Dynamics of malaria-dengue fever and its optimal control

Article history: Received: 3 June 2019 Accepted: 20 December 2019 Available Online: 5 June 2020 The mosquito-borne infectious diseases like malaria and dengue are putative as important tropical infections and cause high morbidity and mortality around the world. In some cases, simultaneous coexistence of both the infections in one individual is seen which is very hard to distinguish as both diseases have almost similar symptoms. In this proposed article, dynamical system of non-linear differential equations is constructed with the help of mathematical modeling, which describe dynamics of the spread of these infectious diseases separately and concurrently. Basic reproduction number is evaluated to understand dynamical behaviour of the model. Local and global stability criteria have been deliberated rigorously. Control parameters are used to perceive effect of medication on these prevalent tropical diseases. Numerical simulations are used to observe effect of control parameters graphically.


Introduction
The present era witneses the globalization of infectious diseases that occurs frequently by an unprecedented level. In this "globalized" environment of interdependent trade, travel, migration, and international economic markets, many factors now play an important role in the emergence and spread of infectious disease, which necessitates a coordinated, global response [17]. Mosquitoes are one of the deadliest insects in the world, with their ability to carry and spread disease to humans causes millions of deaths every year. Mosquito-borne infectious disease is accepted as one of the important tropical infections and is the focused topic in tropical medicine [23]. There are several tropical mosquito borne infections. Malaria and dengue are the two common mosquito infections that are easily spread and cause high morbidity and mortality for many patients around the world. Malaria is caused by Plasmodium parasites, which spreads through the bites of infected female Anopheles mosquitoes, called 'malaria vectors' [18]. Dengue is single positive-stranded RNA virus of the family Flaviviridae which is ingested by female mosquitoes (Aedes mosquito) during feeding [22]. The virus then infects the other mosquito and humans over its incubation period. Due to tremendous progress in malaria and dengue infection, the disease burden remains high mostly in subtropical and tropical areas [21]. Presence of infection in the body results in weakness in immune system, it increases the probability that individual gets infected by another infections. Hence there is a possibility that both malaria and dengue infection can be present in the individual at the same time (e.g., [4], [6], [8], [13], [21], [24], [30] or [13]). This scenario is called concurrent malaria-dengue infection. This overlapping of two different infections can result in more severe situations where both diagnosis and treatment of a patient may become difficult [10]. Initially, two cases of concurrent malaria and dengue infection were identified in July, 2005 and November, 2006 [4]. Malaria and dengue fever represent 2 major public health concerns in South America, whose 92% of area is covered by Amazon rain forest. According to the report in a French territory in South America, 0.99% from overall febrile patients are infected by malaria and dengue concurrently [4]. Malaria vectors and dengue vectors are habited in the forest [20] and in the city [7] respectively. Hence, overlapping of the habitat cannot be easily available and therefore concurrent malaria dengue infection cases are less in number. Mathematical models relevant to the concurrent infections helps the researchers, biologists and public Dynamics of malaria-dengue fever and its optimal control 167 health personnel to adopt improved and most effective strategies to control the diseases. Aldila D. and Agustin M. R. developed a nine-dimensional mathematical model to understand the spread of dengue and chikungunya in a closed population [1]. Isea R. & Lonngren K. E. presented two preliminary models that consist of the individual transmission dynamics of dengue, Chikungunya or Zika, and any possible coinfection between two diseases in the same population [12]. Sharomi et. al. developed a deterministic model which incorporates many of the essential biological and epidemiological features of HIV and tuberculosis and the synergistic interaction between them [25]. Silva C. J. & Torres D. F. proposed a population and introduced optimal treatment strategies for co-infection transmission dynamics of TB and HIV [26]. Some cases are reported where patients have symptoms of malaria and dengue both at the same time. In such situations, higher mortality rate is observed. On the basis of this observation, a mathematical model is constructed in the present work. Also two optimal controls are applied in the model in such a way that it helps to analyse malaria-dengue concurrent case and effect of recovery rate on the disease transmission. The paper is organized as follows. The malaria-dengue model construction will be discussed in section 2. Section 3 focuses on formulating basic reproduction number for concurrent malaria-dengue infection, moreover the equilibrium points of the given model are calculated. Local and global stability of all four equilibrium points are proved in section 4. Optimal control theory is introduced and applied to the model in section 5. The model is analysed numerically and graphically in the next section which provides better explanation of the analytic results.

Mathematical modeling
The environmental stress also damages the immune system and makes the individual weak to resist various kinds of infections. Motivated from this concurrent disease problem, we have proposed a compartmental model to analyze the spread of malaria and dengue infections individually and concurrently. The model subdivides the human population () N into four mutually-exclusive compartments, namely susceptible individuals () S , malaria infected individuals () M , dengue infected individuals () D and corresponding to two infectious agent class of recovered individuals is  represents the rate of the malaria infection giving rise to the dengue infection due to weak immunity. 5  and 6  are the rates at which the population infected by malaria and dengue are recovered respectively.  is assumed as a natural death rate and D  be the dengue infection related death rate. In Figure 1 the schematic diagram of the transmission of disease is shown. Here a concurrent disease case in which individual first get affected by dengue and then by malaria is ignored.
The initial conditions of the system (1)

Basic reproduction number (
Endemic equilibrium point ( )

Stability analysis
This section includes stability results of all the equilibrium points of the proposed malaria-dengue model.

Local stability
Local stability of all the equilibrium points has been established by following theorems.
Clearly all the eigenvalues are negative if Eigenvalues of the Jacobian matrix ( ) 1 JE are:  2  is negative, real part of both these eigenvalues are negative and when 2  is positive, real part of both the eigenvalues 1 Clearly, all the eigenvalues are negative under these conditions. Hence, the theorem.
Clearly, 2 a a a n n n n n n n n n n n Where, 14 7 x  = ,

Global stability
To perform the global stability analysis of the disease free equilibrium we use the method developed by [5].

Global stability of disease-free equilibrium point (
0 E ) The model system can be written as follows: Here 0 (0) (0) 1 () X X X = represents the number of uninfected individuals and denotes the number of infected individuals. According to this notation the disease-free equilibrium point is denoted by 00 ( , 0) ES = . Now as per the method given in [5], following two conditions will ensure global stability of the diseasefree equilibrium point. [H1] [H2] Here, 0 ( , ) 0 G X Z  hence, the conditions (H1) and (H2) stated above are satisfied.

Global stability of dengue-free equilibrium point (
1 E ) The model system can be written as individuals. According to this notation the Dengue free equilibrium point is denoted by The following two conditions will ensure global stability of the dengue-free equilibrium point: ( ,0) EX = is globally asymptotically stable. [H4]

Global stability of malaria-free equilibrium point (
2 E ) The model system can be written as: individuals. According to this notation the Dengue free equilibrium point is denoted by The following two conditions will ensure global stability of the malaria-free equilibrium point: (2) ( ,0) EX = is globally asymptotically stable.

Global stability of endemic equilibrium point ( * E )
We analyze global stability of an endemic equilibrium point through a geometric approach described in [16], [27] and [15]. To use this method let we modify our system (1) as follow: . Further suppose that be a solution to the following system, Px be a matrix valued function on K and let Here, the matrix f P is: The second additive compound matrix obtain from the Jacobian matrix M is [2] M , ( ) And if (8) is stable then also the second compound equation ( ) [2] () is stable, moreover  belong to a set in which 1 P − is bounded. A set K is absorbing with respect to (7) if solution exist for all 0 t  and 1 ( , ) To prove global stability through this approach we use techniques developed in [19].  (1 )   (1 )

Optimal control
Mosquitoes are the most prolific killers of humans in the animal kingdom. One of the most ancient and deadly diseases that mosquitoes transmit are malaria and dengue. It has been hypothesized due to influences on immune responses that infection with malaria can alter to the course of infection of the dengue. An effective way to protect the people from dengue who are already affected by malaria is to control vector. Also medication pays a major role to control spread of vector borne diseases. In present dynamical model, two bounded Lebesgue integrable controls are introduced say 1 u and 2 u . 1 u control is to minimize concurrent infection cases by vector control and 2 u is a treatment control which helps to improve recovery rate. After applying control system (1) will take form as follow: Ju  for the mathematical model along with the optimal control is given by: w are constant of the control rates applied as vector control and treatment control, from which the optimal control condition is normalized. Now, we will calculate the values of control variables from 0 t = to tT = such that

Numerical simulation
Bifurcation analysis helps to demonstrate the qualitative information about the equilibrium point.   Figure 3 shows the flow of malaria-dengue model with time. It is observed that human immunity is more sensitive towards dengue compare to malaria infection moreover compare to dengue, recovery rate of malaria is higher. Hence we can say that medication is more effective on malaria infected compare to dengue infected. Compare to dengue, spread of malaria is easy to control by improving medication. Under proper medication both the diseases can be controlled in 7-8 weeks.    The simulation in figure 6 interprets that that chances to get infected by malaria decreases by 50% after applying control  figure  7(c) that combine effect of both the control is even more effective which shoews only medication is not enough to minimize dengue infaction case, different acts which minimize concurrent infaction case also have a significant effect.  u control gives better result which suggest that madication plays a major contribution to control the concurrent infection. Hence better medication facility and avaibility is good approach to control outbreak of malaria-dengue infections in endemic areas.

Conclusion
The fight against most deadly mosquito-borne diseases malaria and dengue is a challenge to the world. In the present study, the system of dynamical model for two different mosquito borne diseases is studied through the use of mathematical modeling. Moreover, Optimal control theory is also applied on the model to visualise the effect of controles on it. The model have four equilibrium points for four different possible cases including disease free society, case when only one individual infection is present and the case when both the diseases are present concurrently in society. It is proved that all four equilibrium points are local and globally asymptotically stable under some parametric conditions. The formula of basic reproduction number 0 () R used to calculate threshold value of the model. In this article, the basic reproduction number is formulated for malaria and dengue combinely, hence it is unaffected by parameter 4  . Threshold value increases as value of parameters 1  and 2  is increases, and it decreases as 5  and 6  increases. Which simply means threshold value can be controled by improving recovery rates of both the diseases. Bifurcation analysis indicates that minimum rate of diseases spread is 12%. Threshold value signifies that there is 14.9% chance to get infected by malaria and dengue concurrently. In numerical simulation we have observed the effect of optimal controls individually as well as concurrently and more stability is observed when we apply both the controls at same time. Also it is analysed that 61% improvement in recovery rate is observed under the effect of both optimal controls, which suggest that vector control by using insecticide, treated mosquito nets and indoor residual spraying and medication to improve recovery are the main way to prevent and reduce malaria and dengue transmission.