodents infected with septicemic plague caused by one infected rodent with bubonic plague ( 54 ), the expected number of new cases of rodents infected with septicemic plague caused by one infected rodent with septicemic plague ( 55 ), the expected number of new cases of rodents infected with septicemic plague caused by one infected flea ( 57 ), the expected number of new cases of fleas infested with Yersinia pestis caused by one infected human with bubonic plague ( 71 ), the expected number of new cases of fleas infested with Yersinia pestis caused by one infected human with septicemic plague ( 72 ), the expected number of new cases of fleas infested with Yersinia pestis caused by one infected rodent with bubonic plague ( 74 ), and the expected number of new cases of fleas infested with Yersinia pestis caused by one infected rodent with septicemic plague ( 75